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 1,201–1,274 of 1,274
A Topology-Loss Aware Framework for Addressing Oversegmentation in Brain Tumor Segmentation
Kumar, S., & Pilania, U. (2025) · doi:10.1109/cises66934.2025.11265618
“TDA’s core tool, persistent homology, tracks topological changes across image filtrations, producing persistence diagrams that summarize structural properties robustly [15].”
Cellular deconvolution of the brain with topological magnetic resonance image analysisPreprint
Vazquez, L. A., Fromandi, M. B., Hagemann, T. L., Risgaard, R. D., Guerrero-Gonzalez, J. M., Singh, A. P., Frautschi, P. C., Hurley, S. A., Sousa, A. M. M., Dean, D. C., III, et al. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2025.12.10.693426
Diagnostics of Semantic Word Spaces
Isheden, G. (2013) · KTH Publication Database DiVA (KTH Royal Institute of Technology)
Computing multifractal spectra via simplicial measures
Gedeon, T., & Berwald, J. (2011) · Montana State University ScholarWorks (Montana State University)
On the support of betti tables of multiparameter persistent homology modules
Guidolin, A., & Landi, C. (2025) · The Quarterly Journal of Mathematics · doi:10.1093/qmath/haaf021
“One of the main concepts in Topological Data Analysis is persistent homology , a tool to capture topological information at multiple scales and provide meaningful topological summaries of the data, as surveyed, for example, in [Ghr07, Car09, EM12].”
Exploring Use of Data Topology in Improving Gan-Generated Tabular Synthetic Data
Sen, G., Sultanova, N., Mustafina, J., et al. (2025) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-981-96-7749-8_11
Toward structure-preserving quantum encodings
Parzygnat, A. J., Bradley, T.-D., Vlasic, A., & Pham, A. (2025) · Physical Review Research · doi:10.1103/rph8-g15q · Cited by 2
Structural Uncertainty Visualization of Morse Complexes for Time-Varying Data Prediction
Lyu, W., Gupta, S., Chen, C., et al. (2025) · doi:10.1109/topoinvis68599.2025.00008
“• We demonstrate the effectiveness of MC-U both qualitatively and quantitatively through experiments on various 2D time-varying scientific datasets, in comparison with pixel-wise uncertainty models.”
Inferring a Cell Structure on the Space of Cyclooctane Conformations
Bauer, U., & Lenzen, F. (2026) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-032-01279-1_14
From Darwin to teleonomy: A categorical final-cause calculus for evolution
Patrascu, A. T. (2026) · Biosystems · doi:10.1016/j.biosystems.2025.105687
Topological Deep Learning for Coral Reefs
Aguilar-Ortíz, J., Díaz-Parra, O., Ruíz-Vanoye, J. A., et al. (2026) · IGI Global eBooks · doi:10.4018/979-8-3373-5881-9.ch008
Küntzelmann, K., Rozo-Pardo, L., Schäfer, M., Weidner, J., Falková, I., Toufar, J., Toufarová, L., Bestvater, F., Hausmann, M., & Falk, M. (2026) · Nanoscale · doi:10.1039/d5nr05100b · Cited by 1
Schäfer, M., Muhtadi, R., Schumann, S., Bestvater, F., Eberlein, U., Hildenbrand, G. L., Scherthan, H., & Hausmann, M. (2026) · Biomolecules · doi:10.3390/biom16010142
“The similarity between the components or holes in point clouds (protein distribution) was assessed using the Jaccard index [75], determining the overlap of one barcode [76] with another and vice versa.”
Time Series Object Detection with Periodic Boundary Persistent Homology
Oda, H. (2025) · doi:10.1109/ieem63636.2025.11357658
“These births and deaths are often recorded in a form of “barcode” [13,14], where we draw bars corresponding to the intervals in the decomposition.”
Robinson, M. (2026) · Mathematical engineering · doi:10.1007/978-3-032-08634-1_2
Robust Short-Term OEE Forecasting in Industry 4.0 via Topological Data AnalysisPreprint
Anapa, K., Güzel, İ., & Yozgatlıgil, C. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6207839
Algorithms and Complexity Results for K-Theoretic Persistent Homology
Maruyama, Y. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-17801-5_5
Emergent Latent Structure via Co-Reactivity to Shared ConstraintsPreprint
ursachi, A. (2026) · Research Square · doi:10.21203/rs.3.rs-8823860/v1
Classification of epilepsy MEG data using topological data analysis and artificial neural network
Han, Y., Chung, C. K., & Kim, J. H. (2025) · Korean Journal of Applied Statistics · doi:10.5351/kjas.2025.38.2.189
In the Beginning Was ... Gaussian Topology?!
Adler, R. J. (2026) · The Mathematical Intelligencer · doi:10.1007/s00283-026-10503-8
A DNN Biophysics Model with Topological and Electrostatic Features
Sliheet, E., Talha, M. A., & Geng, W. (2026) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.5c03104
“The importance of persistence is that given a parameterized family of spaces, topological features which persist over a significant parameter ranges are considered as signal while those with short-lived features are treated as noise [53].”
Topological Metric for Unsupervised Embedding Quality Evaluation
Shestov, A., Klenitskiy, A., Denisova, D., et al. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-21300-6_51
Statistical Contrastive Learning for Spatio-Temporal Anomaly Detection
Zhen, Z., Chen, Y., & Gel, Y. R. (2026) · Data Science in Science · doi:10.1080/26941899.2026.2648108
Dynamical and topological tools for (modern) music analysis
Bergomi, M. G. (2015) · theses.fr (ABES) · doi:10.70675/7f9d64bezcdc1z4ecbza15fzf3aac6e60bfe · Cited by 12
Revealing the transport mechanisms from a single trajectory in living cells
Lanoiselée, Y. (2018) · theses.fr (ABES) · doi:10.70675/27b830f7zb71fz42b4z8987zb09fdac67f02 · Cited by 1
“A second development would be to find an appropriate window size ∆ directly from the data, or to consider all possible ∆ in the spirit of persistent homology [104].”
Kernel methods for high dimensional data analysis
de Vitis, A. C. (2019) · doi:10.70675/9040fbf6z5a1bz40fbzad75z2994359bc377
“Bar codes are a new type of descriptors that can be used to further analyze the data [38], [39], [40], [41],[42].”
Structure learning to unravel mechanisms of the immune system
Todorov, H. (2020) · theses.fr (ABES) · doi:10.70675/69543fb0zccf0z4c4dzbd34z6aa4482b1101
Beaufils, B. (2020) · theses.fr (ABES) · doi:10.70675/803ee8dbzbb85z4d09za979z798c923c9cb9
Reconstruction of univariate functions from directional persistence diagrams
Ferrà, A., Casacuberta, C., & Pujol, O. (2026) · Applicable Algebra in Engineering Communication and Computing · doi:10.1007/s00200-025-00710-9
Martin, C. B. (2023) · theses.fr (ABES) · doi:10.70675/8eea9293zdaadz4766z9baez15cec27b3b07
Topology enhances classical figure detection
Oda, H. (2025) · International Conference on Electrical, Computer and Energy Technologies (ICECET) · doi:10.1109/icecet63943.2025.11472121
Analyse biogéométrique de l'évolution des protéines et des traits d'histoire de vie
Dagher, L. B. (2024) · theses.fr (ABES) · doi:10.70675/3a39f4f2zb57dz4fb3z8990zbd5d3fe1a6bf
Hu, C., Czajkowsky, D. M., Liang, J., & Shao, Z. (2026) · Biophysical Journal · doi:10.1016/j.bpj.2026.04.010
Higher-order functional connectivity and topological analysis of multichannel fNIRS data
Luo, H., Luo, Y., & Duan, L. (2026) · Biomedical Optics Express · doi:10.1364/boe.589685
“Persistent homology analysis is a topological data analysis technique [21] and has been recently applied to brain connectome studies [19,22].”
Lin, Z., Zulkepli, N. F. S., Kasihmuddin, M. S. M., & Gobithaasan, R. U. (2026) · Knowledge-Based Systems · doi:10.1016/j.knosys.2026.115963
Topology Meets Attribution: A Novel Explainability Approach for Transparent Medical Decision Support
Galis, F., & Onchiş, D. (2025) · International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) · doi:10.1109/synasc69064.2025.00075
“Key advantages of persistent homology include its ability to handle high-dimensional, non-linear relationships and its stability against small perturbations of data [6].”
InfluentialGao, X., Wang, X., Liu, Y., et al. (2026) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp55912.2026.11462478
Adaptive Topological Constraint Enhanced Pedestrian Trajectory Prediction
Zhao, Z., Lai, J., Chen, S., et al. (2026) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp55912.2026.11461902
A patch-grouped topological fusion approach for enhanced multivariate numeric time-series modeling
Lin, Z., Zulkepli, N. F. S., Kasihmuddin, M. S. M., et al. (2026) · Applied Soft Computing · doi:10.1016/j.asoc.2026.115362
Topological biomarkers for seizure state transitions
Fernández, X., & Mateos, D. (2026) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0326217
Persistent homology of hypergraphs via barycentric subdivision
Aktas, M. E., Nguyen, T., Riza, R., et al. (2026) · Journal of Complex Networks · doi:10.1093/comnet/cnag013
On Enhancing Anomaly Detection in X-Ray Baggage Inspection Using Persistence Homology
Ahmed, A., Velayudhan, D., Hassan, T., et al. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-19978-2_11
Lin, Z., Zulkepli, N. F. S., Kasihmuddin, M. S. M., et al. (2026) · Complex & Intelligent Systems · doi:10.1007/s40747-026-02315-2
“This quadratic cost is standard for metric-based simplicial complex construction and is well documented in the computational topology literature [62, 63].”
Domínguez-Monterroza, A., Jiménez-Martín, A., & Caballero, A. M. (2026) · PLoS ONE · doi:10.1371/journal.pone.0347767
PH-SAM2: Persistent Homology-Guided Prompting of SAM2 for Zero-Shot Medical Image Segmentation
Bachri, M., Gazawy, Q., & Shami, A. A. (2026) · Lecture notes in networks and systems · doi:10.1007/978-3-032-26211-0_9
Fixation Sequences as Time Series: A Topological Approach to Dyslexia Detection
Huber, M., Reich, D. R., & Jäger, L. A. (2026) · ACM Symposium on Eye Tracking Research and Applications (ETRA) · doi:10.1145/3797246.3803045
“In recent years, persistent homology, a key tool within TDA, has been increasingly applied to a wide variety of data sets (for accessible introductions, see [8, 19]).”
Topological Analysis of Composite Ageing via Dual Anisotropic Filtrations and Persistent Homology
Canot, H., Durand, P., Frénod, E., et al. (2026) · International Journal of Topology · doi:10.3390/ijt3020011
“Topological Data Analysis (TDA) has emerged as a powerful mathematical framework [23–27] for characterising complex structures in materials science, particularly when classical geometric or statistical descriptors fail to capture global organisation and multi-scale phenomena.”
Homology-Preserving Dimensionality Reduction via Adaptive Mapper and Landmark Isomap
Khourashahi, S., Jahanshahi, I., Wang, B., & Yan, L. (2026) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2026.3703433
“1(c)), or equivalently, persistence barcodes [44] (Fig.”
Diankatu, M., Duque, J., de Vasconcelos, J. B., & Filipe, V. M. D. J. (2026) · IFIP advances in information and communication technology · doi:10.1007/978-3-032-28939-1_5
Furuya, K., Umeno, Y., Ikura, M., Ikura, Y., Nishikawa, S., & Ikura, T. (2026) · Genes to Cells · doi:10.1111/gtc.70127
“From each cell, we extracted topological descriptors, including H1 barcodes representing persistence and persistence diagrams (PDs) encoding birth–death information (Figure 1C) (Ghrist 2008).”
Layer-Wise Persistent Entropy of CT Scan Point Clouds for Lung Tumor Classification
Jose, C. J., Baiju, A. P., Roy, R., Harikrishnan, A., Sanju, R., Kumar, P., Sherly, K. K., Jacob, R., & Sreekumar, G. (2026) · AppliedMath · doi:10.3390/appliedmath6060095
“Topological Data Analysis (TDA) has emerged as a powerful mathematical framework for capturing the geometric and topological characteristics of tumor structures, with persistent homology [3,4] serving as a central method.”
Explainable anomaly detection and localization for ADHD in MRI using topological features
Wang, P., Duan, J., Xing, Y., Xu, R., Xu, A., Chen, H., Hu, S., Wang, T., & Liu, S. (2026) · Pattern Recognition · doi:10.1016/j.patcog.2026.114347
Information networks: principles, axioms, and higher-order structures
Pisarchik, A. N. (2026) · The European Physical Journal Special Topics · doi:10.1140/epjs/s11734-026-02469-7
Deneko, E., Hysenlliu, M., Dhoska, K., & Annuk, A. (2026) · Buildings · doi:10.3390/buildings16152963
Charting higher-order models of brain function beyond pairwise interactions
Santoro, A., Neri, M., Poetto, S., Orsenigo, D., Diano, M., Gatica, M., & Petri, G. (2026) · Nature Communications · doi:10.1038/s41467-026-75959-w · Cited by 1
Improving topological detection of weather regimes in climate dynamical systems
Anbouhi, S. (2026) · Climate Dynamics · doi:10.1007/s00382-026-08296-9
“For further background on persistent homology, we refer the reader to (Chazal et al. 2021; Edelsbrunner et al. 2002; Ghrist 2008 ; Edelsbrunner and Harer 2010; Otter et al. 2017).”
Geometry and Persistent-Cluster Theory for Air Blast Damage PredictionPreprint
Marjanishvili, S. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7227932
Rojas, L., Moraga, P., Peña, Á., et al. (2026) · Coatings · doi:10.3390/coatings16080921
Alsaadawi, R., & Mukhopadhyay, N. D. (2026) · Statistical Analysis and Data Mining The ASA Data Science Journal · doi:10.1002/sam.70105
Martindale, T., Omwenga, M. M., & Seah, Y. Y. (2026) · International Conference on Artificial Intelligence, Robotics and Control (AIRC) · doi:10.1109/airc69745.2026.11631313
Meng, C. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7289255
McMahon, E. (2026) · Neuroscience Informatics · doi:10.1016/j.neuri.2026.100288
The friendship quotient: A measure of local neighborhood persistence in reconstructed phase space
Bhattacharya, J. (2026) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2026.119079
Zargham, M. (2026) · INCOSE International Symposium · doi:10.1002/iis2.70203
“Ghrist’s work spans both theoretical foundations and accessible exposi-tion—from the barcodes paper (Ghrist, 2008) introducing persistent homology as a tool for topological data analysis (TDA) to the comprehensive text-book Elementary Applied Topology (Ghrist, 2014) that makes these methods…”
Alsulami, S. (2026) · AIMS Mathematics · doi:10.3934/math.20261180
TopoFormer : a topological-geometric neural network for accurate protein solubility prediction
Kumar, S., Harman, O., Song, S., Kim, Y., Paton, R. S., & Kim, S. (2026) · Digital Discovery · doi:10.1039/d6dd00219f
Al-Khalayleh, N., Jouini, M. S., Kirane, M., Najman, L. A., & Bouchaala, F. (2026) · Research Square · doi:10.21203/rs.3.rs-10592448/v1
Self-supervised Spline Autoencoder for Smooth Latent Dynamics in Echocardiography
Kwan, L. C. J., & Chung, A. C. S. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-38069-2_59
Shymkovych, V., & Hryshyn, O. (2026) · International Journal of Data Science and Analytics · doi:10.1007/s41060-026-01304-5
“Here, shape refers to connected components, cycles, cavities, nesting of groups, boundaries of regions, and the ways in which local parts of a system are coordinated with one another [73, 74].”
Enhancing structural to functional brain network prediction using topological graph learning
Rajakaruna, S. N. K., Dey, A. K., & Chatterje, S. (2026) · Frontiers in Neuroscience · doi:10.3389/fnins.2026.1939985
Li, Z. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0349045
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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