The DOI matches its Crossref record; 77 of the 78 references it cites were found in a registry or an index; 15 of the 44 sources its claims cite were checked for support.
Topology and data
Open accessAbstract
An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data. Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data. The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, i.e. finite sets of points equipped with a distance function. It then adapts tools from the various branches of geometry to the study of point clouds. The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise. Here are some of the key points which come up when applying these geometric methods to data analysis. • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, i.e. to understand how it is organized on a large scale. For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms. Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key.
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Claims the paper makes
53 No basis given · 38 Own interpretation · 23 Own result · 17 No full text of the sources · 9 Supported · 5 Partly supported · 5 Sources not found · 1 Couldn’t confirm · 1 Not checked yet · accuracy not yet measured
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Much of current research efforts are focused in this direction (see, e.g., [43] and [39]), and functoriality provides the right general mathematical framework for addressing them.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
What the sources say
- Could not verifycites [43] von Luxburg, Ulrike, 2008, Consistency of spectral clustering, Ann. Statist. sourceExperimental — accuracy not yet measured
The Introduction discusses questions about whether clustering algorithms converge, and Section 5.2 constructs operators for analyzing spectral clustering. Neither passage addresses functoriality or presents it as a framework for resolving ambiguities in clustering; searches for those concepts found no supporting passage in our copy. (LLM, conf=0.76)
- Not matchedcites [39] kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Could not verify
Every source the paper cites for this claim was checked and supports it.
In the paper
“We argue that, as suggested in [46], it has a role to play in the study of point cloud data as well, and we give two illustrations of how this could happen, within the context of clustering.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
What the sources say
- Supportedcites [46] McCullagh, Peter, 2002, What is a statistical model?, Ann. Statist. source
Section 8, “Extensive response variable” (page unavailable), describes a point process as a countable set of points and treats Poisson point-process models as natural transformations of functors. That supports the cited suggestion that functoriality could be useful for point-cloud data; the citing authors present clustering as their own proposed application. (LLM, conf=0.86)
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Niyogi, Smale, and Weinberger in [52] provide sufficient hypotheses which guarantee that this is possible for Riemannian manifolds.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.3. Persistent homology.
What the sources say
- Partly supportedcites [52] Niyogi, Partha, 2008, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. sourceExperimental — accuracy not yet measured
Section 2 (Preliminaries; page unavailable) considers compact Riemannian submanifolds sampled as data. Section 3 (An Outline of Our Main Results; page unavailable) gives conditions under which the samples recover the manifold’s homology with high confidence. It does not guarantee reconstruction of the manifold itself, so the claim is broader than the result. (LLM, conf=0.88)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The result of this construction is a database M of ca.4.5 × 10 6 points on a 7-dimensional ellipsoid in R 8 .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.4. Example: Natural image statistics.
What the sources say
- Partly supportedcites [41] lee A.B. Lee, K.S. Pedersen, and D. Mumford, The nonlinear statistics of high-contrast patches in natural images, International Journal of Computer Vision (54), No. 1-3, August 2003, pp. 83-103. sourceExperimental — accuracy not yet measured
Section 3 (Preprocessing; page unavailable) says the preprocessed points lie on a 7-dimensional ellipsoid in R⁹, which is transformed into a 7-dimensional sphere in R⁸. Section 2 (Optical and Range Data Sets; page unavailable) reports about 4.2 million optical patches, not approximately 4.5 million points in a database M. The source supports the general dimensional construction, but not the stated count and geometry together. (LLM, conf=0.85)
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“Their method is to prove that the Čech complex associated to a covering by balls of a fixed radius is homotopy equivalent to the underlying manifold.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.3. Persistent homology.
What the sources say
- Supportedcites [52] Niyogi, Partha, 2008, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. source
Section 3 states that, when the sample is sufficiently dense and the ball radius meets the stated bound, the manifold is a deformation retract of the union of the balls. Section 3.1 constructs the nerve of that ball cover and invokes the Nerve Lemma because the balls are convex. Together, these support the claimed homotopy equivalence under the method’s conditions; the paper calls the nerve a simplicial complex rather than a Čech complex. (LLM, conf=0.88)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“However, it turns out that there is a classification theorem (see [64]) for a subcategory of the category of N-persistence F -vector spaces, where F is a field.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
What the sources say
- Supportedcites [64] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
Section 3.1 (p. 11) establishes an equivalence between finite-type persistence modules and finitely generated graded modules over R[t]. Corollary 3.1 in Section 3.2 (p. 12) then classifies the isomorphism classes of finite-type persistence modules over a field F by finite sets of P-intervals. Finite-type modules are the subcategory referred to in the claim. (LLM, conf=0.94)
plain- Asserted: passes — The authors put it forward as true.
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“In [9], such a context is developed which is not dependent on the choice of a cost function but is rather dependent on”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Supportedcites [9] memolicarlsson G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg , Preprint, March 2008. source
The Discussion (p. 16) presents persistence and functoriality as organizing principles and proposes defining clustering methods through sets of constraints, rather than a chosen cost function. The Results (p. 10) explicitly compares the framework’s properties with Kleinberg’s axiomatic conditions. (LLM, conf=0.87)
plain- Asserted: passes — The authors put it forward as true.
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“It can be shown that p Σ is a homotopy equivalence (see [65]).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 4.3. Tree based persistence.
What the sources say
- Supportedcites [65] Zomorodian, Afra, 2008, Localized homology, Comput. Geom. source
Lemma 1 (global; page unavailable) states that projection from a blowup complex to the covered space is a homotopy equivalence when a simplicial complex is covered by subcomplexes. The paper uses π_X and X_U rather than pΣ and M(|Σ|,S), but states the projection result attributed to it. (LLM, conf=0.88)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“This notion of clustering is closely related to clique clustering algorithms in network clustering [53].”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Supportedcites [53] clique G. Palla, I. Derènyi, I. Farkas, and T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society, Nature, Volume 435, 9 June 2005, pp. 814-818. source
The paper defines network communities through connected sets of adjacent k-cliques (p. 2) and describes an algorithm that finds cliques and clusters them using a clique-overlap matrix (p. 3). It therefore supports the citation as an example of clique-based network clustering; the comparison to the citing paper’s equivalence relation E is the citing authors’ own framing. (LLM, conf=0.88)
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“There is a unique M gen -functorial persistent clustering algorithm Ξ which is scale invariant and which satisfies the requirement that Ξ(E) = P .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Partly supportedcites [9] memolicarlsson G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg , Preprint, March 2008. sourceExperimental — accuracy not yet measured
Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Partly supported
- It may state two separate claims as one. 39
- Probably not a substantial claim. 41
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Although clustering is clearly a very important part of data analysis, the ways in which it is formulated and implemented are fraught with ambiguities.In particular, the arbitrariness of various threshhold choices and lack of robustness are difficulties one confronts.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
What the sources say
- Partly supportedcites [43] von Luxburg, Ulrike, 2008, Consistency of spectral clustering, Ann. Statist. sourceExperimental — accuracy not yet measured
The Introduction describes clustering as widely used and says its consistency guarantees are unsatisfactory. The discussion following Proposition 24 shows that unnormalized spectral clustering can fail and that its reliability can be difficult to assess. These passages support the general concern about reliability, but they do not identify arbitrary threshold choices as a difficulty or establish a lack of robustness for clustering generally; no page numbers are available in our copy. (LLM, conf=0.82)
- Not matchedcites [39] kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“In particular, in [41], indications were found that the data was concentrated around an annulus.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.4. Example: Natural image statistics.
What the sources say
- Supportedcites [41] lee A.B. Lee, K.S. Pedersen, and D. Mumford, The nonlinear statistics of high-contrast patches in natural images, International Journal of Computer Vision (54), No. 1-3, August 2003, pp. 83-103. source
Section 5.1 describes the ideal-edge manifold using a circular orientation parameter and a bounded displacement parameter, giving it an annulus-like parametrization. Section 5.2 finds evidence that optical data points concentrate near that manifold. The paper does not use the word “annulus,” but the citing sentence is a reasonable, hedged paraphrase. (LLM, conf=0.82)
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“By using persistence, one is able to develop a systematic way of addressing such questions.That methodology has been described in [65].”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 4.3. Tree based persistence.
What the sources say
- Supportedcites [65] Zomorodian, Afra, 2008, Localized homology, Comput. Geom. source
Section 1.2 describes a method for localizing topological attributes relative to a cover. Section 4.4 gives the persistence algorithm that computes representatives of homology classes, supplying descriptions of those attributes. Together, these passages support the paraphrase about systematically addressing where features come from. (LLM, conf=0.90)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“The algorithm Ξ is the algorithm which associates to a finite metric space (X, d X ) the persistent set {π 0 (V R(X, ))} ≥0 .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Supportedcites [9] memolicarlsson G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg , Preprint, March 2008. source
Example 2.1 (Section 2, p. 3) associates a finite metric space with a persistent set whose blocks at scale r contain exactly the points connected by chains of distances at most r. Those blocks are the connected components of the Vietoris–Rips complex at that scale, as the claim describes. (LLM, conf=0.90)
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Also, results about barcodes under the general heading have now begun to appear (see [15] and [16]).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Additional information
What the sources say
- Supportedcites [15] Cohen-Steiner, David, 2007, Stability of persistence diagrams, Discrete Comput. Geom. source
The Introduction (p. 2) says the paper makes concrete statements about the stability of barcodes. The Applications section (p. 16) states a Barcode Theorem, making this paper an example of the barcode results the citing sentence says have begun to appear. (LLM, conf=0.90)
- Not matchedcites [16] stabilitytwo D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz functions have 𝐿_{𝑝}-stable persistence, Found. Comput. Math., to appear.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is readily checked that θ is a functor from N pers (Ab) to the category of graded Z[t]-modules, and is in fact an equivalence of categories, since an inverse functor can be given by M * → {M n }, where the morphisms ψ mn are given by multiplication by t n-m .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A first observation is that the points are scattered throughout the 7-sphere, in the sense that no point on S 7 is very far from the set, but that the density appears to vary a great deal.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.4. Example: Natural image statistics.
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We note that there are a number of short lines, and one long one.According to the philosophy mentioned above, this suggests that the first Betti number should be estimated to be one.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.4. Example: Natural image statistics.
hedgedcorrelationalSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We note that in this case, there are many short segments and 5 longer segments.It is again the case that this barcode is stable over the choices of landmark points and samples from M. This suggests that the first Betti number of the putative underlying space should be five.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Primary circle
hedgedcorrelationalSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that while the two smaller holes have now been “filled in”, a new hole has been introduced in the lower right-hand portion of the figure.Consequently, if we were to compute the homology of this complex, we would obtain a first Betti number of two.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We note that the vertical circles intersect the horizontal primary circle in two points, and do not intersect each other, so the picture in question produces a natural embedding of the three circle model in the Klein bottle.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We note that the β 0 barcode (the upper picture) clearly shows the single component, the β 1 shows two lines from threshhold parameter values .15 to .35, and finally the β 2 barcode shows a single line on roughly the same interval.This gives β 0 = 1, β i = 2, and β 2 = 1.These are the mod 2 Betti numbers for the Klein bottle.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At both scales, there is a central dense core, and two”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 3.5. Examples.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These two are distinguished by mod 3 homology, and we have performed the computation to show that the mod 3 homology is consistent with the Klein bottle and not with the torus.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We note that given only the Mapper output, one might suspect that the small feature (the array of orange nodes) could simply be an artifact, but examination of the data shows that they correspond to essentially distinct contact maps.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 3.5. Examples.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Some experimentation with the three circle model results in the following picture in which the horizontal segments form the primary circle and the vertical segments form the secondary circles, as in the diagram.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We have informally confirmed that the indicated patches are the ones which occur in the high density portions of the data set.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By far the most frequently occurring signatures were (1, 1, 0) and (1, 0, 1), corresponding to a circle and a sphere, respectively.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.5. Example: Electrode array data from primary visual cortex.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that the two secondary circles each intersect the primary circle in two points, and that they do not intersect each other.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note also that the data are obtained by simply examining the states occurring in the simulation, and that it does not include any dynamic information which would show how the states are traversed in the folding process.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 3.5. Examples.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The set S exhibits the three circle model clearly, but enlarging it still does not exhibit the non-trivial β 2 which would be characteristic of the Klein bottle.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is easy to check (a) that the factorization is a homeomorphism and (b) that the orbit space is homeomorphic to a Klein bottle.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Three circle model in the data
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A M iso -functorial clustering algorithm determines an choice p ι ∈ P G ι ι for every ι, and conversely an arbitrary choice of such p ι 's determines an M iso -functorial clustering algorithm.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Each of these notions creates the structure of a category whose objects are the finite metric spaces, since one can readily observe that each of the classes of morphisms is closed under composition and contains the identity.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Some work in clustering theory has been done in trying to determine the optimal choice of , but it is now well understood that it is much more informative to maintain the entire dendrogram of the set, which provides a summary of the behavior of clustering under all possible values of the parameter at once.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the biological context, notions of distance are constructed using some intuitively attractive measures of similarity (such as BLAST scores or their relatives), but it is far from clear how much significance to attach to the actual distances, particularly at large scales.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Topology studies only properties of geometric objects which do not depend on the chosen coordinates, but rather on intrinsic geometric properties of the objects.As such, it is coordinate-free.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Moreover, it is understood that most of the information about topological spaces can be obtained through diagrams of discrete sets, via a process of simplicial approximation.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We then demonstrate that persistence can be generalized in several different directions, providing more structure and information about the data sets in question.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If the clusterings are consistent, i.e. if the clusters in C(X 1 ) and C(X 2 ) in C(X 1 ∪ X 2 ) correspond well under these maps, then one can regard that as evidence that the subsample clusterings actually correspond to clusterings on the full data set X.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
hedgedcorrelationaln = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For any topological space X with a finite number of path components, β 0 (X) is the number of path components.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We begin with a discussion of persistent homology, which is a mathematical formalism which permits us to infer topological information from a sample of a geometric object, and we show how it can be applied to particular data sets arising from natural image statistics and neuroscience.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Functoriality is central in algebraic topology in that the functoriality of homological invariants is what permits one to compute them from local information, and that functoriality is at the heart of most of the interesting applications within mathematics.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We then show that the philosophy of functoriality can be used to reason about the nature of clustering methods, and we conclude by speculating about theorems one might hope to prove and discussing how the subject might develop more generally.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“One important goal of data analysis is to allow the user to obtain knowledge about the data, i.e. to understand how it is organized on a large scale.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Although we often receive data in the form of vectors of real numbers, it is frequently the case that the coordinates, like the metrics mentioned above, are not natural in any sense, and that therefore we should not restrict ourselves to studying properties of the data which depend on any particular choice of coordinates.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1. Introduction
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The second method is more efficient, since it is precisely adapted to the complex at hand.Furthermore, it contains complete information about the original R-vector space.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In interpreting the output, one now finds that long intervals in the output barcode indicate the presence of a homology class which “persists” over a long range of parameter values, while short intervals indicate cycles which are “born” at a given parameter value and then “die” at a nearby parameter value.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This is due to the fact that for finite metric spaces, it is generically the case that each value of the distance is taken only for one pair of points, so one does not have any points which are equidistant between a pair of landmarks.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.2. Building coverings and complexes.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Intuitively, we regard these two holes as coming from faulty sampling or other errors in the recovery of the data.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=3
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The pictures and discussion above allow us to formulate the intuition that long intervals correspond to large scale geometric features in the space and that short intervals correspond to noise or inadequate sampling.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 1=2
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Both the the Čech and Vietoris-Rips complexes typically produce simplices in dimensions much higher than the dimension of the space.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.2. Building coverings and complexes.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It turns out that H simp k (X, Z) is always canonically isomorphic to the singular homology of the space |X|.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Remark.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The conclusion is that for simplicial complexes, homology is algorithmically computable, although standard methods for computing Smith normal form can often be painfully slow.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Remark.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If further is sufficiently small, then all the balls will be geodesically convex, and the complex will compute the homology of M correctly.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.3. Persistent homology.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that if the centers actually lie on a submanifold M ⊆ R n , and the set X is sufficiently dense in M , then this complex is the Čech complex attached to a covering of M .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.3. Persistent homology.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is clearly the case that if we have 0 ≤ ≤ , then we have an inclusion W s (X, L, ) ↩→ W s (X, L, ) and similarly for W s V R , W w , and W w V R .”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.2. Building coverings and complexes.
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, for finite metric spaces, it generically produces degenerate (i.e.discrete) complexes, with no one dimensional simplices.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.2. Building coverings and complexes.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In rough terms, δ k for large values of k computes density using points in large neighborhoods of x, and for small values uses small neighborhoods.So, δ k for large k corresponds to a smoothed out notion of density, and for small k corresponds to a version which carries more of the detailed structure of the data set.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.4. Example: Natural image statistics.
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“It was observed in [38] that in an informal sense, the images in the different conditions appeared to be quite similar, and a statistical analysis strongly suggested that the behavior of V1 in the spontaneous condition was consistent with a behavior which consisted in moving through a family of evoked images corresponding to responses to angular boundaries, without any particular order.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.5. Example: Electrode array data from primary visual cortex.
What the sources say
- Not checked — no full text of the sourcecites [38] kenet T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, Spontaneously emerging cortical representations of visual attributes, Nature 425 (2003), 954-956. source
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“This result is consistent with what was obtained by using ISOMAP in [61].”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 3.5. Examples.
What the sources say
- Not checked — no full text of the sourcecites [61] isomapJ.B. Tenenbaum, V. de Silva and J.C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290 (2000) (December), pp. 2319-2323. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The arrays of neurons studied in [59] are from the primary visual cortex or V1 in Macaque monkeys.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 2.5. Example: Electrode array data from primary visual cortex.
What the sources say
- Not checked — no full text of the sourcecites [59] visionG. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach, Topological Structure of Population Activity in Primary Visual Cortex, Journal of Vision, Volume 8, Number 8, Article 11, pp. 1-18, 2008. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“since the intuition is that features which appear at several levels in such a multiresolution diagram would be more intrinsic to the data set than those which appear at a single level.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Definition 3.2.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The core consists of normal or near-normal patients, and the two flares consist of patients with the two different forms of diabetes.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 3.5. Examples.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Z[m, n]-persistence vector spaces are examples of quiver representations, a highly developed area of algebra.See [29] for a complete description.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Corollary 4.6.
What the sources say
- Not checked — no full text of the sourcecites [29] Gabriel, P., 1997 source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These functions are clearly invariants of the isomorphism class of the module M , and we refer to them as the dimension and rank invariants, respectively.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Definition 4.3.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If the question is instead whether or not an element arises from a union of k elements of S, then we do not obtain complete information this way, but we do obtain partial information in that we can preclude the possibility that a class arises from such a union.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 4.3. Tree based persistence.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“We note that the correspondence X → π 0 (X) can actually be viewed as a functor (see [44]) from the category of topological spaces to the category of sets, in the sense that a continuous map f : X → Y induces a map of sets π 0 (f ) : π 0 (X) → π 0 (Y )”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Not matchedcites [44] Mac Lane, Saunders, 1998
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As we have seen above, algebraic topology is capable of producing signatures which indicate the presence of topological features within a space.As it stands, however, it is not capable of describing the source of the feature, i.e.where in the space the hole or other feature is located.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 4.3. Tree based persistence.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Theorem 4.5.Let M denote any Z[m, n]-persistent F -vector space, so that every vector space V i is finite dimensional.Then there is an isomorphism”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Example.
What the sources say
- Not checked — no full text of the sourcecites [29] Gabriel, P., 1997 source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“As Kleinberg points out, it is possible to do so in a number of different ways by specifying cost functions on particular clusterings, and prove a uniqueness result for optimal choices of clusterings with respect to this cost function”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Not matchedcites [39] kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“This machinery is not available in the multivariable case, but it turns out that it can be replaced by the Gröbner basis methodology (see [18] or [47]).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Definition 4.3.
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Any finite Z-persistence F -vector space M can be decomposed as”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Corollary 4.6.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The properties of trees guarantee that ≤ T is a partial ordering.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 4.3. Tree based persistence.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
In the paper
“In the paper [59], this was shown by simulation, which is of course always an option.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
What the sources say
- Not checked — no full text of the sourcecites [59] visionG. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach, Topological Structure of Population Activity in Primary Visual Cortex, Journal of Vision, Volume 8, Number 8, Article 11, pp. 1-18, 2008. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Not only is the correspondence π 0 (-) a functor from spaces to sets, it is equipped with a natural surjective map of sets η X : X → π 0 (X).”
- Also stated in: § 5. Reasoning about clustering · plain
“Further, these surjective maps have the property that for every continuous map f : X → Y , the diagram of sets (below) commutes.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
- Also stated in: § 5. Reasoning about clustering · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“While the observation is of course suggestive, to prove that it indicates structure in the data set one must show that segments of the indicated length cannot occur in a random model associated to a null hypothesis.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This means that Kleinberg's conditions also make sense in this context.”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
In the paper
“We point out that in the context of ordinary homology (i.e.not persistent homology), M. Penrose and others have developed a theory of “geometric random graphs” and have proved various results concerning the Betti numbers of complexes attached to such graphs (see [54]).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Additional information
What the sources say
- Not checked — no full text of the sourcecites [54] Penrose, Mathew, 2003 source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is clear that a C-functorial clustering algorithm is a clustering algorithm in the sense of Kleinberg, since the surjective map of sets from X to χ(X, d X ) yields a partition of X, namely the partition whose blocks are the sets η -1 X (z), as z ranges over all elements of χ(X, d X ).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > 5. Reasoning about clustering
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
None of the sources cited for this claim have been checked yet.
In the paper
“Each such interval can be considered as a point in the set D = {(x, y) ∈ R 2 |x ≤ y}, and the output is therefore a finite subset of the set D. As such, we now have a finite spatial point process (see [20], [3]).”
§ TOPOLOGY AND DATA GUNNAR CARLSSON > Additional information
What the sources say
- Not matchedcites [20] Daley, D. J., 1988
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [3] Baddeley, Adrian, 2007, Spatial point processes and their applications source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper relies on this without support.
There is no classification theorem for R-persistence abelian groups.
“However, we do not have such a theorem.”
The paper relies on this without support.
There is still no classification theorem for graded Z[t]-modules.
“Now, there is still no classification theorem for graded Z[t]-modules.”
The paper relies on this without support.
In studying data from experiments, one typically cannot assume that the data lies along a submanifold.
“If one is interested in studying data from experiments, though, one typically cannot assume that the data lies along a submanifold.”
The paper relies on this without support.
Even if one could assume that the data lies along a manifold, one is usually not in a position to verify that the stringent hypotheses of [52] are satisfied.
“Further, even if one could assume that the data lies along a manifold, one is usually not in a position to verify that the stringent hypotheses of [52] are satisfied.”
The paper relies on this without support.
The analysis of populations of neurons is amenable to geometric analysis.
“The second problem appears to be very amenable to geometric analysis, since it will involve the activities of several neurons at once.”
The paper relies on this without support.
The mechanism by which V1 carries out low-level tasks such as edge and line detection is not understood.
“However, the mechanism by which it carries out these tasks is not understood.”
The paper relies on this without support.
The classification of finitely generated modules over multivariable polynomial rings is much more complicated than the corresponding result for single-variable polynomial rings, and no reasonable parametrization is known.
“As is well known to algebraists, the classification of finitely generated modules over multivariable polynomial rings is much more complicated than the corresponding result for single variable polynomial rings, and in fact no reasonable parametrization is known.”
The paper relies on this without support.
In the case of more than one variable, the classification of multigraded modules depends on the field in question.
“examples show that in the case of more than one variable, the classification of multigraded modules definitely depends on the field in question.”
The paper relies on this without support.
The classification in the multivariable case is parametrized by points in moduli varieties over the ground field.
“In fact, the classification in the multivariable case is parametrized by points in moduli varieties over the ground field”
The paper relies on this without support.
Hierarchical clustering algorithms do not report single partitions but rather dendrograms, which are roughly speaking R+-persistent sets.
“This is not an unreasonable thing to do in view of the fact that hierarchical clustering algorithms do not report single partitions but rather dendrograms, which are roughly speaking R + -persistent sets.”
Citations
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“We note that the correspondence X → π0(X) can actually be viewed as a functor (see [44]) from the category of topological spaces to the category of sets, in the sense that a continuous map f : X → Y induces a map of sets π0(f) : π0(X)→ π0(Y ), satisfying certain obvious conditions on composite maps and identity maps.”
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Cited by
Sorted by citations
Items 1,101–1,200 of 2,331
Decomposing filtered chain complexes: Geometry behind barcoding algorithms
Chachólski, W., Giunti, B., Jin, A., & Landi, C. (2022) · Computational Geometry · doi:10.1016/j.comgeo.2022.101938 · Cited by 10
“Starting from the 1990s, new methods of topological shape analysis arose [15, 21, 22, 37], and developed in what is nowadays known as persistent homology [8, 17], with applications varying from quantum physics [36] to medicine [7], biology [39], and geology [26] (see [25] for an extended list of references to applicat…”
BackgroundManuchehrfar, F., Li, H., Ma, A., & Liang, J. (2022) · The Journal of Physical Chemistry B · doi:10.1021/acs.jpcb.2c07015 · Cited by 5
Polytopal Complex Construction and Use in Persistent Homology
Singh, R. P., & Wilsey, P. A. (2022) · IEEE ... International Conference on Data Mining workshops · doi:10.1109/icdmw58026.2022.00087 · Cited by 5
“TDA is an approach to data science that identifies topological features of data to provide insight into the topological invariants of the space [12].”
BackgroundPersistent Homology for RNA Data Analysis
Xia, K., Liu, X., & Wee, J. (2023) · Methods in molecular biology · doi:10.1007/978-1-0716-2974-1_12 · Cited by 5
Machine composition of Korean music via topological data analysis and artificial neural network
Tran, M. L., Lee, D., & Jung, J.-H. (2023) · Journal of Mathematics and Music · doi:10.1080/17459737.2023.2197905 · Cited by 5
“We refer readers to [13, 6] for more details of TDA through persistent homology.”
BackgroundMethodInfluentialTopoDimRed: a novel dimension reduction technique for topological data analysis
Wang, S., & Chen, B. (2023) · Информатика Экономика Управление - Informatics Economics Management · doi:10.47813/2782-5280-2023-2-2-0201-0213 · Cited by 5
First Demonstration Experiment for Energy-Trading System EDISON-X Using the XRP Ledger
Ikeda, Y., Ohki, Y., Marquardt, Z., et al. (2023) · doi:10.7566/jpscp.40.011008 · Cited by 5
Cleveland, E., Zhu, A., Sandstede, B., & Volkening, A. (2023) · SIAM Journal on Applied Dynamical Systems · doi:10.1137/22m1543082 · Cited by 8
“We provide an informal introduction to persistent homology and α-shapes here; see [17,20,28,39,73,88] and [27,29], respectively, for more technical details.”
BackgroundRutkowski, T. M., Abe, M., Sugimoto, H., et al. (2023) · Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society · doi:10.1109/embc40787.2023.10340508 · Cited by 5
“Topology application in data analysis [19] allows for describing and classifying noisy and highly-dimensional datasets by extracting topological invariants distinguishing the complex datasets [20].”
Vector summaries of persistence diagrams for permutation-based hypothesis testing
Islambekov, U., & Pathirana, H. (2024) · Foundations of Data Science · doi:10.3934/fods.2024002 · Cited by 6
“Topological data analysis (TDA) provides tools to describe the shape structure underlying data [2, 3].”
MethodQuantum topological data analysis via the estimation of the density of states
Scali, S., Umeano, C., & Kyriienko, O. (2024) · Physical Review A · doi:10.1103/physreva.110.042616 · Cited by 11
Tracking the topology of neural manifolds across populations
Yoon, I. H. R., Henselman‐Petrusek, G., Yu, Y., Ghrist, R., Smith, S. L., & Giusti, C. (2024) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2407997121 · Cited by 6
Suay‐García, B., Climent, J., Pérez‐Gracia, M. T., & Falcó, A. (2025) · Expert Opinion on Drug Discovery · doi:10.1080/17460441.2025.2477625 · Cited by 6
“This methodology has been applied across various fields, including economics, chemical engineering, industry, and biosciences [26–30].”
Topological data analysis of pattern formation of human induced pluripotent stem cell colonies
Hartsock, I., Park, E., Toppen, J., Bubenik, P., Dimitrova, E., Kemp, M. L., & Cruz, D. A. (2025) · Scientific Reports · doi:10.1038/s41598-025-90592-1 · Cited by 9
“TDA is a fast-growing field which provides methods for summarizing the shape of complex data 14 and has found useful applications in several fields, including cosmology 15 , material science 16 , neuroscience 17 , anomaly detection 18 , and C. elegans behavior 19 .”
The Offset Filtration of Convex Objects
Halperin, D., Kerber, M., & Shaharabani, D. (2015) · Lecture notes in computer science · doi:10.1007/978-3-662-48350-3_59 · Cited by 6
“k Since its introduction in [14], persistent homology has become an active area of research, including theoretical, algorithmic, and application results; we refer to the textbook [13] and the surveys [15, 7] for an overview. The information gathered by persistence is usually displayed either in terms of a barcode (as…”
BackgroundUsing Effective Subnetworks to Predict Selected Properties of Gene Networks
Gunaratne, G. H., Gunaratne, P. H., Seemann, L., & Török, A. (2010) · PLoS ONE · doi:10.1371/journal.pone.0013080 · Cited by 5
“, persistent homology [38,39]) or graph theoretic (e.”
BackgroundOliveira, J. L., Naveda, L., Gouyon, F., et al. (2012) · EURASIP Journal on Audio Speech and Music Processing · doi:10.1186/1687-4722-2012-18 · Cited by 5
“organized according (or in synchrony) to the projected musical cues it is likely that the projection of points generate clusters in space, or point clouds (Figure 4d), which can be clustered, discriminated and organized in different geometries or representations [24].”
BackgroundUnravelling the geometry of data matrices: effects of water stress regimes on winemaking
Hsieh, F., Hsueh, C., Heitkamp, C., et al. (2015) · Journal of The Royal Society Interface · doi:10.1098/rsif.2015.0753 · Cited by 5
“This bears similarity with the topological concept of shapes, as introduced by Carlsson [10] to analyse data, or by Mémoli [11] for shape comparison.”
BackgroundMethodBreathing detection based on the topological features of IR sensor and accelerometer signals
Erden, F., & Çetin, A. E. (2016) · 2018 52nd Asilomar Conference on Signals, Systems, and Computers · doi:10.1109/acssc.2016.7869685 · Cited by 5
“When all the complete simplices are filled in similarly, we obtain the Vietoris-Rips complex [17].”
MethodTopological Data Analysis for Extracting Hidden Features of Client Data
Schebesch, K. B., & Stecking, R. W. (2017) · Operations research proceedings · doi:10.1007/978-3-319-42902-1_65 · Cited by 5
“TDA is indeed based on concepts of algebraic topology [1], a both venerable and”
BackgroundThe Topological Data Analysis of Time Series Failure Data in Software Evolution
Costa, J. P., & Grbac, T. G. (2017) · doi:10.1145/3053600.3053604 · Cited by 5
“Persistent homology identifies a global structure by inferring high-dimensional structure from low-dimensional representations and studying proper-26 ties of a (often) continuous space by the analysis of a discrete sample of it, assembling discrete points into global structure [2].”
BackgroundConstructing Shape Spaces from a Topological Perspective
for the, A. D. N. I., Hofer, C. D., Kwitt, R., Niethammer, M., Höller, Y., Trinka, E., & Uhl, A. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-59050-9_9 · Cited by 5
“Recently, ideas from topological data analysis [6] have emerged to study 2D/3D objects, either in the form of computing topological invariants of features [16,24,23], or by directly analyzing the object of interest [26].”
BackgroundTime-delay lifts for physiological signal exploration: An application to ECG analysis
Fraser, B. A., Wachowiak, M. P., & Wachowiak-Smolíková, R. (2017) · Conference proceedings - Canadian Conference on Electrical and Computer Engineering · doi:10.1109/ccece.2017.7946619 · Cited by 5
“Recently, topological data analysis (TDA) has been proposed as a promising approach to analysis of these complex data [2].”
BackgroundBump hunting by topological data analysis
Sommerfeld, M., Heo, G., Kim, P., Rush, S. T. A., & Marron, J. S. (2017) · Stat · doi:10.1002/sta4.167 · Cited by 5
“We give a brief review of the main ideas here and refer to the growing literature for details (Bubenik & Kim, 2007; Ghrist, 2008; Carlsson & Zomorodian, 2009; Carlsson, 2009; Chazal et al., 2011).”
BackgroundGovernance Modeling: Dimensionality and Conjugacy
Mazzéga, P., Lajaunie, C., & Fieux, E. (2018) · InTech eBooks · doi:10.5772/intechopen.71774 · Cited by 5
On the Ethics of Biodiversity Models, Forecasts and Scenarios
Mazzéga, P. (2018) · Asian Bioethics Review · doi:10.1007/s41649-018-0069-5 · Cited by 6
“However, a topological approach can extract some of the fundamental properties of the system from these data (Carlsson 2009; Mazzega et al. 2018).”
BackgroundIsa, S. N. M., Shukor, S. A. A., Rahim, N. A., Ma’arof, I., Yahya, Z., Zakaria, A., Abdullah, A. H., & Wong, R. (2018) · IEEE Conference on Systems, Process and Control · doi:10.1109/spc.2018.8704136 · Cited by 5
“Geometric topology and filtering were used in data structure to study point cloud big data [33, 34].”
MethodGraph based analysis for gene segment organization In a scrambled genome
Hajij, M., Jonoska, N., Kukushkin, D., et al. (2020) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2020.110215 · Cited by 5
New families of stable simplicial filtration functors
Chowdhury, S., Clause, N., Mémoli, F., Sánchez, J. Á., & Wellner, Z. (2020) · Topology and its Applications · doi:10.1016/j.topol.2020.107254 · Cited by 5
“For the pair (p1, q1), the two [0, ∞) intervals are matched, and the interval [5, 10) for P is matched with the interval [4, 8) for Q, yielding CΨecc,0(p1, q1) = 1.”
BackgroundInfluentialPersistent homology in tourism: Unlocking the possibilities
Chong, W. K., & Rudkin, S. (2020) · Tourism Management · doi:10.1016/j.tourman.2020.104132 · Cited by 5
Fibers of Failure: Classifying Errors in Predictive Processes
Carlsson, L. S., Vejdemo‐Johansson, M., Carlsson, G., & Jönsson, P. G. (2020) · Algorithms · doi:10.3390/a13060150 · Cited by 5
“More detailed expositions can be found in [55,56].”
BackgroundSelf-citationUnsupervised Environmental Sound Classification Based On Topological Persistence
Cao, Y., Zhang, S., Yan, F.-J., Li, W., Sun, F., & Sun, H. (2019) · IEEE International Conference on Signal, Information and Data Processing (ICSIDP) · doi:10.1109/icsidp47821.2019.9173135 · Cited by 5
“Recent years have witnessed the successes of TDA applied in natural image statistics [8], [9], shape modelling [10], [12], chaotic signals [6], [11] and etc.”
BackgroundTopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data
Doraiswamy, H., Tierny, J., Silva, P. J. S., Nonato, L. G., & Silva, C. (2020) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2020.3030441 · Cited by 4
“One way of interpreting this layout is through the use of the equivalence between the 0-homology filtration of the Rips complex and hierarchical clustering using the single-linkage criterion [13].”
MethodForman-Ricci curvature and persistent homology of unweighted complex networks
Roy, I., Vijayaraghavan, S., Ramaia, S. J., & Samal, A. (2020) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2020.110260 · Cited by 5
Topological Data Analysis and UNICEF Multiple Indicator Cluster Surveys
Anderson, J. R., Memić, F., & Volić, I. (2022) · Journal of Quantitative Economics · doi:10.1007/s40953-022-00288-w · Cited by 5
“For overviews of TDA and its applications, see Carlsson (2009, 2020) and Otter et al.”
MethodWu, H., Tao, X., & Zheng, Z. C. (2021) · Physics of Fluids · doi:10.1063/5.0033509 · Cited by 5
Topological data analysis of Korean music in Jeongganbo: a cycle structure
Tran, M. L., Park, C., & Jung, J.-H. (2023) · Journal of Mathematics and Music · doi:10.1080/17459737.2022.2164626 · Cited by 5
“our node denition, in Tables 6, 7 and 8 in Appendix. 2 2. Topological data analysis In this section we brie y explain TDA, particularly persistent homology. For more detailed explanation we refer to [6]. The main motivation of using TDA is to study the cycle structure in music data. In this paper, we focus on the one-…”
BackgroundPersistence of Conley--Morse Graphs in Combinatorial Dynamical Systems
Dey, T. K., Mrożek, M., & Slechta, R. (2022) · SIAM Journal on Applied Dynamical Systems · doi:10.1137/21m143162x · Cited by 7
Statistical and topological summaries aid disease detection for segmented retinal vascular images
Nardini, J. T., Pugh, C. W. J., & Byrne, H. M. (2023) · Microcirculation · doi:10.1111/micc.12799 · Cited by 5
“PH quantifies the lifetimes of topological features, here connected components and loops, in a filtration using barcodes and persistence diagrams [19, 20, 21, 42, 43].”
BackgroundMethodInfluentialLukmanov, R., de Koning, C., Schmidt, P. K., Wacey, D., Ligterink, N. F. W., Gruchola, S., Grimaudo, V., Neubeck, A., Riedo, A., Tulej, M., et al. (2022) · Frontiers in Space Technologies · doi:10.3389/frspt.2022.718943 · Cited by 5
“Here, we will provide only a short description of the algorithm; for a full account of the inner workings of the algorithm, we refer the reader to the original publication (Singh et al., 2007) and an introductory article on topological data analysis (Carlsson, 2009).”
Configuration spaces of labeled points on a circle with two anchors
Kozlov, D. N. (2022) · Topology and its Applications · doi:10.1016/j.topol.2022.108147 · Cited by 5
User dynamic topology-information-based matrix factorization for e-government recommendation
Sun, N., Chen, T., Luo, Q., & Ran, L. (2022) · Applied Soft Computing · doi:10.1016/j.asoc.2022.108993 · Cited by 6
Functional neuroimaging in the era of Big Data and Open Science: A modern overview
Lazar, N. A. (2023) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1609 · Cited by 6
“Topological data analysis (TDA; Carlsson, 2009; Chazal & Michel, 2021; Ghrist, 2008) is one such direction which is gaining popularity in the neuroimaging field.”
BackgroundReconstructing functions from random samples
Ferry, S., Mischaikow, K., & Nanda, V. (2014) · Journal of Computational Dynamics · doi:10.3934/jcd.2014.1.233 · Cited by 5
“The use of algebraic topological methods for the analysis of nonlinear data has become a subject of considerable interest with a wide variety of promising applications [3, 5, 7, 10].”
BackgroundGeometric Quantification of Cost Uncertainty Propagation: A Case Study
Schwabe, O., Shehab, E., & Erkoyuncu, J. A. (2015) · Procedia CIRP · doi:10.1016/j.procir.2015.08.078 · Cited by 5
Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2016) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2016.1260472 · Cited by 5
“…Letscher, and Zomorodian (2002); Cohen-Steiner, Edelsbrunner, and Harer (2005); Bubenik and Kim (2007); Chazal and Oudot (2008); Edelsbrunner and Harer (2008); Ghrist (2008); Carlsson (2009); Carlsson and Zomorodian (2009); Edelsbrunner and Harer (2010); Chazal et al. (2011); Turner et al. (2012).”
BackgroundInfluentialHidden multidimensional social structure modeling applied to biased social perception
Maletić, S., & Zhao, Y. (2017) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2017.11.069 · Cited by 4
A regularity index for dendrites - local statistics of a neuron's input space
Anton-Sanchez, L., Effenberger, F., Bielza, C., Larrañaga, P., & Cuntz, H. (2018) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1006593 · Cited by 5
Real-Time Motion Planning in Changing Environments Using Topology-Based Encoding of Past Knowledge
Fisher, R., Rosman, B., & Ivan, V. (2018) · IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) · doi:10.1109/iros.2018.8593879 · Cited by 5
“1For a full treatment of computational topology in path planning, see Carlsson [17], Edelsbrunner [18] and LaValle [1].”
MethodData Reduction and Feature Isolation for Computing Persistent Homology on High Dimensional Data
Verma, R. R., Malott, N. O., & Wilsey, P. A. (2021) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata52589.2021.9671839 · Cited by 5
“Persistent Homology (PH) is one of the principal components of Topological Data Analysis (TDA) [1].”
BackgroundThe Complexity of Some Topological Inference Problems
Weinberger, S. (2013) · Foundations of Computational Mathematics · doi:10.1007/s10208-013-9152-1 · Cited by 5
“In recent years, there has been a growing amount of work, with a number of notable successes (see, e.g., the surveys [6, 14, 19] and the book [38] for an overview, and the recent paper [29] for an outstanding example) in applying geometric or topological methods to this problem.”
BackgroundADVANCES IN MULTIDIMENSIONAL SIZE THEORY
Cerri, A., & Frosini, P. (2011) · Image Analysis & Stereology · doi:10.5566/ias.v29.p19-26 · Cited by 4
“, 2002; Carlsson and Zomorodian, 2007). More precisely, the (multidimensional) size function l(M ,~φ) coincides with the 0-th rank invariant associated with the (multi)filtration induced on M by ~φ . For a formal definition of rank invariant the reader is referred to Carlsson and Zomorodian (2007).”
BackgroundChen, L. M. (2015) · Springer eBooks · doi:10.1007/978-3-319-25127-1_6 · Cited by 5
Topological Signals of Singularities in Ricci Flow
Alsing, P. M., Blair, H. A., Corne, M., Jones, G., Miller, W. A., Mischaikow, K., & Nanda, V. (2017) · Axioms · doi:10.3390/axioms6030024 · Cited by 5
“Since our treatment of these topics is cursory, we refer the interested reader to [18,19,21,29] for overviews of persistent homology and its applications and to [17,30] for various technical details pertaining to Ricci flow and its discrete analogues.”
Modeling of persistent homology
Agami, S., & Adler, R. J. (2019) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2019.1615091 · Cited by 5
“) corresponding to all the bars is called the persistence diagram. (We shall assume that the reader is familiar with these concepts. Recent excellent and quite different books and reviews by Carlsson [6, 7], Edelsbrunner and Harer [10, 11, 12], Zomorodian [17], Oudot [14] and Ghrist [13] not only give give broad expos…”
BackgroundNakajima, T. (2019) · Entropy · doi:10.3390/e21020216 · Cited by 5
Understanding l4-based Dictionary Learning: Interpretation, Stability, and Robustness
Zhai, Y., Mehta, H., Zhou, Z., & Ma, Y. (2020) · International Conference on Learning Representations · Cited by 6
Lima, A. D. C. (1927) · Zenodo (CERN European Organization for Nuclear Research) · doi:10.5281/zenodo.24037 · Cited by 5
Topological Signatures of Singularities in Simplicial Ricci FlowPreprint
Alsing, P. M., Blair, H. A., Corne, M., Jones, G., Miller, W. A., Mischaikow, K., & Nanda, V. (2015) · arXiv (Cornell University) · Cited by 5
Illustrations of Data Analysis Using the Mapper Algorithm and Persistent Homology
Kraft, R. (2016) · KTH Publication Database DiVA (KTH Royal Institute of Technology) · Cited by 5
Crawford, L. A., Monod, A., Chen, A., Mukherjee, S., & Rabadán, R. (2016) · arXiv (Cornell University) · Cited by 5
Generalized Cluster Trees and Singular MeasuresPreprint
Chen, Y. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1611.02762 · Cited by 5
“The cluster tree is also related to the topological data analysis (Carlsson, 2009; Edelsbrunner and Morozov, 2012; Wasserman, 2016).”
Jeitziner, R., Carrière, M., Rougemont, J., Oudot, S., Hess, K. P., & Brisken, C. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1801.01841 · Cited by 5
“Topological data analysis (TDA) is used to reduce dimensions and to recognize patterns ([4], [10]).”
MethodSpectral Sparsification of Simplicial Complexes for Clustering and Label PropagationPreprint
Osting, B., Palande, S., & Wang, B. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.08436 · Cited by 5
“…in activity for the development of data analysis methods that focus on simplicial complexes, including methods based on computational topology [8, 20, 23, 27], higher-order random processes [5, 26], generalized Cheeger inequalities [28, 54], isoperimetric inequalities [45], high-dimensional…”
MethodMeasuring Human Activity Spaces from GPS Data with Density Ranking and Summary CurvesPreprint
Chen, Y., & Dobra, A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.05017 · Cited by 5
Weighted persistent homology for biomolecular data analysisPreprint
Meng, Z., Anand, D. V., Lu, Y., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1903.02890 · Cited by 5
“Further, different PH based functions are proposed for the visualization, representation and modeling of topological information.(15,22,24) The persistent Betti number (PBN) or Betti curve is one of them.”
Signal processing on simplicial complexesPreprint
Ji, F. Q., Kahn, G., & Tay, W. P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2004.02392 · Cited by 5
“They are already used in topological data analysis [22], representation of surfaces in high dimensions, and modeling of complex networks [23].”
BackgroundCovid-19 clinical data analysis using Ball MapperPreprint
Dłotko, P., & Rudkin, S. (2020) · medRxiv · doi:10.1101/2020.04.10.20061374 · Cited by 5
“Starting from pioneering works as in Carlsson (2009) and Nicolau et al. (2011) we will build up on the recent developement presented in D lotko (2019a) to localize clusters of patients that may be important from the perspective of a healthcare provider.”
BackgroundMethodDecoding asymptomatic COVID-19 infection and transmissionPreprint
Wang, R., Hozumi, Y., Yin, C., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2007.01344 · Cited by 5
“TML-MP/) which is 12 briefly reviewed as following and its detail can be found in the literature [6]. TML-MP applies element specific persistent homology, which reveals essential biological information [8,14]. The method employs the element-specific persistent homology [7] and other biological and chemical properties as…”
MethodMultiscale Topology Characterises Dynamic Tumour Vascular NetworksPreprint
Stolz, B. J., Kaeppler, J., Markelc, B., Mech, F., Lipsmeier, F., Muschel, R. J., Byrne, H. M., & Harrington, H. A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.08667 · Cited by 7
“To compare this multiscale topological data analysis to standard descriptors (which are single scale) requires its compression; TDA gives additional information that surpasses manual counting and existing descriptors.”
BackgroundInfluentialEntropy and Diversity: The Axiomatic ApproachPreprint
Leinster, T. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.02113 · Cited by 24
“(For expository accounts of persistent homology, see Ghrist [113] or Carlsson [59].”
ToDD: Topological Compound Fingerprinting in Computer-Aided Drug DiscoveryPreprint
Demir, A., Coşkunuzer, B., Segovia-Domínguez, I., Chen, Y., Gel, Y. R., & Kızıltan, B. (2022) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2022.11.08.515685 · Cited by 5
“Persistent homology (PH) is a key approach in TDA, allowing us to extract the evolution of subtler patterns in the data shape dynamics at multiple resolution scales which are not accessible with more conventional, non-topological methods [17].”
BackgroundFiniteness of rank invariants of multidimensional persistent homology groups
Cagliari, F., & Landi, C. (2010) · Applied Mathematics Letters · doi:10.1016/j.aml.2010.11.004 · Cited by 4
On estimating the perimeter using the alpha-shape
Ery Arias-castro, A., & Rodríguez‐Casal, A. (2017) · Annales de l Institut Henri Poincaré Probabilités et Statistiques · doi:10.1214/16-aihp747 · Cited by 4
“This includes, for example, estimating of the number of connected components [2], the intrinsic dimensionality [17] and, more generally, the homology [5,6,20,30,37], the Minkowski content [7], as well as the perimeter and area [3,29].”
BackgroundOn shapes recognition in topological data analysis
Janes, F., Mwanzalima, M., & Mpimbo, M. (2025) · Research in Mathematics · doi:10.1080/27684830.2025.2573577 · Cited by 4
Domínguez-Monterroza, A., Caballero, A. M., & Jiménez-Martín, A. (2026) · IEEE Open Journal of the Computer Society · doi:10.1109/ojcs.2026.3651318 · Cited by 4
“Persistent homology, a key TDA technique, extracts topological features such as connected components, loops, and voids that persist across multiple scales, offering novel insights into the geometric complexity of speech signals [37], [19].”
TopoAttn-Nets: Topological Attention in Graph Representation Learning
Chen, Y., Sizikova, E., & Gel, Y. R. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-26390-3_19 · Cited by 4
“In particular, topological data analysis (TDA) offers rigorous mathematical tools to explore structural shape properties of the graph-structured data [4,9,14].”
BackgroundMild explocivity, persistent homology and cryptocurrencies' bubbles: An empirical exercise
Arvanitis, S., & Detsis, M. (2023) · AIMS Mathematics · doi:10.3934/math.2024045 · Cited by 4
“We work either with pairs, or with the ensemble of cryptocurrencies, thus constructing via the relevant time series, point clouds inside R(2) or R(4), respectively-see Carlsson (2009) [4], and (2014) [5].”
Structural Feature Extraction via Topological Data Analysis
Wang, B., Feng, B., Lv, L., Li, S., & Pan, F. M. (2025) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.5c01831 · Cited by 4
Topological information retrieval with dilation-invariant bottleneck comparative measures
Cao, Y., Monod, A., Vlontzos, A., Schmidtke, L., & Kainz, B. (2023) · Information and Inference A Journal of the IMA · doi:10.1093/imaiai/iaad022 · Cited by 4
“We now briefly formalize these technicalities; a complete discussion with full details can be found in the literature on applied and computational topology (e.g., Carlsson (2009); Ghrist (2008); Edelsbrunner and Harer (2008)).”
BackgroundFrosini, P., Gridelli, I., & Pascucci, A. (2023) · Entropy · doi:10.3390/e25081150 · Cited by 4
“In the last thirty years Topological Data Analysis (TDA) developed as a useful mathematical theory for analyzing data, benefiting from the reduction of dimensionality guaranteed by topology[3, 5, 11].”
Barcodes distinguishing morphology of neuronal tauopathy
Beers, D., Goniotaki, D., Hanger, D. P., Goriely, A., & Harrington, H. A. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.043006 · Cited by 7
Simplicial complex entropy for time series analysis
Guzmán-Vargas, L., Zabaleta-Ortega, Á., & Guzmán-Sáenz, A. (2023) · Scientific Reports · doi:10.1038/s41598-023-49958-6 · Cited by 5
Persistent homotopy groups of metric spaces
Mémoli, F., & Zhou, L. (2024) · Journal of Topology and Analysis · doi:10.1142/s1793525324500018 · Cited by 4
On Vietoris–Rips complexes of finite metric spaces with scale 2
Feng, Z., & Nukala, N. C. P. (2024) · Journal of Homotopy and Related Structures · doi:10.1007/s40062-024-00340-x · Cited by 4
“Along with the development of topological data analysis [10, 6], determining the homotopy types of Vietoris-Rips complex of finite metric spaces has become crucial in applied topology.”
Topological Interpretability for Deep Learning
Spannaus, A., Hanson, H. A., Tourassi, G. D., & Penberthy, L. T. (2024) · Platform for Advanced Scientific Computing Conference (PASC) · doi:10.1145/3659914.3659935 · Cited by 4
“This result is is non-trivial, we omit the technical details here; see [31, 7, 9, 15] and references contained therein.”
LightGBM robust optimization algorithm based on topological data analysis
Yang, H., Qin, G., Liu, Z. Y., Hu, Y., & Dai, Q. (2024) · International Conference on Computer and Multimedia Technology (ICCMT) · doi:10.1145/3675249.3675350 · Cited by 6
Fan, S., Xu, H., Fu, S., et al. (2024) · IEEE/ACM International Symposium on Quality of Service (IWQoS) · doi:10.1109/iwqos61813.2024.10682857 · Cited by 4
Abid, M., & Saqlain, M. (2024) · Management Science Advances. · doi:10.31181/msa1120241 · Cited by 4
Bulk-Hole Correspondence and Inner Robust Boundary Modes in Singular Flatband Lattices
Song, L., Gao, S., Xia, S., Liang, Y., Tang, L., Song, D., Leykam, D., & Chen, Z. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.063803 · Cited by 4
The asymptotics of the expected Betti numbers of preferential attachment clique complexes
Siu, C., Samorodnitsky, G. P., Yu, C. L., et al. (2025) · Advances in Applied Probability · doi:10.1017/apr.2024.66 · Cited by 4
“[Aktas et al., 2019, Carlsson, 2009, Chazal and Michel, 2021] For instance, it was observed that the way holes emerge in (biological) neural networks helps distinguish different stimuli to the brains of different animals.”
Singh, M. K., Kumar, A., Pant, S., Chaube, S., Misra, K., Singh, J. P., & Kotecha, K. V. (2025) · Future Batteries · doi:10.1016/j.fub.2025.100059 · Cited by 4
Lee, J., Mun, J., Choo, M., & Park, S. (2025) · npj Digital Medicine · doi:10.1038/s41746-025-01635-w · Cited by 5
Malinowski, J., Kostrzewa, M., Balcerek, M., Tomczuk, W., & Szwabiński, J. (2025) · Journal of Physics Photonics · doi:10.1088/2515-7647/add826 · Cited by 5
“Euler characteristics computed for complexes generated with the Vietoris–Rips filtration [68] combined with the order of points within the trajectory (points that occur later in trajectory, occur later in filtration) have been used as additional features for our regressor.”
Human perception faithful curve reconstruction based on persistent homology and principal curve
Chen, Y., Lin, H., & Xing, Y. (2025) · Graphical Models · doi:10.1016/j.gmod.2025.101267 · Cited by 4
Topology-Driven Trajectory Synthesis with an Example on Retinal Cell Motions
Gu, C., Guibas, L., & Kerber, M. (2014) · Lecture notes in computer science · doi:10.1007/978-3-662-44753-6_24 · Cited by 4
“Persistence diagrams are currently a popular research topic with many theoretical and practical contributions; we refer to the surveys [1, 6] for contemporary overviews.”
BackgroundSurmounting the Cartesian Cut: Klein Bottle Logophysics, The Dirac Algebra and the Genetic Code
Rapoport, D. L. (2011) · NeuroQuantology · doi:10.14704/nq.2011.9.4.493 · Cited by 4
“This robustness to the influence of noise is apparent also in the KB topographic map of the sensorium in the human body (Rapoport, 2011b, Werner, 1968) and the visual recognition of arbitrary patterns that show an intrinsic universal gestalt, which is the KB (Ghrist, 2010; Carlson, 2009).”
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies
Chepushtanova, S., Kirby, M., Peterson, C., & Ziegelmeier, L. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-39441-1_21 · Cited by 4
“It is a computational approach to topology that allows one to answer basic questions about the structure of point clouds in data sets at multiple scales [3].”
BackgroundNetwork Topology vs. Geometry: From persistent Homology to Curvature
Saucan, E., & Jost, J. (2017) · Cited by 5
Survey of TOPDRIM applications of Topological Data Analysis
Merelli, E., Rucco, M., Tesei, L., Piangerelli, M., Mamuye, A. L., & Quadrini, M. (2016) · Cited by 4
Metadata
- Authors
- Carlsson, G.
- Affiliations
- Stanford University (US)
- Corresponding author
- Carlsson, Gunnar
- ORCID
- Carlsson, Gunnar
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2009-01-29
- Volume
- 46
- Issue
- 2
- Pages
- 255-308
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W1991566301
- Cited by
- 2,302
- References
- 78
- Citations by year
- 2026: 215 · 2025: 216 · 2024: 158 · 2023: 176 · 2022: 159 · 2021: 215 · 2020: 227 · 2019: 151 · 2018: 163 · 2017: 132 · 2016: 128 · 2015: 95 · 2014: 99 · 2013: 65 · 2012: 46
- FWCI
- 33.43
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- Funders
- National Science Foundation (0354543)
- Defense Advanced Research Projects Agency
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- point clouds, distance measures, high-dimensional data, biological data, microarray data, high-throughput data
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