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.
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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.
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- 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)
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- 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.
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- 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.
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- 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.
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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 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.
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- 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.
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- 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.
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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
“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.
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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 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.
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- 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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. 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.
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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
“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.
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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.
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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
“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.
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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.
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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
“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.
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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.
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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.
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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.
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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
“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.
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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
“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.
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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
“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.
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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.
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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.
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- 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 701–800 of 1,274
Lau, A., & Yin, C. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1007.1880 · Cited by 4
Edge Contractions and Simplicial HomologyPreprint
Dey, T. K., Hirani, A. N., Krishnamoorthy, B., et al. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1304.0664 · Cited by 4
“For example, the Vietoris-Rips complex has been recognized as a versatile data structure for inferring homological attributes from point cloud data [12].”
BackgroundSketches of a platypus: persistent homology and its algebraic foundationsPreprint
Vejdemo‐Johansson, M. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1212.5398 · Cited by 4
“Persistent homology is a technique that has sparked the birth of a new field of research; while several introductory texts have been written [7,36,48,65], and several good survey articles have been published [16, 37], most if not all target computer scientists or data scientists with an interest in topology.”
On the Vanishing of Homology in Random \v{C}ech ComplexesPreprint
Bobrowski, O., & Weinberger, S. (2015) · arXiv (Cornell University) · Cited by 4
Computing the persistent homology of range images with alpha shapes
Martynov, I. (2008) · LUTPub (LUT University) · Cited by 4
Limit Theorems for the Sum of Persistence BarcodesPreprint
Owada, T. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1604.04058 · Cited by 3
“rigorous description of persistent homology may refer to [19], [40] and [18], while [1] and [21] provide an elegant review of the topics in an accessible way for non-topologists.”
BackgroundPersistent Homology of Asymmetric Networks: An Approach based on Dowker FiltrationsPreprint
Chowdhury, S., & Mémoli, F. (2016) · arXiv (Cornell University) · Cited by 7
Multiscale persistent functions for biomolecular structure characterizationPreprint
Xia, K., Li, Z., & Mu, L. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1612.08311 · Cited by 4
Topological Distances between Networks and Its Application to Brain ImagingPreprint
Lee, H., Ma, Z., Wang, Y., & Chung, M. K. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1701.04171 · Cited by 10
“The binary network B (X ) is a simplicial complex consisting of 0-simplices (nodes) and 1-simplices (edges) [28,30].”
BackgroundTesting to distinguish measures on metric spacesPreprint
Blumberg, A. J., Bhaumik, P., & Walker, S. G. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.01152 · Cited by 4
“, see [17, 8]) forms a metric space which is not Euclidean.”
BackgroundCellular Sheaves And Cosheaves For Distributed Topological Data Analysis
Yoon, I. H. R. (2018) · ScholarlyCommons (University of Pennsylvania) · Cited by 5
Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimesPreprint
Owada, T., & Thomas, A. M. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1809.05758 · Cited by 4
“The first [13] is an essential and succinct overview.”
BackgroundBayesian Inference for Persistent HomologyPreprint
Maroulas, V., Nasrin, F., & Oballe, C. (2019) · arXiv (Cornell University) · Cited by 4
Topological fingerprints reveal protein-ligand binding mechanismPreprint
Cang, Z., & Wei, G. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1703.10982 · Cited by 4
“It reveals the topological connectivity in biomolecular complexes in terms of topological fingerprints (TFs), 14–17 which are recorded as the barcodes 18 of biomolecular topological invariants over the filtration.”
BackgroundTopological Persistence in Geometry and AnalysisPreprint
Polterovich, L., Rosen, D., Samvelyan, K., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.04044 · Cited by 4
“• In practice, long bars in the barcode of the Rips complex carry more reliable information about homology of M than short bars which can be interpreted as a “topological” noise (see [34]).”
BackgroundTowards Persistence-Based Reconstruction in Euclidean SpacesPreprint
Chazal, F., & Oudot, S. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0712.2638 · Cited by 4
“[18] V. de Silva and R. Ghrist.”
MethodTopological Data Analysis in Text Classification: Extracting Features with Additive InformationPreprint
Gholizadeh, S., Savle, K., Seyeditabari, A., & Zadrozny, W. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2003.13138 · Cited by 4
“In the procedure we described above, the information structure based on thresholding distances is called Vietoris-Rips Filtration [14].”
BackgroundMethodTowards quantum advantage for topological data analysisPreprint
Gyurik, C., Cade, C., & Dunjko, V. (2020) · arXiv (Cornell University) · Cited by 4
Extra-fine sheaves and interaction decompositionsPreprint
Bennequin, D., Peltre, O., Sergeant-Perthuis, G., & Vigneaux, J. P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.12646 · Cited by 4
The Emergence of Higher-Order Structure in Scientific and Technological Knowledge Networks
Gebhart, T., & Funk, R. J. (2023) · Academy of Management Proceedings · doi:10.5465/amproc.2023.240bp · Cited by 4
“We can construct the Vietoris-Rips filtration from these distances and its homology will be identical [Ghrist, 2008].”
MethodMavrogiannis, C., DeCastro, J., & Srinivasa, S. S. (2023) · The International Journal of Robotics Research · doi:10.1177/02783649231188740 · Cited by 4
“Techniques from topological data analysis (Ghrist 2007) have offered valuable insights in many real-world domains (Chazal and Michel 2021).”
tramoTDA: A trajectory monitoring system using Topological Data Analysis
Esteve, M., & Falcó, A. (2024) · SoftwareX · doi:10.1016/j.softx.2024.101953 · Cited by 4
Change Points Detection in Financial Market Using Topological Data AnalysisPreprint
Yao, J. Q., Li, J., Wu, J., Yang, M., & Wang, X. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5196633 · Cited by 4
(2009) · Banach Center Publications · doi:10.4064/bc84 · Cited by 4
Stratifying the space of barcodes using Coxeter complexes
Brück, B., & Garin, A. (2022) · Journal of Applied and Computational Topology · doi:10.1007/s41468-022-00104-7 · Cited by 3
“For the reader who is completely new to this, we refer to the reviews [10, 18, 20].”
Remarks on the oscillation energy of Legendrian isotopies
Cant, D. (2023) · Geometriae Dedicata · doi:10.1007/s10711-023-00825-8 · Cited by 3
“4 In the sense of topological data analysis; see [34] for further details and [40, 54, 60, 61, 67] and the references therein for applications to symplectic geometry.”
BackgroundA topological data analysis study on murine pulmonary arterial trees with pulmonary hypertension
Miller, M., Johnston, N., Livengood, I., et al. (2023) · Mathematical Biosciences · doi:10.1016/j.mbs.2023.109056 · Cited by 3
DVT-Net: A Multimodal Deep Vascular Topology Network for Disease Prediction
Tian, Y., Wu, G. K., Bearelly, S., et al. (2023) · IEEE International Symposium on Biomedical Imaging (ISBI) · doi:10.1109/isbi53787.2023.10230374 · Cited by 3
“, connected components and loops) that are characteristic of a dataset [29].”
BackgroundIdentification of Autism Spectrum Disorder Using Topological Data Analysis
Zhang, X., Gao, Y., Zhang, Y., Li, F., Li, H., & Lei, F. (2024) · Journal of Imaging Informatics in Medicine · doi:10.1007/s10278-024-01002-3 · Cited by 3
“For more details about the TDA, persistent homology and its representation, we can refer to [24, 25, 32].”
Singular-Value-Decomposition-Based Matrix Surgery
Ghafuri, J., & Jassim, S. (2024) · Entropy · doi:10.3390/e26080701 · Cited by 3
Topological Fingerprints for Audio Identification
Reise, W., Fernández, X., Dominguez, M., Harrington, H. A., & Beguerisse-Díaz, M. (2024) · SIAM Journal on Mathematics of Data Science · doi:10.1137/23m1605090 · Cited by 8
Topology-informed derivative-free metaheuristic optimization method
Wen, C.-M., & Ierapetritou, M. (2024) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2024.108973 · Cited by 4
Zhu, J., Gao, S., Tao, C., Liu, Y., & Li, H. (2025) · Transactions in GIS · doi:10.1111/tgis.70081 · Cited by 3
“Topology is the branch of mathematics that analyzes and extracts insights from shapes (Ghrist 2008).”
RETRACTED CHAPTER: Hyper Klein Bottle Logophysics Ontopoiesis of the Cosmos and Life
Rapoport, D. L. (2013) · Springer eBooks · doi:10.1007/978-3-319-02039-6_22 · Cited by 3
RetractedUSING EPSILON HULLS TO CHARACTERIZE AND CLASSIFY TOTALLY DISCONNECTED SIERPIŃSKI RELATIVES
TAYLOR, T. D. (2015) · Fractals · doi:10.1142/s0218348x15500152 · Cited by 3
González-Dı́az, R., Jiménez, M.-J., & Krim, H. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-38221-5_20 · Cited by 3
“vertices within some specified distance (see [8]).”
Overview of Basic Methods for Data Science
Chen, L. M. (2015) · Springer eBooks · doi:10.1007/978-3-319-25127-1_2 · Cited by 3
On the nonlinear analysis of optical flow
Xia, S., & Yin, Y. (2016) · Topological Methods in Nonlinear Analysis · doi:10.12775/tmna.2016.054 · Cited by 3
“We also study Kopt 7 ( w, t ) for w = 180 , 220 , 260 , 280, and t = 10 , 11 , 12, we ran many experiments on them, which give similar results as for Kopt 7 (200 , 10), thus we may conclude that the Klein bottle characteristics of the spaces X 3 , X 5 , and X 7 gradually disappears as the size of…”
BackgroundResultPersistent homology in graph power filtrations
Parks, A., & Marchette, D. J. (2016) · Royal Society Open Science · doi:10.1098/rsos.160228 · Cited by 3
“[3], these homology groups can often provide valuable insights about the underlying phenomena represented by the data.”
BackgroundMicrolocal Analysis and Beyond
Schapira, P. (2021) · Cambridge University Press eBooks · doi:10.1017/9781108854429.005 · Cited by 4
“j = d k for j 6= k. To the righthand side of (3.7) we may associate a family of barcodes as follows. For each d j 2Z, replace [a j;b j)[d j] with the vertical interval [0;b j) centered at a j2R (see [Ghr08]). 3.3 The functor hom In [KS90, xIV.4], Sato’s microlocalization functor is generalized as follows. The functor…”
BackgroundInfluentialPreserving Graph Utility in Anonymized Social Networks? A Study on the Persistent Homology
Gao, T., & Li, F. (2017) · IEEE International Conference on Mobile Ad-Hoc and Sensor Systems (MASS) · doi:10.1109/mass.2017.20 · Cited by 3
“Furthermore, although the H1 bars in the dK-3 result of ca-HepPh graph is less than the bars in the original graph, the anonymized result is not more stable because it has a bar [6, 7).”
BackgroundComputational Geometry, Topology and Physics of Visual Scenes
Peters, J. F. (2019) · Intelligent systems reference library · doi:10.1007/978-3-030-22192-8_1 · Cited by 3
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
Learning from the Shape of Data
Rosenstock, S. (2021) · Philosophy of Science · doi:10.1086/715078 · Cited by 3
“Figure 2: Example of a homological barcode, from Ghrist (2008).”
BackgroundSpatial representability of neuronal activity
Akhtiamov, D., Cohn, A. G., & Dabaghian, Y. A. (2021) · Scientific Reports · doi:10.1038/s41598-021-00281-y · Cited by 3
Two-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].”
BackgroundCharacterizing emerging features in cell dynamics using topological data analysis methods
Dawson, M., Dudley, C., Omoma, S., Tung, H.-R., & Ciocanel, M.-V. (2022) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2023143 · Cited by 3
“Čech complexes are more computationally expensive than VR since VR complexes are flag complexes, which makes them easier to store and use previous computations [20].”
BackgroundMethodRepresenting the Topology of Complex Networks Based on Graph Embedding
Li, R., Liu, Z., Zeng, Y., et al. (2022) · International Conference on Networking and Network Applications (NaNA) · doi:10.1109/nana56854.2022.00062 · Cited by 3
“Index Terms—Complex Networks, Graph Embedding, Persistent Homology I. INTRODUCTION Topological Data Analysis (TDA) [1] is a method that focuses on studying the topology of data in order to extract meaningful information.”
MethodInfluentialTime-Inhomogeneous Diffusion Geometry and Topology
Huguet, G., Tong, A., Rieck, B., Huang, J., Kuchroo, M. R., Hirn, M., Wolf, G., & Krishnaswamy, S. (2023) · SIAM Journal on Mathematics of Data Science · doi:10.1137/21m1462945 · Cited by 3
Multivariate central limit theorems for random clique complexes
Temčinas, T., Nanda, V., & Reinert, G. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00146-5 · Cited by 3
“Methods from applied and computational algebraic topology have recently found substantial applications in the analysis of nonlinear and unstructured datasets (Ghrist 2008; Carlsson 2005).”
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
“Later work (Ghrist et al, 2018; Curry et al, 2018) proved injectivity in all dimensions, for both the PHT and ECT, for a very general class of subsets that includes simplicial complexes.”
BackgroundInfluentialAbdullahi, 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
Persistent homology based Bottleneck distance in hypergraph products
Babu, A., & John, S. J. (2024) · Applied Network Science · doi:10.1007/s41109-024-00617-3 · Cited by 3
“Two efficient tools for displaying persistent Betti numbers (Pears 1975), which give information about the lifetime and persistence of topological features, are persistence diagrams (Pears 1975; Edelsbrunner and Harer 2022) and barcodes (Carlsson et al. 2004; Ghrist 2008).”
MethodProjected pooling loss for red nucleus segmentation with soft topology constraints
Fu, G., Jurdi, R. E., Chougar, L., Dormont, D., Valabrègue, R., Lehéricy, S., & Olivier Colliot, A. (2024) · Journal of medical imaging · doi:10.1117/1.jmi.11.4.044002 · Cited by 4
“32 More details on PH can be found in Refs.”
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
“ite subset of points in a Euclidean space Rk, where the dimension kcan be large, and the number of points is often very large. Many excellent surveys and introductions are available in the literature [7, 28, 15, 34, 19, 33]. Decomposition plays a central role both in the theory and in the applications of persistent ho…”
BackgroundMethodTopological data analysis as image steganalysis technique
Asaad, A., Jassim, S., & Rashid, R. D. (2018) · doi:10.1117/12.2309767 · Cited by 3
Geometric 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
Distributions of Matching Distances in Topological Data Analysis
Zheng, X. (2020) · SIAM Undergraduate Research Online · doi:10.1137/18s017302 · Cited by 3
“We give here a brief introduction to persistent homology in both the one- and two-parameter settings; more detailed surveys of the subject are found in [6, 7].”
BackgroundHajij, M., Assiri, B., & Rosen, P. (2020) · Advances in intelligent systems and computing · doi:10.1007/978-3-030-63089-8_47 · Cited by 3
Li, J., Bian, C., Luo, H., et al. (2020) · Journal of Neural Engineering · doi:10.1088/1741-2552/abc7ef · Cited by 3
“This procedure can be characterized using a barcord , where β 0 barcode describes the persistence of connected components, i.e. the zero-dimensional features of PH (PH-0), and β 1 barcode describes the persistence of cycle structures, i.e. the one-dimensional features of PH (PH-1) [10].”
Regularization of Persistent Homology Gradient ComputationPreprint
Corcoran, P., & Deng, B. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.05804 · Cited by 4
“, xm} in R, the corresponding Rips complexRr for a specified r ≥ 0 radius value is defined as follows: Rr(X) = {σ ⊆ X : ∀x, y ∈ σ, ‖x− y‖ ≤ r} (1) The k-simplices in this simplicial complex equal unordered (k+1)-tuples of points which are pairwise within distance r [14].”
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
“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 topological and geometric features of point clouds.”
MethodA new non-archimedean metric on persistent homology
Güzel, İ., & Kaygun, A. (2022) · Computational Statistics · doi:10.1007/s00180-021-01187-z · Cited by 3
“…topological features as the filtration scale parameter increases can be presented in various different ways by barcodes (Carlsson et al. 2005) and (Ghrist 2008), by persistence diagrams (CohenSteiner et al. 2007), by landscapes (Bubenik 2015), by images (Adams et al. 2017), by terraces (Moon et…”
BackgroundMethodTopological learning for acoustic signal identification
Marchese, A., & Maroulas, V. (2016) · 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
The 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…”
MethodOn 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
“…segments, which are called barcodes, on the ( )F t 2t M R∈ ⊂ ( )F t u [ ]{ }( ) , .uA t M F t u= ∈ ∈ ∞ 1( )F u− u ≥ v .uA A⊆ v M uA u uA u 1 2,u u 0β 1β F GEOMAGNETISM AND AERONOMY Vol. 57 No. 8 2017 ON THE PROGNOSTIC EFFICIENCY OF TOPOLOGICAL DESCRIPTORS 1089 axis of the levels (Ghrist, 2008).”
BackgroundObservability of Complex Systems by Means of Relative Distances Between Homological Groups
Ochoa, J. G. D. (2020) · Frontiers in Physics · doi:10.3389/fphy.2020.465982 · Cited by 3
Goethals, P. L., Scala, N. M., & Bennett, D. (2022) · Chapman & Hall/CRC eBooks · doi:10.1201/9780429354649 · Cited by 3
“Perhaps the most widely recognized and utilized tool in TDA is persistent homology [70, 26, 9, 23, 27, 57].”
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
“Point-cloud data and their topological features [9].”
Detecting Traffic Incidents Using Persistence Diagrams
Weber, E., Harding, S. N., & Przybylski, L. (2020) · Algorithms · doi:10.3390/a13090222 · Cited by 3
Measuring and Localing Homology ClassesPreprint
Freedman, D., & Chen, C. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0705.3061 · Cited by 2
“topological features are especially important in high dimensional data analysis, where purely geometric tools are often decient, and full-blown space reconstruction is expensive and often ill-posed [3, 16]. Once we are able to compute topological features, a natural problem is to rank the features according to their i…”
BackgroundLau, A., & Yin, C. (2009) · Second International Meeting for Applied Geoscience & Energy · doi:10.1190/1.3255425 · Cited by 3
Statistical 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
“Reviews of the Topological Data Analysis workflow may be found in Carlsson [1], Ghrist [6], and Nanda and Sazadanovi´c [7].”
BackgroundA quantitative structure comparison with persistent similarityPreprint
Xia, K. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.03572 · Cited by 3
A Quantum Circuit to Construct All Maximal Cliques Using Grover Search AlgorithmPreprint
Wie, C. R. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1711.06146 · Cited by 3
“Whereas, in the persistent homology [9], the Betti numbers of a various order can be plotted as a function of distance between nodes, resulting in a barcode-like plot [10].”
BackgroundSandoz, 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. (2018) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/431106 · 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
“ings by approximations called Rips complexes, see Appendix D. 10In topological data analysis, persistent 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 intro…”
MethodThe 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
Interval graph limitsPreprint
Diaconis, P. W., Holmes, S., & Janson, S. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1102.2841 · Cited by 3
“For an application of rectangle intersections, see Rim and Nakajima [36] and for sphere intersections see Ghrist [16].”
MethodModules over posets: commutative and homological algebraPreprint
Miller, E. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1908.09750 · Cited by 3
“des were invented by Abeasis and Del Fra, who called them “diagrams of boxes” [AD80, ADK81]. Subsequent terminology for objects equivalent to these diagrams of boxes include bar codes themselves (see [Ghr08]) and planar depictions discovered effectively simultaneously in topological data analysis, where they are called…”
BackgroundDong, Z., Lin, H., & Zhou, C. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1909.08417 · Cited by 3
“These multidimensional homological features are represented as a persistence diagram (PD) or a barcode [3].”
BackgroundInfluentialA 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
“Alternatively, we may show the birth and the death of loops with barcodes where each hole is represented by a bar from its birth to its death [5, 6, 7].”
BackgroundKuchroo, M. R., DiStasio, M., Calapkulu, E., Ige, M., Zhang, L., Sheth, A. H., Menon, M., Xing, Y., Gigante, S., Huang, J., et al. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.01.19.427286 · Cited by 3
Outlier-robust subsampling techniques for persistent homologyPreprint
Stolz, B. J. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.14743 · Cited by 5
“Our method relies on computing the Vietoris–Rips filtration [4, 12] locally, in a small neighbourhood around each data point, where the afore mentioned computational problems of PH vanish.”
MethodFast Topological Clustering with Wasserstein DistancePreprint
Songdechakraiwut, T., Krause, B. M., Banks, M. I., Nourski, K. V., & Veen, B. D. V. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2112.00101 · Cited by 3
“There are no topological features of higher dimensions in the 1-skeleton, in contrast to well-known Rips (Ghrist, 2008) and clique complexes (Otter et al., 2017).”
BackgroundMethodResultInfluentialPersistent Topological Structures and Dynamics in Tuberculosis Delay ModelPreprint
Elfouly, M. A., & Abouelenien, R. (2025) · Research Square · doi:10.21203/rs.3.rs-7077315/v1 · Cited by 3
Computation of persistent homology on streaming data using topological data summaries
Moitra, A., Malott, N. O., & Wilsey, P. A. (2023) · Computational Intelligence · doi:10.1111/coin.12597 · Cited by 2
Topological data analysis of Chinese stocks’ dynamic correlations under major public events
Guo, H., Ming, Z., & Xing, B. (2023) · Frontiers in Physics · doi:10.3389/fphy.2023.1253953 · Cited by 2
Hemodynamic cortical ripples through cyclicity analysis
Abraham, I., Shahsavarani, S., Zimmerman, B., Husain, F. T., & Baryshnikov, Y. M. (2024) · Network Neuroscience · doi:10.1162/netn_a_00392 · Cited by 4
“The coupled pairs whose span d − b is high are said to be persistent features in the data (see Ghrist, 2008 for more details).”
Art’s hidden topology: A window into human perception
Dmitruk, E., Bajno, B., Kot, L., Dreszer, J., Bałaj, B., Ratajczak, E. K., Hajnowski, M., Janik, R. A., Kuś, M., Kadir, S., et al. (2026) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1014156 · Cited by 2
“We have just described informally a tool used in computational topology [34], persistent homology [10].”
Effective data reduction algorithm for topological data analysis
Choi, S., Oh, J., Park, J. R., Yang, S. Y., & Yun, H. (2025) · Applied Mathematics and Computation · doi:10.1016/j.amc.2025.129302 · Cited by 5
Topological optimal transport for geometric cycle matching
Zhang, S. Y., Stumpf, M. P. H., Needham, T., & Barbensi, A. (2025) · Journal of Applied and Computational Topology · doi:10.1007/s41468-025-00207-x · Cited by 2
“In a nutshell, the PH algorithm proceeds by building a nested sequence of discrete spaces describing the input data at increasingly coarse scales, known as a filtration of simplicial complexes.”
Che, T., Zhang, L., Zeng, D., et al. (2025) · Medical Image Analysis · doi:10.1016/j.media.2025.103624 · Cited by 2
Change Point Detection in Financial Market Using Topological Data Analysis
Yao, J. X., Li, J., Wu, J., Yang, M., & Wang, X. (2025) · Systems · doi:10.3390/systems13100875 · Cited by 2
A topological data analysis method for revealing dynamic changes in psychotherapy microprocesses
Luo, X., & Zhang, M. (2026) · Frontiers in Psychology · doi:10.3389/fpsyg.2025.1711782 · Cited by 2
“Temporal Mapper is a tool based on Topological Data Analysis (TDA), a growing field in applied mathematics aiming to quantify the “shape” of high-dimensional objects (see Ghrist, 2007; Carlsson, 2009, and Munch, 2017 for an introduction).”
Trace fossils constrain the perceptual ranges of the earliest motile animals
Wang, Z., & Shi, T. (2026) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2609730123 · Cited by 2
Singh, Y., Himeur, Y., Farrelly, C., Kadzue, P. K.-M. T., Rozenblit, J., Kunovac, A., Pappas, I. C., Choudhary, A., Salehi, S., Atalla, S., et al. (2026) · Bioengineering · doi:10.3390/bioengineering13060638 · Cited by 3
“…structure, symmetry, long-range dependencies, and low-label learning, but they also increase computational complexity and validation requirements [7–35 Future research should emphasize standardized graph construction, transparent reporting of preprocessing and validation design, external testing…”
InfluentialZheng, S., Wang, B., Zhang, M., Tan, J., Lv, H.-B., Liu, R., Shi, Z.-Z., Li, X., Wang, R., Zhang, S. Q., et al. (2026) · Chemical Reviews · doi:10.1021/acs.chemrev.5c01098 · Cited by 4
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
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