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.
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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- 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)
hedgedstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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- 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.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- 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)
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- 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)
plainstates an association- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- 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.
plainstates an association- 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.
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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)
plainstates a cause- 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.
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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)
plainstates an association- 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.
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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
boostedstates an association- 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.
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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.
hedgedstates an associationSingle 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
hedgedstates an associationSingle 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
hedgedstates a cause- 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
plainstates a cause- 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
boostedstates an association- 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.
hedgedstates an association- 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
hedgedstates an association- 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.
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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
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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
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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
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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
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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
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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
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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
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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
plainstates a cause- Asserted: passes — The authors put it forward as true.
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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
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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.
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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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
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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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
hedgedstates an association- Asserted: passes — The authors put it forward as true.
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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.
plainstates an association- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
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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.
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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.
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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.
plainstates a cause- Asserted: passes — The authors put it forward as true.
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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.
plainstates an association- Asserted: passes — The authors put it forward as true.
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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.
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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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.
plainstates a cause- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
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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.
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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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
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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
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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
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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.
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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.
plainstates an association- Asserted: passes — The authors put it forward as true.
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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
plainstates an association- Asserted: passes — The authors put it forward as true.
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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.
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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.
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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.
plainstates a cause- Asserted: passes — The authors put it forward as true.
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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.
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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.
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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.
hedgedstates a cause- Asserted: passes — The authors put it forward as true.
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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
plainstates a cause- 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.
plainstates a cause- 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
plainstates an association- 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
plainstates an association- 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
boostedstates a cause- 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 1,301–1,400 of 2,331
Hypothesis testing for shapes using vectorized persistence diagrams
Moon, C., & Lazar, N. A. (2023) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1093/jrsssc/qlad024 · Cited by 6
“A primary TDA tool is persistent homology, which analyzes topological features of data in various scales (Edelsbrunner and Harer, 2008; Carlsson, 2009).”
BackgroundTransformations, Mappings, and Data Summaries
Whitaker, R., & Hotz, I. (2020) · Springer eBooks · doi:10.1007/978-3-030-34444-3_6 · Cited by 3
Learning from the Shape of Data
Rosenstock, S. (2021) · Philosophy of Science · doi:10.1086/715078 · Cited by 3
“(See Zomorodian, Afra and Carlsson, Gunnar (2008) for details on this localization method).”
BackgroundSome New Methods to Build Group Equivariant Non-expansive Operators in TDA
Quercioli, N. (2021) · Springer proceedings in mathematics & statistics · doi:10.1007/978-981-16-0174-3_19 · Cited by 3
Interpolated functional manifold for functional near-infrared spectroscopy analysis at group level
Ávila-Sansores, S. M., Rodríguez‐Gómez, G., Tachtsidis, I., et al. (2020) · Neurophotonics · doi:10.1117/1.nph.7.4.045009 · Cited by 3
“However, the analysis of the geometry and topology of a cloud of points faces important challenges, even though strong mathematical formalisms are already available.(5) Consequently, this approach has remained largely underused and, as far as we are aware, mostly confined to the analysis of functional connectivity acr…”
BackgroundThe Application of Topological Data Analysis in Practice and Its Effectiveness
Cheng, L. (2020) · E3S Web of Conferences · doi:10.1051/e3sconf/202021403034 · Cited by 3
Unsupervised Sentence-embeddings by Manifold Approximation and Projection
Kayal, S. (2021) · Conference of the European Chapter of the Association for Computational Linguistics · doi:10.18653/v1/2021.eacl-main.1 · Cited by 4
Quantitative measure of memory loss in complex spatiotemporal systems
Kramár, M., Kovalčinová, L., Mischaikow, K., et al. (2021) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0033419 · Cited by 3
“9 In particular, we use persistence diagrams 10,11 to quantify the changes in topology of an interaction force network as the force level changes from infinity to zero.”
MethodGraph Approximations to Geodesics on Metric Graphs
Vandaele, R., Saeys, Y., & Bie, T. D. (2021) · Proceedings - International Conference on Pattern Recognition/Proceedings/International Conference on Pattern Recognition · doi:10.1109/icpr48806.2021.9412448 · Cited by 3
“Furthermore, the field of Topological Data Analysis [18] provides theoretical guarantees that proximity graphs preserving the geodesic distances well, also preserve important topological characteristics of these spaces, or any totally bounded metric space in general [19, Th. 7.5.].”
BackgroundInfluentialSymplectic rigidity and quantum mechanics
Polterovich, L. (2018) · EMS Press eBooks · doi:10.4171/176-1/6 · Cited by 3
Cliques Are Bricks for k-CT Graphs
Snåšel, V., Dráždilová, P., & Platoš, J. (2021) · Mathematics · doi:10.3390/math9111160 · Cited by 3
“Limited-length cycles are crucial in algebraic topology applications [11,12]; they are necessary to calculate the topological properties of data [13,14].”
BackgroundA Sheaf and Topology Approach to Detecting Local Merging Relations in Digital Images
Hu, C.-S., & Chung, Y.-M. (2021) · IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) · doi:10.1109/cvprw53098.2021.00496 · Cited by 4
“Persistent homology (PH), one of the main tools in TDA, tracks changes of towered topological spaces induced by the original objects [27, 21, 15, 25, 49, 6].”
BackgroundTwo-parameter Persistence for Images via Distance Transform
Hu, C.-S., Lawson, A., Chung, Y.-M., et al. (2021) · IEEE/CVF International Conference on Computer Vision Workshops (ICCVW) · doi:10.1109/iccvw54120.2021.00464 · Cited by 3
“Topological data analysis (TDA) is a rising field at the intersection of Mathematics, Statistics, and Machine Learning [9, 3, 33, 5].”
BackgroundSpectral Permutation Test on Persistence Diagrams
Wang, Y., Chung, M. K., & Fridriksson, J. (2022) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp43922.2022.9747399 · Cited by 3
“The fact that the overall topological changes hold more sig-nificance over fleeting structures in PH makes the approach particularly robust under the presence of noise and artifacts, thus revealing more topological insight than traditional meth-ods [8].”
BackgroundAlarm Sound Detection Using Topological Signal Processing
Fireaizen, T., Ron, S., & Bobrowski, O. (2022) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp43922.2022.9747228 · Cited by 3
“For (2)-(8) we used 20[ms] windows, and took the maximum, minimum, average, median, and standard deviation over all windows.”
MethodThe Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous
Ery Arias-castro, A., Bruno Pelletier, B., & Pudlo, P. (2012) · Advances in Applied Probability · doi:10.1017/s000186780000598x · Cited by 3
TAaCGH Suite for Detecting Cancer—Specific Copy Number Changes Using Topological Signatures
Aslam, J., Ardanza-Trevijano, S., Xiong, J., et al. (2022) · Entropy · doi:10.3390/e24070896 · Cited by 3
“The existence results given here are for Vietoris–Rips and Čech complexes, but essentially the same arguments work for the various forms of witness complexes described in [31].”
ResultToda, A., Hiwa, S., Tanioka, K., & Hiroyasu, T. (2022) · IEEE Congress on Evolutionary Computation (CEC) · doi:10.1109/cec55065.2022.9870341 · Cited by 4
“Mapper is one of TDA tools [16]–[18].”
MethodBayır, M. A., Shamsi, K., Kaynak, H., & Akçora, C. G. (2022) · IEEE Access · doi:10.1109/access.2022.3229008 · Cited by 3
“We then use TDA Mapper [13] to transform these high-dimensional feature vectors into a topological cluster network.”
MethodInvestigating the influence of topology on elastic properties in spinodal microstructures
Golnary, F., & Asghari, M. (2023) · Modelling and Simulation in Materials Science and Engineering · doi:10.1088/1361-651x/acfd48 · Cited by 3
“Another tool that can be used to decompose the space into simpler parts and compute the Betti numbers is the Mayer–Vietoris sequence [39].”
Tree representations of brain structural connectivity via persistent homology
Li, D., Nguyen, P. H., Zhang, Z., & Dunson, D. B. (2023) · Frontiers in Neuroscience · doi:10.3389/fnins.2023.1200373 · Cited by 3
“In this paper, we propose a new representation of the brain connectome that is inspired by ideas in computational topology, a field focused on developing computational tools for investigating topological and geometric structure in complex data (Carlsson, 2009; Edelsbrunner and Harer, 2010).”
BackgroundMethodInfluentialMultiscale transforms for signals on simplicial complexes
Saito, N., Schonsheck, S. C., & Shvarts, E. (2023) · Sampling Theory Signal Processing and Data Analysis · doi:10.1007/s43670-023-00076-4 · Cited by 3
“For a more thorough review, see [5, 17].”
Convolutional persistence transforms
Solomon, E., & Bendich, P. L. (2024) · Journal of Applied and Computational Topology · doi:10.1007/s41468-024-00164-x · Cited by 3
“As (Carlsson et al, 2008) observed, the space of local patches in natural images has very high codimension in the space of all possible patches, and hence in practice the CPT may not require a very high-dimensional collection of filter functions to be effective at distinguishing images.”
BackgroundAbdullahi, M. S., Suratanee, A., Piro, R. M., & Plaimas, K. (2024) · Mathematics · doi:10.3390/math12050725 · Cited by 3
Topological data analysis and image visibility graph for texture classification
Pal, R., Kumar, S., & Singh, M. K. (2024) · International Journal of Systems Assurance Engineering and Management · doi:10.1007/s13198-024-02272-4 · Cited by 3
Mountaineer: Topology-Driven Visual Analytics for Comparing Local Explanations
Solunke, P., Guardieiro, V., Rulff, J., Xenopoulos, P., Chan, G. Y.-Y., Barr, B., Nonato, L. G., & Silva, C. T. (2024) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2024.3418653 · Cited by 3
“However, in the real world, datasets are not continuous manifolds but are usually sets of high-dimensional data points [26].”
BackgroundTopological variable neighborhood search
Filipović, V., & Kartelj, A. (2024) · Journal Of Big Data · doi:10.1186/s40537-024-01017-1 · Cited by 3
“Information from biological systems and from “in silico” experiments is usually less structured and can be organized via a concept of “cluster-ing” [35] into a dynamical (filtered) simplicial complex.”
Dagher, L. B., Madern, D., Malbos, P., & Brochier‐Armanet, C. (2025) · Molecular Biology and Evolution · doi:10.1093/molbev/msae271 · Cited by 3
“1) (Barannikov 1994; Carlsson 2009).”
Classification of COVID-19 via Homology of CT-SCAN
Iqbal, S., Ahmed, H. M., Qaiser, T., Qureshi, M. I., & Rajpoot, N. (2025) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2025.110226 · Cited by 10
“…(de Silva & Ghrist, 2007; Lee et al. , 2012), computational biology (Kasson et al. , 2007; Yao et al. , 2009; Wang & Wei, 2016), data analysis (Carlsson, 2009; Liu et al. , 2012; Rieck et al. , 2012), image analysis (Carlsson et al. , 2009; Frosini & Landi, 2013; Bendich et al. , 2010),…”
On the Structural Theorem of Persistent Homology
Meehan, K., Pavlichenko, A., & Segert, J. (2018) · Discrete & Computational Geometry · doi:10.1007/s00454-018-0042-9 · Cited by 5
“either provides an algorithm to decompose a given object as a direct sum of indecomposables, nor a classification of indecomposables. 2.2. Persistence Objects and Filtered Objects. Persistence objects [5] and filtered objects [17] are described by categorical diagrams. Suppose that X is a linear Abelian category (and th…”
BackgroundInfluentialTopological data analysis as image steganalysis technique
Asaad, A., Jassim, S., & Rashid, R. D. (2018) · doi:10.1117/12.2309767 · Cited by 3
Klemelä, J. (2018) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1436 · Cited by 3
“Carlsson (2009) gives a review of concepts of topological data analysis.”
BackgroundTopological Interpretation of Interactive Computation
Merelli, E., & Wasilewska, A. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-31514-6_12 · Cited by 3
A Persistent Entropy Automaton for the Dow Jones Stock Market
Piangerelli, M., Tesei, L., & Merelli, E. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-31517-7_3 · Cited by 3
A torus model for optical flow
Adams, H., Bush, J., Carr, B., Kassab, L., & Mirth, J. (2019) · Pattern Recognition Letters · doi:10.1016/j.patrec.2019.11.029 · Cited by 3
“For more information on homology see [5, 27], for introductions to persistent homology see [16, 23, 24, 41], and for applications of persistent homology to sensor networks, machine learning, biology, medical imaging, see [4, 9, 12, 13, 20, 32, 22, 38, 40].”
BackgroundGeometric Metrics for Topological Representations
Som, A., Ramamurthy, K. N., & Turaga, P. K. (2020) · Springer eBooks · doi:10.1007/978-3-030-31351-7_15 · Cited by 3
Dubuc, B., Ebrahimkhanlou, A., Sitaropoulos, K., et al. (2020) · doi:10.1117/12.2557322 · Cited by 3
Hajij, M., Assiri, B., & Rosen, P. (2020) · Advances in intelligent systems and computing · doi:10.1007/978-3-030-63089-8_47 · Cited by 3
Tape surfaces characterization with persistence images
Frahi, T., Argerich, C., Yun, M., et al. (2020) · AIMS Materials Science · doi:10.3934/ms.2020.4.364 · Cited by 3
“) and the change in their topological features as considered in [13, 14, 20, 21].”
BackgroundIto, D., Shirasawa, R., Iino, Y., Tomiya, S., & Tanaka, G. (2020) · PLoS ONE · doi:10.1371/journal.pone.0239933 · Cited by 3
Simple homotopy theory and nerve theorem for categories
Tanaka, K. (2021) · Topology and its Applications · doi:10.1016/j.topol.2021.107609 · Cited by 3
Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology
Landi, C., & Scaramuccia, S. (2021) · Journal of Combinatorial Optimization · doi:10.1007/s10878-021-00729-x · Cited by 3
“In the pipeline of TDA, one starts from a point cloud representing data with unknown structure.”
BackgroundMethodThe Topology of Projective Codes and the Distribution of Zeros of Odd Maps
Adams, H., Bush, J., & Frick, F. (2023) · The Michigan Mathematical Journal · doi:10.1307/mmj/20216170 · Cited by 3
“These complexes are ubiquitous in computational topology and topological data analysis [15].”
BackgroundA statistical approach to knot confinement via persistent homology
Celoria, D., & Mahler, B. I. (2022) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2021.0709 · Cited by 3
“see e.g. [1, 34] and the review [20]). We refer to the comprehensive survey [21] for further references and several related notions. Our main tool to address these matters is persistent homology (PH) [17, 4]; this is a relatively new technique in topological data analysis, commonly used to detect insightful topologica…”
BackgroundRasetti, M. (2021) · Contemporary systems thinking · doi:10.1007/978-3-030-71877-0_10 · Cited by 3
Individual Topology Structure of Eye Movement Trajectories
Onuchin, A. A., & Kachan, O. (2022) · Studies in computational intelligence · doi:10.1007/978-3-031-19032-2_5 · Cited by 3
“For rigorous treatment, we refer the reader to ([11, 14]) due to the lack of space.”
BackgroundInfluentialExtracting Persistent Clusters in Dynamic Data via Möbius Inversion
Kim, W., & Mémoli, F. (2023) · Discrete & Computational Geometry · doi:10.1007/s00454-023-00590-1 · Cited by 24
“This identification allows us to put formigrams into the context of the theories of persistence [17, 37], lattices [8], and categories [14, 58].”
MethodEmrani, S., & Krim, H. (2014) · IEEE Global Conference on Signal and Information Processing · doi:10.1109/globalsip.2014.7032359 · Cited by 3
“On the other hand if we use the proposed model for estimation of a non-periodic gene, we get a much higher error as RMSE for the one shown in Figure 1 is 0.4877.”
MethodInfluentialApproximation algorithms for Max Morse Matching
Rathod, A., Masood, T. B., & Natarajan, V. (2016) · Computational Geometry · doi:10.1016/j.comgeo.2016.10.002 · Cited by 3
“More recently, homology and persistent homology have been appraised to be a more widely applicable computational invariant of topological spaces, arising from practical data sets of interest [5].”
BackgroundNaveda, L., & Santana, I. (2014) · University of Michigan Library Repository · doi:10.13140/2.1.2879.6168 · Cited by 3
Delaunay simplices pruning based clustering
Razafindramanana, O., & Venturini, G. (2013) · Cited by 3
Driving behaviour analysis using topological features
Hossny, M., Mohammed, S., Nahavandi, S., Nelson, K. J., & Hossny, M. (2016) · Conference proceedings/Conference proceedings - IEEE International Conference on Systems, Man, and Cybernetics · doi:10.1109/smc.2016.7844736 · Cited by 3
“al [1], [2] uses the foundations of algebraic topology [3] to find robust models capable of presenting nonlinear behaviour within a”
BackgroundDecomposition theory of modules: the case of Kronecker algebra
Asashiba, H., Nakashima, K., & Yoshiwaki, M. (2017) · Japan Journal of Industrial and Applied Mathematics · doi:10.1007/s13160-017-0247-y · Cited by 5
“We refer the reader to [4,5,9] and [7] for details on persistent homology and/or topological data analysis.”
BackgroundThe representation theorem of persistence revisited and generalized
Corbet, R., & Kerber, M. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-018-0015-3 · Cited by 10
“The usefulness of this paradigm in the context of real-world data sets has led to the term topological data analysis; see the surveys (Carlsson 2009; Ghrist 2008; Edelsbrunner and Morozov 2012; Vejdemo-Johansson 2014; Kerber 2016) and textbooks (Edelsbrunner and Harer 2010; Oudot 2015) for various…”
MethodZhang, T., Broderick, S., & Rajan, K. (2018) · Springer eBooks · doi:10.1007/978-981-10-7617-6_7 · Cited by 3
On the Prognostic Efficiency of Topological Descriptors for Magnetograms of Active Regions
Knyazeva, I., Urtiev, F. A., & Makarenko, N. G. (2017) · Geomagnetism and Aeronomy · doi:10.1134/s0016793217080126 · Cited by 3
“(The ranks of homological groups are called such; they can be calculated by methods of computational topology (Ghrist, 2014; Carlsson, 2009).”
BackgroundInfluentialNonparametric Analysis of Non-Euclidean Data on Shapes and Images
Bhattacharya, R. N., & Oliver, R. (2018) · Sankhya A · doi:10.1007/s13171-018-0127-9 · Cited by 3
“One uses persistent homology (Carlsson, 2009) to identify these features.”
BackgroundBergomi, M. G., & Vertechi, P. (2020) · Theory and applications of categories · doi:10.70930/tac/rqcp3ycf · Cited by 6
“e Vietoris-Rips ltration [6] on (X, d) isG-invariant and therefore induces a (R,≤)-indexed diagram in FinSimp.”
BackgroundBlaser, N., & Brun, M. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-29726-8_17 · Cited by 3
Integration of the Kenzo System within SageMath for New Algebraic Topology Computations
Cuevas-Rozo, J., Divasón, J., Marco-Buzunáriz, M. Á., et al. (2021) · Mathematics · doi:10.3390/math9070722 · Cited by 3
“For example, we can construct the cartesian product of the Eilenberg–MacLane space K(Z/2Z, 2) and the loop space of the sphere S3 and compute its homology groups as follows (the Python function range(n) returns the sequence of numbers (0, 1, . . . , n− 1)): sage: from s a g e . interfaces.kenzo import EilenbergMacLane…”
BackgroundInfluentialA Sensorimotor Perspective on Contrastive Multiview Visual Representation Learning
Laflaquière, A. (2021) · IEEE Transactions on Cognitive and Developmental Systems · doi:10.1109/tcds.2021.3086267 · Cited by 3
“This research topic is particularly active in the field of Computer Vision, where high-dimensional input images are expected to contain limited and well-structured information (see, for instance, the manifold hypothesis [8]).”
BackgroundSentiment Analysis to Assess Educational Methodologies in a Competency-Based Educational System
Juárez-Ordóñez, S., Medina, R. B., & Herrera, L. M. (2021) · doi:10.1109/ieeeconf53024.2021.9733755 · Cited by 3
On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs
Shukla, S. (2023) · SIAM Journal on Discrete Mathematics · doi:10.1137/22m1481440 · Cited by 5
“In topological data analysis, it has been used to analyse data with persistent homology [10, 14, 41, 42].”
MethodMusical stylistic analysis: a study of intervallic transition graphs via persistent homology
Mijangos, M., Bravetti, A., & Padilla-Longoria, P. (2023) · Journal of Mathematics and Music · doi:10.1080/17459737.2023.2232811 · Cited by 4
“We make this idea precise as follows (Otter et al. 2017; Carlsson 2009).”
MethodComputing Persistent Homology by Spanning Trees and Critical Simplices
Shi, D., Chen, Z., Ma, C., et al. (2023) · Research · doi:10.34133/research.0230 · Cited by 3
“In the era of big data [1], higher-dimensional data and their analysis connect together the fields of data science [2,3], network science [4], and computational topology [5,6].”
BackgroundPersistent homology of collider observations: When (w)hole matters
Beuria, J. (2023) · Physics Letters B · doi:10.1016/j.physletb.2023.138188 · Cited by 3
Spherical coordinates from persistent cohomology
Schonsheck, N., & Schonsheck, S. C. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00141-w · Cited by 3
“Recently, algorithms that can preserve various topological properties have been of particular interest [6, 12, 14].”
Constructing Simplicial Complexes over Topological Spaces
Doktorova, M. N. (2012) · Cited by 3
Homotopical decompositions of simplicial and Vietoris Rips complexes
Chachólski, W., Jin, A., Scolamiero, M., & Tombari, F. (2021) · Journal of Applied and Computational Topology · doi:10.1007/s41468-021-00066-2 · Cited by 3
“lculable and geometrically informative. These two features are key reasons why invariants derived from homology are fundamental in Algebraic Topology in general and in Topological Data Analysis (TDA) [8] in particular. Calculability is a consequence of the fact that homology converts homotopy push-outs into Mayer-Viet…”
BackgroundDetecting Traffic Incidents Using Persistence Diagrams
Weber, E., Harding, S. N., & Przybylski, L. (2020) · Algorithms · doi:10.3390/a13090222 · Cited by 3
“These features are often summarized using Betti numbers or a persistence diagram [23].”
MethodTopological Methods for the Representation and Analysis of Exploration Data in Oil Industry
Artamonov, O. (2010) · Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology) · Cited by 3
Learning domain abstractions for long lived robots
Rosman, B. (2014) · ERA · Cited by 3
Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial ComplexesPreprint
Mathew, R., Newman, I., Rabinovich, Y. G., et al. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1507.04471 · Cited by 3
Brooks, M. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1604.07361 · Cited by 3
“The power of persistent homology [22] for topological data analysis [11] derives in part from its context in algebraic topology; however in this paper take a simpler approach for more limited results.”
BackgroundStatistical Methods in Topological Data Analysis for Complex, High-Dimensional DataPreprint
Medina, P., & Doerge, R. W. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1607.05150 · Cited by 3
“For each of these shape features a Betti number quantifies the distinct number of shape features that appear in a mathematical space [1].”
BackgroundNonparametric Inference via Bootstrapping the Debiased EstimatorPreprint
Cheng, G., & Chen, Y. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1702.07027 · Cited by 3
“The bootstrap consistency given in Theorem 4 shows that our method may be very useful in topological data analysis (Carlsson, 2009; Edelsbrunner and Morozov, 2012; Wasserman, 2018).”
MethodA quantitative structure comparison with persistent similarityPreprint
Xia, K. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.03572 · Cited by 3
Morse theory in topological data analysisPreprint
Adams, H., Atanasov, A., & Carlsson, G. (2011) · arXiv (Cornell University) · Cited by 4
Self-citationKusano, G. (2018) · arXiv (Cornell University) · Cited by 3
Persistent Homology and Euler Integral TransformsPreprint
Ghrist, R., Levanger, R., & Mai, H. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1804.04740 · Cited by 3
MOG: Mapper on Graphs for Relationship Preserving ClusteringPreprint
Hajij, M., Wang, B., & Rosen, P. (2018) · arXiv (Cornell University) · Cited by 3
Decoding of neural data using cohomological feature extractionPreprint
Rybakken, E., Baas, N. A., & Dunn, B. A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1711.07205 · Cited by 2
“ersistent homology is often used instead, but we will need cohomology for the decoding step. We refer to Ghrist, 2008; Edelsbrunner and Harer, 2010 for introductions to persistence, and Ghrist, 2014; Carlsson, 2009 for introductions to applied topology in general. Figure 5: Left: A ltration together with its rst persi…”
The Fiber of the Persistence Map for Functions on the IntervalPreprint
Curry, J. M. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1706.06059 · Cited by 3
Distributing Persistent Homology via Spectral SequencesPreprint
Torras-Casas, Á. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.05228 · Cited by 3
“uch as the study of geometric structure of sets of points lying in Rn, see [15,18]. This introduced the field of Topological Data Analysis which, very soon, was applied to a multitude of problems, see [6,19] for a survey article and an introduction. Among others, persistent homology has been applied to study coverage i…”
BackgroundFunctional limit theorems for the Euler characteristic process in the critical regimePreprint
Thomas, A. M., & Owada, T. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.00751 · Cited by 3
“isticofsublevel(orsuperlevel)setsofrandomfieldshavefound broad applications [1, 8]. In TDA, the technique of capturing the dynamic evolution of topology is generally studied in persistent homology—see [7] for a good introduction. Persistenthomologyoriginatedincomputationaltopology[10]andhasreceivedmuchattention as a us…”
MethodHarnessing the power of Topological Data Analysis to detect change points in time seriesPreprint
Islambekov, U., Yuvaraj, M., & Gel, Y. R. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.12939 · Cited by 3
Persistent Homology of Graph EmbeddingsPreprint
Solanki, V., Rubin‐Delanchy, P., & Gallagher, I. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.10238 · Cited by 3
“These observations may be inferred using the tools of topological data analysis, in particular those of persistent homology (for a general overview see [9]).”
MethodDetection and skeletonization of single neurons and tracer injections using topological methodsPreprint
Wang, D., Magee, L., Huo, B., Banerjee, S., Li, X., Jayakumar, J., Lin, M. K., Ram, K., Wang, S., Wang, Y., et al. (2020) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.03.21.000323 · Cited by 2
A Novel Method of Extracting Topological Features from Word EmbeddingsPreprint
Gholizadeh, S., Seyeditabari, A., & Zadrozny, W. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2003.13074 · Cited by 3
“Persistent homology [1, 2, 3, 4] is a technique in TDA for multi-scale analysis of data clouds.”
MethodVisualising the Evolution of English Covid-19 Cases with Topological Data Analysis Ball MapperPreprint
Dłotko, P., & Rudkin, S. (2020) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.2004.03282 · Cited by 5
“Where summary statistics speak of trends, and maps of cases help get a visual handle on the spatial scale, Topological Data Analysis (TDA) after Carlsson (2009) and particularly the Ball Mapper (BM) algorithm of D(cid:32)lotko (2019) can quickly highlight patterns within the characteristics of…”
BackgroundIntroductory Topological Data AnalysisPreprint
Sheffar, D. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2004.04108 · Cited by 3
“Notice a solid disc is homotopy equivalent to a point, and hence their Betti number sequences are identical, they share a homology group, and this is pretty suggestive of the value of this sort of analysis [3].”
BackgroundMethodOn the notion of weak isometry for finite metric spacesPreprint
Gregorio, A. D., Fugacci, U., Mémoli, F., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.03109 · Cited by 3
“One of the main purposes of Topological Data Analysis (TDA) is to gather topological and geometric information from these datasets [4], in order to be able to perform comparisons between the phenomena that generated the measurements.”
BackgroundLandscapes of data sets and functoriality of persistent homologyPreprint
Chachólski, W., Gregorio, A. D., Quercioli, N., & Tombari, F. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2002.05972 · Cited by 3
“Paraphrasing Gunnar Carlsson [3]: for obtaining qualitative information and reducing dependance on measures and distances, “we are forced to make functorial geometric constructions and analyze their behaviour on maps even to obtain information about single point clouds”.”
BackgroundIsometric Gaussian Process Latent Variable Model for Dissimilarity DataPreprint
Jørgensen, M., & Hauberg, S. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.11741 · Cited by 3
“We argue that it can be chosen in a geometrically founded way using Topological Data Analysis (Carlsson, 2009).”
MethodA unified framework for equivalences in social networksPreprint
Otter, N., & Porter, M. A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.10733 · Cited by 3
“3See, for instance, the discussion in [6], from which we quote the following excerpt: “Functoriality has proven itself to be a powerful tool in the development of various parts of mathematics, such as Galois theory within algebra, the theory of Fourier series within harmonic analysis, and the applicaton of algebraic t…”
BackgroundMethodTime-Resolved Topological Data Analysis of Market InstabilitiesPreprint
Katz, Y. A., & Biem, A. (2020) · SSRN Electronic Journal · doi:10.2139/ssrn.3581869 · Cited by 3
On Qualitative Shape Inferences: a journey from geometry to topologyPreprint
Zucker, S. W. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.08622 · Cited by 3
“The examples shown up till now were all done using techniques from computational topology [21, 11].”
MethodA Topological Framework for Deep LearningPreprint
Hajij, M., & Istvan, K. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.13697 · Cited by 5
Classification of COVID-19 via Homology of CT-SCANPreprint
Iqbal, S., Ahmed, H., Qaiser, T., Qureshi, M. I., & Rajpoot, N. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.10593 · Cited by 3
“…(de Silva & Ghrist, 2007; Lee et al. , 2012), computational biology (Kasson et al. , 2007; Yao et al. , 2009; Wang & Wei, 2016), data analysis (Carlsson, 2009; Liu et al. , 2012; Rieck et al. , 2012), image analysis (Carlsson et al. , 2009; Frosini & Landi, 2013; Bendich et al. , 2010),…”
The Better Turbulence Index? Forecasting Adverse Financial Markets Regimes with Persistent HomologyPreprint
Baitinger, E., & Flegel, S. (2019) · SSRN Electronic Journal · doi:10.2139/ssrn.3490567 · Cited by 3
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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