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 1,101–1,200 of 1,274
Notes on pivot pairingsPreprint
Giunti, B. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2101.00451
“Persistent homology summarises a dataset’s information in the form of barcode [6, 14], whose efficient computation is then crucial to TDA.”
BackgroundDomain adaptation by topology regularization
Weeks, D., & Rivera, S. (2021) · doi:10.1117/12.2585705
“10 More details can be found in the text by Edelsbrunner and Harer.”
BackgroundExtraction and Analysis of Spatial Correlation Micrograph Features for Traceability in Manufacturing
Dachowicz, A. P., Atallah, M. J., & Panchal, J. H. (2020) · Journal of Computing and Information Science in Engineering · doi:10.1115/1.4046891
A Topological Approach for Motion Track Discrimination
Emerson, T., Tymochko, S., Stantchev, G. M., et al. (2022) · Association for Women in Mathematics series · doi:10.1007/978-3-030-95519-9_9
Online High-Dimensional Change-Point Detection using Topological Data AnalysisPreprint
Zheng, X., Mak, S., & Xie, Y. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.00117
“Further details can be found in [7] and [8].”
BackgroundConvergence of persistence diagram in the subcritical regimePreprint
Owada, T. (2021) · arXiv (Cornell University)
Mathematics and Art: Unifying Perspectives
Russell, H. M., & Sazdanović, R. (2021) · Springer eBooks · doi:10.1007/978-3-319-70658-0_125-1
Topological Simplifications of HypergraphsPreprint
Zhou, Y., Rathore, A., Purvine, E., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2104.11214
“In this section, we first introduce the topological profile of a weighted graph, formally known as its barcode [7], [55], which is grounded in persistent homology [56].”
BackgroundTopological Data Analysis of COVID-19 Virus Spike ProteinsPreprint
Chung, M. K., & Ombao, H. (2021) · arXiv (Cornell University)
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.006
Application of Topological Data Analysis in Co-infections and Its Effectiveness
Guo, M. (2020) · International Conference on Big Data and Social Sciences (ICBDSS) · doi:10.1109/icbdss51270.2020.00011
“Besides, the researchers could use persistence barcode and persistence diagram to visualize the birth and death of the hole in the simplicial complexes [6].”
BackgroundMethodBeyond “Evidence-Based” Policymaking
Iyetomi, H. (2021) · Springer eBooks · doi:10.1007/978-981-15-4944-1_15
Persistence Barcoded Vehicular Traffic Videos in a Topology of Data Approach to Shape Tracking
Don, A. P. H., Peters, J. F., & Ramanna, S. (2021) · Advances in intelligent systems and computing · doi:10.1007/978-981-16-1402-6_3
“Each occurrence of a frame shape is represented by a horizontal bar in 2D Ghrist pictographs called barcodes, introduced by R. Ghrist [8, 9] and others [7, 6, 13, 4] and later extended to 3D persistence barcodes in [5].”
BackgroundInfluentialTopological data analysis with metric learning and an application to high-dimensional football data
Meza, D. (2015) · instname:Universidad de los Andes
Spatial representability of neuronal activity
Akhtiamov, D., Cohn, A. G., & Dabaghian, Y. A. (2021) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · doi:10.48550/arxiv.2108.03518
Dong, L., Hang, H., Park, J. G., Mio, W., & Liang, R. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2108.04375
“explanation for this convention may be found in [32,33].”
BackgroundMethodBouazzaoui, H., Elomary, M. A., & Mamouni, M. I. (2021) · Statistics in Transition New Series · doi:10.21307/stattrans-2021-031
“For further details on persistent homology, we refer the reader to these standard references (Edelsbrunner and Harer 2010) and (Ghrist 2008).”
BackgroundDive into Layers: Neural Network Capacity Bounding using Algebraic GeometryPreprint
Yang, J. H., Sang, L., & Cremers, D. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2109.01461
Servis, M. J., Sadhu, B., Soderholm, L., et al. (2021) · doi:10.33774/chemrxiv-2021-6zndd
Harmonic representatives in homology over arbitrary fields
Catanzaro, M. J., & Vose, B. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00117-w
“See [12] for a friendly introduction to C̆ech and Vietoris-Rips complexes, and their applications to persistent homology.”
Background(Thesis) Reservoir Computing With Dynamical SystemsPreprint
Hart, A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2111.14226 · Cited by 2
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.002
Series Expansion of Connectivity Matrices
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.013
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.009
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.014
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.015
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.001
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Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.007
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.004
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.010
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.008
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.011
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.005
Combinatorial Inferences for Networks
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.012
Ludu, A. (2016) · Springer series in synergetics · doi:10.1007/978-3-662-49078-5_1
Topological Conditions for Propagation of Spatially-Distributed Neural ActivityPreprint
Tudoras, À., & Reyes, A. D. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4046865
Higher-Order Description of Brain Function
Expert, P., & Petri, G. (2022) · Understanding complex systems · doi:10.1007/978-3-030-91374-8_17 · Cited by 1
Chung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.016
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Persistent Homology Metrics Reveal Quantum Fluctuations and Reactive Atoms in Path Integral DynamicsPreprint
Hu, Y., Ounkham, P., Maršálek, O., et al. (2021) · ChemRxiv · doi:10.26434/chemrxiv.13618922
Multiscale Methods for Signal Selection in Single-Cell Data
Hoekzema, R. S., Marsh, L., Sumray, O., Carroll, T. M., Lü, X., Byrne, H. M., & Harrington, H. A. (2022) · Entropy · doi:10.3390/e24081116
“The most common tool is persistent homology [33], which computes how invariants, such as connected components in G[α(i)], persist in the larger graph G[α(j)].”
BackgroundHomología persistente para la detección de anomalías
Castro, L. C. R. (2022) · doi:10.11144/javeriana.10554.59464
Spectral aspects of mining complex networks
Torres, L. (2021) · Digital Repository Service (Northeastern University) · doi:10.17760/d20413933
Storch, L. S., & Day, S. (2022) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2022.111274
Comparative study of left atrium epicardial fat tissue pattern using persistent homology approach
Deepa, D., Singh, Y., Mansoor, W., et al. (2022) · BMC Research Notes · doi:10.1186/s13104-022-06173-2
Formation of Machine Learning Features Based on the Construction of Tropical Functions
Chukanov, S. N., & Чуканов, И. С. (2022) · Modeling and Analysis of Information Systems · doi:10.18255/1818-1015-2022-3-200-209
Ulm, M. A., & Brändle, N. (2016) · Proceedings - International Conference on Pattern Recognition/Proceedings/International Conference on Pattern Recognition · doi:10.1109/icpr.2016.7900202
“Although such geometric properties of data manifolds have not been given much attention in the literature, with the notable exception of the work of Carlsson and collaborators [6], [7], [19], they can provide valuable additional insights into the structure of the data.”
BackgroundUnderstanding the Mapping of Encode Data Through An Implementation of Quantum Topological AnalysisPreprint
Vlasic, A., & Pham, A. T. (2022) · Research Square · doi:10.21203/rs.3.rs-2346302/v1
Schenck, H. (2022) · Mathematics of data · doi:10.1007/978-3-031-06664-1_7 · Cited by 1
Topological analysis of unaligned audio and text data
Kozhirbayev, Z., & Yessenbayev, Z. (2022) · Bulletin of L N Gumilyov Eurasian National University Technical Science and Technology Series · doi:10.32523/2616-7263-2022-141-4-116-126
Dłotko, P. (2023) · ACADEMIA - The magazine of the Polish Academy of Sciences · doi:10.24425/academiapas.2022.144686
Comparison of complex dynamical systems based on topological data analysis
Chukanov, S. N. (2023) · Computer Research and Modeling · doi:10.20537/2076-7633-2023-15-3-513-525
Topological Loss on Graph Auto-Encoders
Gao, J., Liu, Z., & Zeng, Y. (2023) · International Conference on Networking and Network Applications (NaNA) · doi:10.1109/nana60121.2023.00088
“Persistence barcode [7], persistence diagram are basic representation of results of TDA [1], they are 2-rank lipschitz stable locally [8].”
BackgroundTopological Estimation of Number of Sources in Linear Monocomponent Mixtures
Kennedy, S., Tummala, M., & McEachen, J. (2023) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco58844.2023.10289812
“We direct the unfamiliar reader to reference [21] for an introduction to the theory and computation of PH.”
An Improved K-Means Algorithm Based on Persistent HomologyPreprint
Yin, X., Peng, N., & Yan, X. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4627285
Can persistence barcode topology reveal tumors’ architecture information?
Singh, B., Mansoor, W., Atalla, S., et al. (2023) · AIP conference proceedings · doi:10.1063/5.0181973
Oda, H., Ng, C. F. S., & Hashizume, M. (2024) · Research Square · doi:10.21203/rs.3.rs-3980735/v1
Portfolio selection via topological data analysis
Sokerin, P., Kuznetsov, K., Makhneva, E., & Zaytsev, A. (2024) · International Conference on Machine Vision · doi:10.1117/12.3023429 · Cited by 3
“Statistical features derived from barcodes can be used as a feature space to analyze graphs, point clouds, and so on [26].”
BackgroundThe Internal Topology of Rocks
Korvin, G. (2024) · Earth and environmental sciences library · doi:10.1007/978-3-031-46700-4_3
Persistent homology based Wasserstein distance for graph networks
Babu, A., & John, S. J. (2024) · Hacettepe Journal of Mathematics and Statistics · doi:10.15672/hujms.1317203
DFT and Persistent Homology for Topological Musical Data Analysis
Callet, V. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-60638-0_23
Persistent Homology and Harmonic Analysis
Gilblas, R. C. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-60638-0_16
Integrating Image Visibility Graph and Topological Data Analysis for Enhanced Texture Classification
Pal, R., Kumar, S., & Singh, M. K. (2024) · Lecture notes in networks and systems · doi:10.1007/978-981-97-3292-0_40
Schrader, P. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4885393
Baldo, H. (2024) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/t.95.2024.tde-04072024-124243 · Cited by 1
Topological Structures in the Space of Treatment-Naïve Patients with Chronic Lymphocytic Leukemia
McGee, R. L., Reed, J., Coombes, C. E., Herling, C., Keating, M. J., Abruzzo, L. V., & Coombes, K. R. (2024) · Cancers · doi:10.3390/cancers16152662
Research on Clothing Image Retrieval Combining Topology Features with Color Texture Features
Zhang, X., Sun, H., & Ma, J. (2024) · Mathematics · doi:10.3390/math12152363
Topological Delaunay Graph for Efficient 3D Binary Image Analysis
Yoshizawa, S., Michikawa, T., & Yokota, H. (2024) · International Journal of Automation Technology · doi:10.20965/ijat.2024.p0632
Use of Riemannian Distance Metric to Verify Topological Similarity of Acoustic and Text Domains
Yessenbayev, Z., & Kozhirbayev, Z. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-72350-6_25
Topological Signal Processing and Learning: Recent Advances and Future ChallengesPreprint
Isufi, E., Leus, G. J. T., Beferull‐Lozano, B., Barbarossa, S., & Lorenzo, P. D. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.5043391
Leveraging persistent homology for liver tumour classification
Ali, D., Peoples, J., Mojtahedi, R., et al. (2025) · SPIE Medical Imaging: Computer-Aided Diagnosis · doi:10.1117/12.3045640
System Observability and $${\varvec{\varPhi}}_{{\varvec{S}}}$$ Complexity
Ochoa, J. G. D. (2025) · Understanding complex systems · doi:10.1007/978-3-031-84709-7_5
On the Space of Generalized Persistence Diagrams
Jose, C. J., & Kumar, P. (2025) · Springer proceedings in mathematics & statistics · doi:10.1007/978-981-96-1505-6_27
Spatial and Sequential Topological Analysis of Molecular Dynamics Simulations of IgG1 Fc Domains
Kleczynski, M., Bergonzo, C., & Kearsley, A. J. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.5c00161
“AVietoris − Rips complex is one way to represent point cloud data as a mathematical object called a simplicial complex, 30 and is the type of simplicial complex typically used in CROCKER plots and matrices.”
Classification of epilepsy MEG data using topological data analysis and artificial neural network
Han, Y., Chung, C. K., & Kim, J. (2025) · Korean Journal of Applied Statistics · doi:10.5351/kjas.2025.38.2.177
Claudi, F., Chandra, S., & Fiete, I. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.05.07.652608
“To quantify the structure of the resulting population states, we use persistent homology, a Topological Data Analysis [66, 67] technique that has been applied with success in neuroscience [68, 3, 44].”
Callet-Feltz, V. (2025) · Computational music science · doi:10.1007/978-3-031-82236-0_1
Callet-Feltz, V. (2025) · Computational music science · doi:10.1007/978-3-031-82236-0_5
Homological Properties and Digital Relative Homology Groups of MA-Spaces
Demir, E. Ü. (2025) · Symmetry · doi:10.3390/sym17060862
Bartosova, M., Saleem, A., Schäfer, M., Messerschmidt, G., Schmitt, C. P., & Hausmann, M. (2025) · Nanoscale · doi:10.1039/d5nr00686d
Claudi, F., Chandra, S., & Fiete, I. (2025) · eLife · doi:10.7554/elife.107224
Decoding AI-Generated Images: A Topological Approach to Source Classification with Noise
Khan, M. S., Rema, V., Mondal, A., et al. (2025) · Lecture notes in networks and systems · doi:10.1007/978-3-031-97992-7_21
Big Data Searching Using Words
Acharjee, S., & Choudhury, R. (2025) · New Mathematics and Natural Computation · doi:10.1142/s179300572550053x
Shami, A. A., Bachri, M., Young, C., & Grissom, D. (2025) · Lecture notes in networks and systems · doi:10.1007/978-3-031-99965-9_37
Parallel computation of interval bases for persistence module decomposition
Gregorio, A. D., Guerra, M., Scaramuccia, S., & Vaccarino, F. (2025) · Applicable Algebra in Engineering Communication and Computing · doi:10.1007/s00200-025-00699-1 · Cited by 4
“A specialization to the case of real coefficient is included in Algorithm 7 in Appendix A.”
Generative AI and Topological Data Analysis of Longitudinal Panel Data
Arfi, B. (2025) · Political Analysis · doi:10.1017/pan.2025.10019
From Darwin to Teleology: A Categorical Final--Cause Calculus for EvolutionPreprint
Patrascu, A. T. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5448115
Interval Decomposition of Persistence Modules over a Principal Ideal Domain
Luo, J., & Henselman‐Petrusek, G. (2025) · Foundations of Computational Mathematics · doi:10.1007/s10208-025-09732-8 · Cited by 5
“While abstract in definition, persistence modules underlie persistent homology (PH) [15, 18, 20], an area of topological data analysis (TDA) that has contributed to applied and theoretical scientific research in domains including medicine [40], neuroscience [12, 46, 47], image processing [16,39],…”
Nested Ensemble Learning with Topological Data Analysis for Graph Classification and Regression
Abaa, I., & Islambekov, U. (2025) · International Journal of Topology · doi:10.3390/ijt2040017
Hu, X., Nickolay Korabel, A., Millard, T. H., & Waigh, T. A. (2025) · Physical review. E · doi:10.1103/mc9z-kw1q
He, Y., Wu, J., & Li, M. Z. (2025) · Chinese Physics B · doi:10.1088/1674-1056/ae1457
Elementary Methods for Persistent Homotopy Groups
Adams, H., Batan, M. A., Pamuk, M., & Varlı, H. (2025) · Discrete & Computational Geometry · doi:10.1007/s00454-025-00781-y
“This theory has many applications and has become a central tool in topological data analysis [11, 19].”
Topological Analysis of Alzheimer’s Progression with Persistence Images
Kande, J. (2025) · Communications in computer and information science · doi:10.1007/978-3-032-06198-0_2
Domínguez-Monterroza, A., Mateos, A., & Martín, A. J. (2025) · PLoS ONE · doi:10.1371/journal.pone.0337620 · Cited by 1
Topology-Based Optimization of Image Symmetry Axes Using Wasserstein Distance
Liu, T., Li, T., Li, L., & He, Y. (2025) · International Congress on Image and Signal Processing (CISP) · doi:10.1109/cisp-bmei68103.2025.11259429
“Persistent homology captures the evolution of connected components, loops, and higher-order structures across multiple scales, yielding stable descriptors in the form of persistence diagrams [13].”
H-VIP: quantifying regional topological contributions of the brain network to cognition
Garai, S., Vo, S., Blank, L., Xu, F., Chen, J., Duong-Tran, D. A., Zhao, Y., Brown, B. C., & Shen, L. (2025) · Frontiers in Radiology · doi:10.3389/fradi.2025.1686780
“In applications, long intervals typically signal topological features that persist across many scales (and are hence considered signal), while very short intervals are often treated as noise [19, 30].”
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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