The DOI matches its Crossref record; 32 of the 32 references it cites were found in a registry or an index; 8 of the 28 sources its claims cite were checked for support.
Barcodes: The persistent topology of data
Open accessAbstract
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— persistent homology —and a novel representation of this algebraic characterization— barcodes . We sketch an application of these techniques to the classification of natural images.
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
17 Own interpretation · 13 Sources not found · 12 No basis given · 5 Supported · 3 Partly supported · 2 Not checked yet · 2 Own result · 1 No full text of the sources · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“R ↩→ C √ 2 ↩→ R √ 2 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.1. Persistence.
What the sources say
- Supportedcites [10] DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358. source
Theorem 2.5 (§2.3; page not provided) gives a chain of inclusions from a Rips complex through a Čech complex to a Rips complex at the larger parameter. Its required scale ratio is √(2d/(d+1)), which is at most √2, so the claimed √2-scaled chain follows for a positive parameter. (LLM, conf=0.90)
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
“Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
What the sources say
- Supportedcites [7] CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271. source
The Main Theorem (Section 3.1, p. 7) says persistence diagrams are stable under small-amplitude perturbations of a function. The Introduction (p. 2) says its assumptions are satisfied by Morse functions, and Applications (p. 16) identifies barcodes as persistence diagrams represented by intervals. (LLM, conf=0.92)
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.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“H * (C; F ) ∼ = i x t i • F [x] ⊕ ⎛ ⎝ j x r j • (F [x]/(x s j • F [x])) ⎞ ⎠ .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
What the sources say
- Supportedcites [22] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
Section 3.2 (p. 12) says that finite-type persistence modules over a field correspond to finitely generated graded modules over F[t] and describes their decomposition into free modules and quotients by powers of t. Section 2.1 (p. 6) identifies these as the free and torsional portions of the decomposition. (LLM, conf=0.91)
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.
- The authors state this as an assumption: Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“However, the Čech complex and various topologically equivalent subcomplexes (e.g., the alpha complex of [13]) are delicate objects to compute, relying on the precise distances between the nodes in E n .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
What the sources say
- Supportedcites [13] EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72. source
The citation identifies the alpha complex as an example of the subcomplexes being discussed. Section 3.3 defines alpha-complex membership using circumsphere radii, and Section 6 (Implementation) describes the need for exact arithmetic and symbolic perturbation to ensure correct computation. The paper need not independently establish the sentence’s broader claim about Čech complexes for this example citation to be supported. (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
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The full data set consists of roughly 8,000,000 points in E 9 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Partly supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. sourceExperimental — accuracy not yet measured
Section 2 (page unavailable) reports about 4.2 million optical patches and 0.79 million range patches—roughly 5 million total, not 8 million. Section 3 (page unavailable) represents each original 3 × 3 patch as a vector in nine-dimensional space, although the preprocessed data lie on a seven-dimensional sphere. (LLM, conf=0.90)
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 witness complex of Carlsson and de Silva [8,9,14] is one solution to this problem.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
What the sources say
- Partly supportedcites [8] D V. de Silva, “A weak definition of Delaunay triangulation”, preprint (2003). sourceExperimental — accuracy not yet measured
Section 6 (p. 8) describes using a subsample as the vertices of a topology-approximating simplicial complex and using weak witnesses to construct it. That supports the witness-complex alternative, but our copy does not discuss Rips complexes or say that their unmanageable size is the problem this construction solves. (LLM, conf=0.82)
- Not checked — no full text of the sourcecites [9] DC V. de Silva and G. Carlsson, “Topological estimation using witness complexes”, in SPBG’04 Symposium on Point-Based Graphics (2004), 157-166. source
- Not matchedcites [14] GO L. Guibas and S. Oudot, “Reconstruction using witness complexes”, in Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms (2007).
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: 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
“By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M on a topological seven-sphere S 7 ⊂ E 8 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. source
Section 3 (Preprocessing; page unavailable) says the authors subtract each patch’s mean, normalize its contrast, and transform the resulting points onto a seven-dimensional sphere in R⁸. The Introduction (page unavailable) says the study focuses on high-contrast data, including regions far from the origin. More precisely, preprocessing uses the logarithms of intensity values. (LLM, conf=0.92)
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.
- 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
“The Čech theorem (or, equivalently, the “nerve theorem”) states that C has the homotopy type of the union of closed radius /2 balls about the point set {x α }.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
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.
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 with β k , the barcode for H k does not give any information about the finer structure of the homology, but merely a continuously parameterized rank.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
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.
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 torsional elements correspond to those homology generators which appear at parameter r j and disappear at parameter r j + s j .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
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. 10
- It may state two separate claims as one. 16
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3. Example: natural images
What the sources say
- Partly supportedcites [20] HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366. sourceExperimental — accuracy not yet measured
The Images section describes a collection photographed with a digital camera, but gives its size as 4,212 images, not 4,167. It says the images came from woods, open landscapes and urban areas, often in consecutive series, rather than describing them as random outdoor scenes. The paper was published in 1998, but the passage does not specify when the photographs were taken. (LLM, conf=0.90)
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.
- Partly supported
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
“For example, in Figure 4, one sees (from a very coarse sampling) that the point cloud likely represents a connected object with one or two significant 'holes' as measured by H 1 and no significant higher homology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
hedgedSingle case- 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
“From the figure, it certainly appears as though an ideal choice of , if it exists, is rare: by the time is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
hedgeddirectSingle case- 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
“So coarse a reduction of natural images (into three-by-three squares of greyscale intensities) still leads to a point cloud of too high a dimension to visualize.Worse still, what structure there is is blurred and foggy: points appear at first to be distributed over the entire S 7 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.2. "Hover through the fog".
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
“The resulting homology H * (C) retains the structure of an R[x]-module, but, unlike the chain module, is not necessarily free.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
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
“Thus, the combinatorics of the 1-skeleton completely determines the complex, and the Rips complex can be stored as a graph and reconstituted instead of storing the entire boundary operator needed for a Čech complex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 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.
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
“The reason is that the Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
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: 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.
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 sufficiently small, the complex is a discrete set; for sufficiently large, the complex is a single highdimensional simplex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
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. 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
“Nor is it easily classified: the Artin-Rees theory from commutative algebra implies that the problem of classifying (finite-type) persistence modules such as H * (C) is equivalent to classifying finitely generated nonnegatively graded R[x]-modules.This is known to be very difficult in, say, Z[x].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
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.
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 general, it is neither a subcomplex of E n nor does it necessarily behave like an n-dimensional space at all (Figure 2).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
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.
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
“From a computational point of view, the Rips complex is less expensive that the corresponding Čech complex, even though the Rips complex has more simplices (in general).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
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.
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
“Despite being both computable and insightful, the homology of a complex associated to a point cloud at a particular is insufficient: it is a mistake to ask which value of is optimal.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2. Algebraic topology for 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.
In the paper
“A close examination of the data point corresponding to the primary circle reveals a pattern of 3-by-3 patches with one light region and one dark region separated by a linear transition.This nodal curve between light and dark is linear and appears in a circular family parameterized by the angle of the nodal line, as shown in Figure 5.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedSingle 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.
- 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
“This necessitates efficient sampling or reduction of the complex with accurate topology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plaindirect- 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.
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 persistent generators, upon close examination, do correspond to meaningful structures in the data, inspiring a sensible parametrization of the global structure of the data set.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
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.
In the paper
“A close examination of these three circles reveals that each intersects the highk primary twice, yet the two secondary circles are disjoint.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
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: 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.
- Not matched
In the paper
“[3] reports that a repetition of the experiment with a camera held at a constant angle π/4 yields a data set whose secondary persistent H 1 generators exhibit a bias towards true vertical and true horizontal: the axis of pixellation appears less relevant than the axis of gravity in natural image data.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
- Not matched
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 worth noting that for chain filtrations arising from realistic data sets, the Rips complexes are of an unmanageable size.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
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: 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
“The computation of persistent relative homology is more subtle, since the ensuing parameterized chain complex C is no longer free as an F [x]-module.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
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
“Following this analogy, the H 1 primary generator fills the role of a next term in the expansion of the homotopy type of the data set, collating the nodal curve between two contrasting patches.The secondary circles, interpolating between single and dual nodal curves, act as higher-order terms in the expansion, in which horizontal and vertical biases arise.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
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.
In the paper
“However, there are fundamental algebraic difficulties in constructing an analogous theory of persistence modules in this setting [24].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
What the sources say
- Not checked — no full text of the sourcecites [24] ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193. 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
“As noted in [3], each secondary circle regulates images with three contrasting regions and interpolates between these states and the primary circle.The difference between the two secondary circles lies in their bias for horizontal and vertical stratification respectively.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
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: 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.
- Not matched
The authors state this as an assumption.
Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
“One assumes a finite-type condition that each C i * is finitely generated as an R[x]-module and that the sequence stabilizes in i (in the case of an infinite sequence of chain complexes).”
Citations
Cites
Density Estimation for Statistics and Data Analysis
Silverman, B. W. (2018) · Routledge eBooks · doi:10.1201/9781315140919 · Cited by 15,938
Density Estimation for Statistics and Data Analysis
(1986) · Springer eBooks · doi:10.1007/978-1-4899-3324-9 · Cited by 8,569
Density Estimation for Statistics and Data Analysis
Dehnad, K., & Silverman, B. W. (1987) · Technometrics · doi:10.2307/1269475 · Cited by 5,479
(2001) · Cited by 5,374
Three-dimensional alpha shapes
Edelsbrunner, H., & Mücke, E. P. (1994) · ACM Transactions on Graphics · doi:10.1145/174462.156635 · Cited by 2,488
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Stability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280
Independent component filters of natural images compared with simple cells in primary visual cortex
van Hateren, J. H., & van der Schaaf, A. (1998) · Proceedings of the Royal Society of London. Series B: Biological Sciences · doi:10.1098/rspb.1998.0303 · Cited by 1,106 · cited 1×
“A collection of 4167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
BackgroundTopological persistence and simplification
Edelsbrunner, H., Letscher, D., & Zomorodian, A. J. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.2000.892133 · Cited by 716 · cited 2×
“Persistence, as introduced by Edelsbrunner, Letscher, and Zomo-rodian [12] and refined by Carlsson and Zomorodian [22], is a rigorous response to this problem.”
Method
Cited by
Sorted by citations
Items 201–300 of 1,274
Modeling and replicating statistical topology and evidence for CMB nonhomogeneity
Adler, R. J., Agami, S., & Pranav, P. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1706885114 · Cited by 34
“, much of TDA has become almost synonymous with an analysis based on some version of persistent homology [19,20,21], represented visually as barcodes, persistence diagrams, or related representations [13,14,25,26,48]. With relatively few exceptions, notably [17,23,41,6,10,42], TDA has not employed statistical methodol…”
BackgroundTopological Signature of 19th Century Novelists: Persistent Homology in Text Mining
Gholizadeh, S., Seyeditabari, A., & Zadrozny, W. (2018) · Big Data and Cognitive Computing · doi:10.3390/bdcc2040033 · Cited by 33
“Here the information structure based on thresholding distances is called Rips Filtration [9].”
MethodRucco, M., Viticchi, G., & Falsetti, L. (2020) · Mathematics · doi:10.3390/math8050770 · Cited by 33
“A topological n-dimensional hole is generated by the so-called homological generator [50,51].”
BackgroundBenzekry, S., Tuszyński, J. A., Rietman, E. A., et al. (2015) · Biology Direct · doi:10.1186/s13062-015-0058-5 · Cited by 33
“Persistent homology is a more involved topological measure of networks [3, 10, 15] an abstract n -simplex { v 0 , ⋯ , v n } of a set of n + 1 called vertices.”
BackgroundOn the vanishing of homology in random Čech complexes
Bobrowski, O., & Weinberger, S. (2016) · Random Structures and Algorithms · doi:10.1002/rsa.20697 · Cited by 31
“The rapidly developing field known as Topological Data Analysis (TDA) focuses on using topological signatures of data in machine learning and statistics (for some introduction see [14,23,44]).”
BackgroundDiscrete curvature on graphs from the effective resistance*
Devriendt, K., & Lambiotte, R. (2022) · Journal of Physics Complexity · doi:10.1088/2632-072x/ac730d · Cited by 32
Topological feature vectors for chatter detection in turning processes
Yesilli, M. C., Khasawneh, F. A., & Otto, A. (2022) · The International Journal of Advanced Manufacturing Technology · doi:10.1007/s00170-021-08242-5 · Cited by 31
“Another promising tool for generating feature vectors for chatter detection comes from a new field with many mature computational tools: Topological Data Analysis (TDA) [11, 12, 13, 14].”
BackgroundMethodInvariants for homology classes with application to optimal search and planning problem in robotics
Bhattacharya, S., Lipsky, D., Ghrist, R., & Kumar, V. (2013) · Annals of Mathematics and Artificial Intelligence · doi:10.1007/s10472-013-9357-7 · Cited by 30
“Recent applications of algebraic topology to the applied sciences has primarily focused on data analysis [6, 16, 31].”
BackgroundSelf-citationA Bayesian Framework for Persistent Homology
Maroulas, V., Nasrin, F., & Oballe, C. (2020) · SIAM Journal on Mathematics of Data Science · doi:10.1137/19m1268719 · Cited by 32
“Overall, topological data analysis (TDA) encompasses a broad set of techniques that explore topological structure in datasets [12,17,22,59].”
BackgroundStatistical Inference Over Persistent Homology Predicts Fluid Flow in Porous Media
Moon, C., Mitchell, S. A., Heath, J. E., & Andrew, M. (2019) · Water Resources Research · doi:10.1029/2019wr025171 · Cited by 31
Dynamics of CLIMP-63 S-acylation control ER morphology
Sandoz, P. A., Denhardt-Eriksson, R. A., Abrami, L., Abriata, L. A., Spreemann, G., Maclachlan, C., Ho, S., Kunz, B., Hess, K. P., Knott, G. W., et al. (2023) · Nature Communications · doi:10.1038/s41467-023-35921-6 · Cited by 31
EULER INTEGRATION OF GAUSSIAN RANDOM FIELDS AND PERSISTENT HOMOLOGY
Bobrowski, O., & Borman, M. S. (2012) · Journal of Topology and Analysis · doi:10.1142/s1793525312500057 · Cited by 31
“For more details and further references see [6, 7, 8].”
BackgroundZhang, Z., Yang, S., Cui, H., Wu, J. J., Schwartz, F., & Qi, H. (2016) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2016.2634531 · Cited by 31
“The simplicial complex constructed by thismanner is calledRips complex [34].”
BackgroundLi, B., Ofori-Boateng, D., Gel, Y. R., et al. (2020) · Sustainable and Resilient Infrastructure · doi:10.1080/23789689.2019.1708182 · Cited by 31
“In addition to the sequence of Betti numbers, the lifespan of topological features can be represented by persistence diagrams and barcodes (Carlsson, 2009; Ghrist, 2008; Kerber et al., 2016; Zomorodian, 2010).”
BackgroundEuler calculus with applications to signals and sensing
Curry, J. M., Ghrist, R., & Robinson, M. (2012) · Proceedings of symposia in applied mathematics · doi:10.1090/psapm/070/589 · Cited by 31
Self-citationA Short Survey of Topological Data Analysis in Time Series and Systems AnalysisPreprint
Gholizadeh, S., & Zadrozny, W. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1809.10745 · Cited by 48
“In TDA, this procedure is often automated by a technique called persistent homology [4, 5, 6, 7].”
BackgroundMethodLeite, L. S. G., Banerjee, S., Wei, Y., et al. (2024) · Wiley Interdisciplinary Reviews Computational Molecular Science · doi:10.1002/wcms.1729 · Cited by 30
A topological similarity measure for proteins
Máté, G., Hofmann, A., Wenzel, N., & Heermann, D. W. (2013) · Biochimica et Biophysica Acta (BBA) - Biomembranes · doi:10.1016/j.bbamem.2013.08.019 · Cited by 29
Information Correlated Lévy Walk Exploration and Distributed Mapping Using a Swarm of Robots
Ramachandran, R. K., Kakish, Z., & Berman, S. (2020) · IEEE Transactions on Robotics · doi:10.1109/tro.2020.2991612 · Cited by 31
“The dimension of Ht(T), defined as the Betti number βt [50], is a topological invariant that represents the number of independent topological features encoded by Ht(T).”
BackgroundMethodFinding hidden order in spin models with persistent homology
Olsthoorn, B., Hellsvik, J., & Balatsky, A. V. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.043308 · Cited by 39
“Persistent homology (PH) is a method that identifies the qualitative features of a finite metric space (also called a point-cloud dataset) – for an introduction see [31, 32].”
BackgroundMethodPersistent homology analysis of ion aggregations and hydrogen-bonding networks
Xia, K. (2018) · Physical Chemistry Chemical Physics · doi:10.1039/c8cp01552j · Cited by 33
A Geometric Perspective on Sparse FiltrationsPreprint
Cavanna, N. J., Jahanseir, M., & Sheehy, D. R. (2015) · arXiv (Cornell University) · doi:10.4230/lipics.socg.2026.5 · Cited by 42
Crawford, L. A., Monod, A., Chen, A., et al. (2019) · Figshare · doi:10.6084/m9.figshare.10269596.v1 · Cited by 30
Wong, E., Palande, S., Wang, B., et al. (2016) · IEEE International Symposium on Biomedical Imaging (ISBI) · doi:10.1109/isbi.2016.7493506 · Cited by 29
Rajput, A. A., & Mostafavi, A. (2023) · Scientific Reports · doi:10.1038/s41598-023-37965-6 · Cited by 29
“In contrast to barcode approach [26], where there is a single birth and death event, motifs can show multiple birth-death patterns.”
BackgroundProbabilistic convergence and stability of random mapper graphs
Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00063-x · Cited by 27
“Popular topological descriptors can be classified into vector-based ones such as persistence diagrams (Edelsbrunner et al. 2002) and barcodes (Ghrist 2008; Carlsson et al. 2004), graph-based ones such as Reeb graphs (Reeb 1946) and their variants merge trees (Beketayev et al. 2014) and contour…”
BackgroundPersistent Clustering and a Theorem of J. Kleinberg
Carlsson, G., & Memoli, F. (2008) · arXiv · doi:10.48550/arxiv.0808.2241 · Cited by 28
“However, there turns out to be (see below) a readily computable analogue of the persistence barcode, [ Ghr08 , ZC04].”
BackgroundNeural manifolds in V1 change with top-down signals from V4 targeting the foveal region
Morales-Gregorio, A., Kurth, A. C., Ito, J., Kleinjohann, A., Barthélemy, F. V., Brochier, T., Grün, S., & van Albada, S. J. (2024) · Cell Reports · doi:10.1016/j.celrep.2024.114371 · Cited by 28
Enhancing financial time series forecasting through topological data analysis
de Jesus, L. C., Fernández‐Navarro, F., & Carbonero-Ruz, M. (2025) · Neural Computing and Applications · doi:10.1007/s00521-024-10787-x · Cited by 28
“At its core, TDA employs persistent homology to unveil the underlying structure of data by identifying and tracking features such as connected components, loops, and voids across different scales [2].”
A Review of Topological Data Analysis and Topological Deep Learning in Molecular Sciences
Wee, J., & Jiang, J. W. (2025) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.5c02266 · Cited by 36
“Topological invariants are often represented as persistence barcodes [64], persistence images [2, 156], persistence landscapes [13, 82], and persistence surfaces [2].”
Multiresolution Topological Simplification
Xia, K., Zhao, Z., & Wei, G. (2015) · Journal of Computational Biology · doi:10.1089/cmb.2015.0104 · Cited by 28
“, Betti numbers, of the underlying copies can be tracked by using either persistent diagrams or barcode representation.(13) Appropriate mathematical apparatus has been deviced to organize simplicial complexes generated via the filtration process into homology groups.”
BackgroundTracing patterns and shapes in remittance and migration networks via persistent homology
Ignacio, P. S., & Darcy, I. K. (2019) · EPJ Data Science · doi:10.1140/epjds/s13688-018-0179-z · Cited by 28
Safarbali, B., & Golpayegani, S. M. R. H. (2019) · Biomedical Signal Processing and Control · doi:10.1016/j.bspc.2019.101563 · Cited by 30
Chen, C., & Freedman, D. (2008) · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · doi:10.4230/lipics.stacs.2008.1343 · Cited by 25
“The problem of computing the topological features of a space has recently drawn much attention from researchers in various fields, such as high-dimensional data analysis [3, 15], graphics [13, 5], networks [10] and computational biology [1, 8].”
BackgroundPersistent homology of unweighted complex networks via discrete Morse theory
Kannan, H., Saucan, E., Roy, I., & Samal, A. (2019) · MPG.PuRe (Max Planck Society) · Cited by 27
Topological Data Analysis in smart manufacturing: State of the art and future directions
Uray, M., Giunti, B., Kerber, M., et al. (2024) · Journal of Manufacturing Systems · doi:10.1016/j.jmsy.2024.07.006 · Cited by 27
Hyun, Y., & Kim, D. (2023) · Computational and Structural Biotechnology Journal · doi:10.1016/j.csbj.2023.01.006 · Cited by 30
“By employing both persistence-based clustering [25] and persistent homology [26], Pike et al. developed the topological mode analysis tool (ToMATo) to quantify complex biological nanostructure [24].”
MethodOffset hypersurfaces and persistent homology of algebraic varieties
Horobeţ, E., & Weinstein, M. (2019) · Computer Aided Geometric Design · doi:10.1016/j.cagd.2019.101767 · Cited by 29
“where ti, rj, and sj are values of the persistence parameter [12].”
BackgroundMethodHausmann, M., Falk, M., Neitzel, C., Hofmann, A., Biswas, A., Gier, T., Falková, I., Heermann, D. W., & Hildenbrand, G. L. (2021) · International Journal of Molecular Sciences · doi:10.3390/ijms22073636 · Cited by 25
“The results are presented as “barcodes” [95] to track the formation and disappearance of components and holes with increasing α (Figure 6C).”
BackgroundPersistent Topological Laplacians—A Survey
Wei, X., & Wei, G. (2025) · Mathematics · doi:10.3390/math13020208 · Cited by 28
“The output of persistent homology consists of arrays of topological invariants computed on various scales, often visualized or represented by persistence diagrams, persistence barcodes [3], persistence images [4,5], or persistence landscapes [6].”
Introduction to correlation networks: Interdisciplinary approaches beyond thresholding
Masuda, N., Boyd, Z. M., Garlaschelli, D., & Mucha, P. J. (2025) · Physics Reports · doi:10.1016/j.physrep.2025.06.002 · Cited by 40
Spatiotemporal Persistent Homology for Dynamic Metric Spaces
Kim, W., & Mémoli, F. (2020) · Discrete & Computational Geometry · doi:10.1007/s00454-019-00168-w · Cited by 40
“In [64], Halverson, Topaz and Ziegelmeier study aggregation models for biological systems by adopting ideas from TDA.”
BackgroundInfluentialRobustness and parameter geography in post-translational modification systems
Nam, K.-M., Gyori, B. M., Amethyst, S. V., Bates, D. J., & Gunawardena, J. (2020) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1007573 · Cited by 27
Topological learning for brain networks
Songdechakraiwut, T., & Chung, M. K. (2023) · The Annals of Applied Statistics · doi:10.1214/22-aoas1633 · Cited by 32
“One representation to capture the persistence of such topological features is in the form of barcodes (Ghrist, 2008; Adler et al., 2010; Lee et al., 2011).”
BackgroundIntrinsic Interleaving Distance for Merge Trees
Gasparovic, E., Munch, E., Oudot, S., et al. (2024) · La Matematica · doi:10.1007/s44007-024-00143-9 · Cited by 26
“There are several types of summaries constructed from topological methods, including vector-based summaries such as persistence diagrams [29] and barcodes [33], as well as graph-based summaries such as merge trees, contour trees [13], and Reeb graphs [44].”
Graph induced complex on point data
Dey, T. K., Fan, F., & Wang, Y. (2013) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/2462356.2462387 · Cited by 26
“Therefore, a considerable effort has ensued to extract topological information from point data in recent years [7, 9, 14, 18].”
A Fractal Dimension for Measures via Persistent Homology
Adams, H., Aminian, M., Farnell, E., Kirby, M. J. L., Mirth, J., Neville, R., Peterson, C., & Shonkwiler, C. (2020) · Abel symposia · doi:10.1007/978-3-030-43408-3_1 · Cited by 24
“e scale increases. We provide a brief introduction to persistent homology in this preliminaries section, but we point the interested reader to [8,28,39] for thorough introductions to homology, and to [17,23,36] for excellent expository articles on persistent homology. Geometric complexes, which are at the heart of the…”
BackgroundStrong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime
Goel, A., Trinh, K. D., & Tsunoda, K. (2018) · Journal of Statistical Physics · doi:10.1007/s10955-018-2201-z · Cited by 26
Topological exploration of artificial neuronal network dynamics
Bardin, J.-B., Spreemann, G., & Hess, K. P. (2019) · Network Neuroscience · doi:10.1162/netn_a_00080 · Cited by 30
Vandaele, R., Mukherjee, P., Selby, H. M., Shah, R. P., & Gevaert, O. (2022) · Patterns · doi:10.1016/j.patter.2022.100657 · Cited by 26
“Complementary to this, the rising field of topological data analysis (TDA) [4], and in particular its main method, persistent homology [5], provides an unparalleled tool to quantify local to global structural information in data.”
BackgroundGraph induced complex on point data
Dey, T. K., Fan, F., & Wang, Y. (2013) · doi:10.1145/2493132.2462387 · Cited by 25
Topology of dynamical reconstructions from Lagrangian data
Charó, G. D., Artana, G. O., & Sciamarella, D. (2020) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2020.132371 · Cited by 25
“the subject is attracting once again the attention of dynamicists who considered applying general pattern recognition concepts [3, 4, 5] to the spatio-temporal patterns emerging in nonlinear systems involving diffusion and convection mechanisms [6, 7, 8, 9, 10, 11].”
BackgroundTopological View of Flows Inside the BOLD Spontaneous Activity of the Human Brain
Don, A. P. H., Peters, J. F., Ramanna, S., & Tozzi, A. (2020) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2020.00034 · Cited by 25
“Each β1 in a persistent barcode (Ghrist, 2008) represents the number of connected barycentric cycles covering an activation area of the brain.”
BackgroundUsing persistent homology as preprocessing of early warning signals for critical transition in flood
Musa, S. M. S. S., Noorani, M. S. M., Razak, F. A., et al. (2021) · Scientific Reports · doi:10.1038/s41598-021-86739-5 · Cited by 25
Holzinger, A. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-09891-3_46 · Cited by 25
“On the other hand, computers can be used to deal with highdimensional data, where we can make use of the benefits of computational topology [24], e.”
BackgroundEsparza, J., Quintanilla, J. P., Cid, E., Medeiros, A. C., Gallego, J. Á., & de la Prida, L. M. (2025) · Neuron · doi:10.1016/j.neuron.2025.01.022 · Cited by 28
Why Topology for Machine Learning and Knowledge Extraction?
Ferri, M. (2018) · Machine Learning and Knowledge Extraction · doi:10.3390/make1010006 · Cited by 23
“, in the set {x ∈ X | f (x) ≤ u}) survive in the sublevel set under v [22,23].”
BackgroundFeasibility of topological data analysis for event-related fMRI
Ellis, C. T., Lesnick, M., Henselman‐Petrusek, G., Keller, B., & Cohen, J. D. (2019) · Network Neuroscience · doi:10.1162/netn_a_00095 · Cited by 23
“A full description of persistent homology is outside of the scope of the present work; interested readers are pointed to standard introductions to the topic (Carlsson, 2009; Ghrist, 2008).”
BackgroundHahn, H., Neitzel, C., Kopečná, O., Heermann, D. W., Falk, M., & Hausmann, M. (2021) · Cancers · doi:10.3390/cancers13215561 · Cited by 23
“With persistent homology [22,23], it is possible to simultaneously”
BackgroundQuantitative analysis of phase transitions in two-dimensional XY models using persistent homology
Giansiracusa, J., & Biagio Lucini, A. (2022) · Cronfa (Swansea University) · Cited by 23
Aspects of topological approaches for data science
Grbić, J., Wu, J., Xia, K., & Wei, G. (2022) · Foundations of Data Science · doi:10.3934/fods.2022002 · Cited by 24
“1] derived from the classical Gabriel Theorem in representation theory [66].”
MethodExploring persistent local homology in topological data analysis
Fasy, B. T., & Wang, B. (2016) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2016.7472915 · Cited by 21
“For more details, see [3, 4] for a gentle introduction to TDA and [5] for algorithmic foundations.”
BackgroundInverse Problems in Topological Persistence
Oudot, S., & Solomon, E. (2020) · Abel symposia · doi:10.1007/978-3-030-43408-3_16 · Cited by 24
“bra (chapter 2 of Atiyah and McDonald’s book [AM69] is an excellent reference). For an introduction to the themes and tools of topological data analysis, the reader can consult the articles of Ghrist [Ghr08] and Carlsson [Car09]. Lastly, a more formal and comprehensive treatment can be found in the texts by Edelsbrunn…”
MethodQuantitative analysis of phase transitions in two-dimensional XY models using persistent homology
Sale, N., Giansiracusa, J., & Biagio Lucini, A. (2022) · Physical review. E · doi:10.1103/physreve.105.024121 · Cited by 36
Comparing feature sets and machine-learning models for prediction of solar flares
Deshmukh, V., Baskar, S., Berger, T. E., Bradley, E., & Meiss, J. D. (2023) · Astronomy and Astrophysics · doi:10.1051/0004-6361/202245742 · Cited by 23
Persistent cohomology and circular coordinates
de Silva, V., & Vejdemo-Johansson, M. (2009) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1542362.1542406 · Cited by 22
Leveraging the mathematics of shape for solar magnetic eruption prediction
Deshmukh, V., Berger, T. E., Bradley, E., & Meiss, J. D. (2020) · Journal of Space Weather and Space Climate · doi:10.1051/swsc/2020014 · Cited by 23
“TDA has been used in many applications, including coverage of sensor networks (de Silva and Ghrist, 2007), structures in natural images (Ghrist, 2008), neural spike train data (Singh et al., 2008), and even the largescale structure of the universe (Xu et al., 2019).”
BackgroundMethodDynamic Topological Data Analysis for Functional Brain Signals
Songdechakraiwut, T., & Chung, M. K. (2020) · doi:10.1109/isbiworkshops50223.2020.9153431 · Cited by 24
“For each sliding window of length w at time t , we have a point cloud consisting of w points we then construct a Vietoris-Rips complex K (cid:15) whose k -simplices correspond to unordered ( k +1) -tuples of points, which are pairwise within distance (cid:15) [3].”
MethodLi, L., Thompson, C., Henselman‐Petrusek, G., Giusti, C., & Ziegelmeier, L. (2021) · Frontiers in Artificial Intelligence · doi:10.3389/frai.2021.681117 · Cited by 22
“Barcodes are among the foremost tools in topological data analysis [29, 22], and they contain a great deal of information about a filtration.”
BackgroundYan, Y., Liu, Y.-S., Li, C., Wang, J., Ma, L., Xiong, J., Zhao, X., & Wang, L. (2022) · IEEE Sensors Journal · doi:10.1109/jsen.2022.3142750 · Cited by 23
Cumulative areawise testing in wavelet analysis and its application to geophysical time series
Schulte, J. A. (2016) · Nonlinear processes in geophysics · doi:10.5194/npg-23-45-2016 · Cited by 22
Supervised Approaches for Protein Function Prediction by Topological Data Analysis
Martino, A., Rizzi, A., & Mascioli, F. M. F. (2018) · doi:10.1109/ijcnn.2018.8489307 · Cited by 22
“2For the sake of ease and in order to suit also novice readers, we omit any heavy mathematical notations, referring the interested reader to [22]–[24] and other references therein for more rigorous definitions.”
BackgroundPersistent homology of unweighted complex networks via discrete Morse theory
Kannan, H., Saucan, E., Roy, I., & Samal, A. (2019) · Scientific Reports · doi:10.1038/s41598-019-50202-3 · Cited by 22
“A Hp-barcode diagram corresponding to a filtration of the clique complex K is a graphical representation containing horizontal line segments, each of which represents a p-hole in K, plotted against the x-axis ranging from 0 to 1 which corresponds to the normalized filtration weights of simplices in K [51].”
MethodRevealing neural correlates of behavior without behavioral measurementsPreprint
Rubin, A., Sheintuch, L., Brande-Eilat, N., Pinchasof, O., Rechavi, Y., Geva, N., & Ziv, Y. (2019) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/540195 · Cited by 22
Through synapses to spatial memory maps via a topological model
Dabaghian, Y. A. (2019) · Scientific Reports · doi:10.1038/s41598-018-36807-0 · Cited by 19
Honeyboost: Boosting honeypot performance with data fusion and anomaly detection
Kandanaarachchi, S., Ochiai, H., & Rao, A. (2022) · Expert Systems with Applications · doi:10.1016/j.eswa.2022.117073 · Cited by 20
“1) Using persistent homology [8], a methodology in topological data analysis, choose a bandwidth appropriate for anomaly detection.”
MethodPersistent homology of tumor CT scans is associated with survival in lung cancer
Somasundaram, E., Litzler, A., Wadhwa, R., et al. (2021) · Medical Physics · doi:10.1002/mp.15255 · Cited by 21
Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes
Owada, T. (2018) · The Annals of Applied Probability · doi:10.1214/17-aap1375 · Cited by 18
“Readers interested in a more rigorous description of persistent homology may refer to [23], [49] and [22], while [1] and [27] provide an elegant review of the topics in an accessible way for non-topologists.”
BackgroundDubuc, B., Ebrahimkhanlou, A., & Salamone, S. (2019) · Smart Materials and Structures · doi:10.1088/1361-665x/ab0e96 · Cited by 21
Holzinger, A. (2014) · Lecture notes in computer science · doi:10.1007/978-3-662-43968-5_19 · Cited by 21
“Ghrist (2007) [31] starts in the beginning of his paper with summarizing three benefits of topology:”
BackgroundCapturing dynamics of time-varying data via topology
Xian, L., Adams, H., Topaz, C. M., & Ziegelmeier, L. (2021) · Foundations of Data Science · doi:10.3934/fods.2021033 · Cited by 34
“[19], development economics [3], urban planning [26], political science [27], natural language processing [55], and much more. Classic works that build and review the fundamental ideas of TDA include [56,31,28,23,8,44]. While TDA was originally developed with the study of static data in mind, in recent years, it has f…”
BackgroundInfluentialOn Topological Data Analysis for Structural Dynamics: An Introduction to Persistent Homology
Gowdridge, T., Dervilis, N., & Worden, K. (2022) · ASME Open Journal of Engineering · doi:10.1115/1.4055184 · Cited by 20
“For more information regarding the mathematics of group theory, and algebraic topology, the authors would advise referring to [5–8], and [1, 2, 9–15] for more on TDA.”
BackgroundA topology-based object representation for clasping, latching and hooking
Stork, J. A., Pokorny, F. T., & Kragić, D. (2013) · IEEE-RAS International Conference on Humanoid Robots (Humanoids) · doi:10.1109/humanoids.2013.7029968 · Cited by 20
“using a persistent homology analysis of P in the form of barcodes [39].”
MethodA Topological Model of the Hippocampal Cell Assembly Network
Babichev, A., Ji, D., Mémoli, F., & Dabaghian, Y. A. (2016) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2016.00050 · Cited by 21
A model of topological mapping of space in bat hippocampus
Hoffman, K., Babichev, A., & Dabaghian, Y. A. (2016) · Hippocampus · doi:10.1002/hipo.22610 · Cited by 15
“The number of inequivalent topological loops of a dimensionality n is known as the nth Betti number, bn, and the list of all Betti numbers, (b0, b1, b2, . . .), provides a convenient “barcode” of the space’s topological structure (Ghrist, 2008).”
BackgroundUnsupervised space–time clustering using persistent homology
Islambekov, U., & Gel, Y. R. (2018) · Environmetrics · doi:10.1002/env.2539 · Cited by 19
“ng out TDA related computations in C++, Python and R. Perseus is another noteworthy persistent homology software [20]. For more details on TDA, the reader may consult survey papers such as [21], [7], [22], [23]. 3 Clustering using Betti Numbers (CBN) Algorithm Let X be a point cloud consisting of npoints belonging to…”
MethodRobust spatial memory maps encoded by networks with transient connections
Babichev, A., Morozov, D., & Dabaghian, Y. A. (2018) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1006433 · Cited by 16
“The topological structure of a steadily growing coactivity complex can be described using persistent homology theory methods [30, 31].”
BackgroundPersistent homology analysis of brain transcriptome data in autism
Shnier, D., Voineagu, M., & Voineagu, I. (2019) · Journal of The Royal Society Interface · doi:10.1098/rsif.2019.0531 · Cited by 17
“The choice of Z/2 coefficients for Hk groups is frequently used [15] as it gives a simplified version of persistent homology.”
BackgroundA higher-dimensional homologically persistent skeleton
Kališnik, S., Kurlin, V. A., & Lešnik, D. (2018) · Advances in Applied Mathematics · doi:10.1016/j.aam.2018.07.004 · Cited by 18
“For excellent introductions to persistent homology, see [16,3,4,13].”
BackgroundČech closure spaces: A unified framework for discrete and continuous homotopy
Rieser, A. (2021) · Topology and its Applications · doi:10.1016/j.topol.2021.107613 · Cited by 20
Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks
Xia, K., Anand, D. V., Saxena, S., & Mu, Y. (2019) · Physical Chemistry Chemical Physics · doi:10.1039/c9cp03009c · Cited by 17
A spectral sequence for parallelized persistence
Lipsky, D., Skraba, P., & Vejdemo-Johansson, M. (2011) · arXiv · doi:10.48550/arxiv.1112.1245 · Cited by 22
“Topological data analysis [3, 15] looks to quantify the qualitative aspects of the geometry of a data set.”
BackgroundA study on validating non-linear dimensionality reduction using persistent homology
Paul, R., & Chalup, S. K. (2017) · Pattern Recognition Letters · doi:10.1016/j.patrec.2017.09.032 · Cited by 19
Deng, R., & Duzhin, F. (2022) · Big Data and Cognitive Computing · doi:10.3390/bdcc6030074 · Cited by 25
“Topological data analysis can provide qualitative and quantitative insights of the data [2] using low dimensional representations [3].”
BackgroundDuan, S., Wu, X., Wang, J., Zou, Y., Jiang, L., & Wei, Y. (2023) · Journal of Nondestructive Evaluation · doi:10.1007/s10921-023-00940-2 · Cited by 19
“That is, the topological features – persistences—of the 2-dimensional phase space are extracted by persistent homology and displayed in the 0-th and 1-th persistent barcodes (PB) [33].”
BackgroundPersistent homology of directed networks
Chowdhury, S., & Mémoli, F. (2016) · Asilomar Conference on Signals, Systems and Computers (ACSSC) · doi:10.1109/acssc.2016.7868997 · Cited by 19
“For the sake of brevity, we simply assert that PH has been successfully applied to numerous problems in the sciences, and point the reader to [1]–[3] for accounts of these applications written by the pioneers of the theory.”
MethodModelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes
Corcoran, P., & Jones, C. B. (2017) · IEEE Access · doi:10.1109/access.2017.2749319 · Cited by 22
“Topological features, such as pathconnected components or holes, which persist over longer periods of time are usually considered of greater significance than those which persist over shorter periods [9].”
BackgroundHuckemann, S., Kim, K., Munk, A., et al. (2016) · MPG.PuRe (Max Planck Society) · Cited by 19
Cycle Registration in Persistent Homology With Applications in Topological Bootstrap
Reani, Y., & Bobrowski, O. (2022) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2022.3217443 · Cited by 25
“TOPOLOGICAL Data Analysis (TDA) refers to a set of methods promoting the use of mathematical topology in analyzing data and networks [1], [2], [3].”
MethodTopological Learning and Its Application to Multimodal Brain Network Integration
Songdechakraiwut, T., Shen, L., & Chung, M. K. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-87196-3_16 · Cited by 20
“Long lifetimes indicate global topological features while short lifetimes indicate small-scale topological features [11,20,29].”
Background
Metadata
- Authors
- Ghrist, R.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Ghrist, Robert
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2007-10-26
- Volume
- 45
- Issue
- 1
- Pages
- 61-75
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2071391326
- Semantic Scholar
- cad0bff789bb0e626245b21a47791ef76a9cd5c2
- Cited by
- 1,236
- References
- 32
- Citations by year
- 2026: 78 · 2025: 98 · 2024: 81 · 2023: 105 · 2022: 93 · 2021: 111 · 2020: 121 · 2019: 121 · 2018: 86 · 2017: 66 · 2016: 76 · 2015: 46 · 2014: 41 · 2013: 37 · 2012: 18
- FWCI
- 16.70
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- ScholarlyCommons (University of Pennsylvania) (submittedVersion)
- Scholarly Commons (University of Pennsylvania) (submittedVersion)
- Funders
- Defense Advanced Research Projects Agency (HR0011-07-1-0002)
- University of Illinois at Urbana-Champaign
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistent homology, barcodes, computational algebraic topology, feature detection, shape recognition, high-dimensional data, point clouds, natural image classification
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-02, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-02 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
