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
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Sorted by citations
Items 301–400 of 1,274
Kim, S., Jung, W. H., Kim, S. N., Jang, J. H., & Kwon, J. S. (2013) · Frontiers in Human Neuroscience · doi:10.3389/fnhum.2013.00302 · Cited by 20
“…| Article 302 | 5 by graph filtration (Lee et al., 2011), the process in which the succeeding network embeds the preceding network with the decreasing threshold of correlation (or the increasing “epsilon” distance in constructing Rips complexes (Ghrist, 2007) in an N-dimensional similarity space).”
BackgroundHuckemann, S., Kim, K.-R., Munk, A., et al. (2016) · Bernoulli · doi:10.3150/15-bej722 · Cited by 19
“In fact, these expand the general scheme of topological persistence introduced by Edelsbrunner et al. [17], cf. also Ghrist [23], Carlsson [8] among many others.”
BackgroundAlgebraic topology for computer vision
Freedman, D. Z., Chen, C., & Freedman, D. Z. (2009) · Cited by 19
Multidimensional Interleavings and Applications to Topological InferencePreprint
Lesnick, M. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1206.1365 · Cited by 19
“See the reviews [22, 26, 4] and the textbook [23] for treatment of the basics, and Sections 2.”
MethodSimplex2Vec embeddings for community detection in simplicial complexesPreprint
Billings, J., Hu, M., Lerda, G., Medvedev, A. N., Mottes, F., Onicas, A. I., Santoro, A., & Petri, G. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.09068 · Cited by 39
“Persistent homology [25, 40] was developed for this purpose of quantifying the shape of data by providing a multiscale description of its topological structure.”
MethodDistinct manifold encoding of navigational information in the subiculum and hippocampus
Nakai, S., Kitanishi, T., & Mizuseki, K. (2024) · Science Advances · doi:10.1126/sciadv.adi4471 · Cited by 25
Kurkin, S., Pisarchik, A. N., Mayorova, L. A., & Hramov, A. E. (2026) · Physics Reports · doi:10.1016/j.physrep.2026.02.004 · Cited by 25
Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis
Cang, Z., Munch, E., & Wei, G. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00057-9 · Cited by 19
“Persistent homology is often visualized by barcodes (Carlsson et al. 2005; Ghrist 2008) where horizontal line segments called bars represent homology generators that survive over different filtration scales.”
BackgroundTopological machine learning for multivariate time series
Wu, C., & Hargreaves, C. A. (2021) · Journal of Experimental & Theoretical Artificial Intelligence · doi:10.1080/0952813x.2021.1871971 · Cited by 17
Topological data analysis of human brain networks through order statistics
Das, S., Anand, D. V., & Chung, M. K. (2023) · PLoS ONE · doi:10.1371/journal.pone.0276419 · Cited by 25
“The collection of intervals from births to deaths that defines persistences are called the barcodes which characterizes the topology of an underlying dataset [14].”
BackgroundRips filtrations for quasimetric spaces and asymmetric functions with stability results
Turner, K. (2019) · Algebraic & Geometric Topology · doi:10.2140/agt.2019.19.1135 · Cited by 18
A new topological descriptor for water network structure
Steinberg, L., Russo, J., & Frey, J. G. (2019) · Journal of Cheminformatics · doi:10.1186/s13321-019-0369-0 · Cited by 17
Limit theorems for process-level Betti numbers for sparse and critical regimes
Owada, T., & Thomas, A. M. (2020) · Advances in Applied Probability · doi:10.1017/apr.2019.50 · Cited by 18
“For surveys of applications of persistent homology, see [15], [10], and [32].”
BackgroundSun, H., Manchester, W. B., & Chen, Y. (2021) · Space Weather · doi:10.1029/2021sw002837 · Cited by 18
“A graph showing the birth-death pair of all loops is called the persistence diagram (Ghrist, 2008; Kaczynski et al., 2004; Munch, 2017; Zomorodian, 2012).”
Servis, M. J., Sadhu, B., Soderholm, L., et al. (2021) · Journal of Molecular Liquids · doi:10.1016/j.molliq.2021.117743 · Cited by 18
Parallel Computation of Persistent Homology using the Blowup Complex
Lewis, R. H., & Morozov, D. (2015) · ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) · doi:10.1145/2755573.2755587 · Cited by 18
“We refer the reader to the several excellent surveys [9, 10, 11, 12, 13] for a more detailed look at what makes persistence so exciting both to mathematicians and practitioners.”
BackgroundPHDP: Preserving Persistent Homology in Differentially Private Graph Publications
Gao, T., & Li, F. (2019) · IEEE Conference on Computer Communications (INFOCOM) · doi:10.1109/infocom.2019.8737584 · Cited by 18
“These intervals are called barcode, where each interval means a component or a hole in the corresponding dimension [11].”
BackgroundInfluentialQuantifying Homology Classes II: Localization and StabilityPreprint
Chen, C., & Freedman, D. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0709.2512 · Cited by 17
Topological data analysis and topological deep learning beyond persistent homology: a review
Su, Z., Liu, X., Hamdan, L. B., Maroulas, V., Wu, J., Carlsson, G., & Wei, G.-W. (2025) · Artificial Intelligence Review · doi:10.1007/s10462-025-11462-w · Cited by 38
Sritharan, D., Wang, S., & Hormoz, S. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2100473118 · Cited by 17
Dobešová, L., Gier, T., Kopečná, O., Pagáčová, E., Vičar, T., Bestvater, F., Toufar, J., Bačı́ková, A., Kopel, P., Fedr, R., et al. (2022) · Pharmaceutics · doi:10.3390/pharmaceutics14010166 · Cited by 16
“Various mathematical algorithms can directly be applied on the coordinate data matrices generated by SMLM (the fundamental concept of SMLM and topological analysis procedures have been described elsewhere [51,59,62,63] and in Methods) to calculate spatial distances and topological arrangements of the molecules”
MethodTopological Simplifications of Hypergraphs
Zhou, Y., Rathore, A., Purvine, E., & Wang, B. (2022) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2022.3153895 · Cited by 21
“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].”
BackgroundA Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science
Hoef, L. V., Adams, H., King, E. J., & Ebert‐Uphoff, I. (2022) · Artificial Intelligence for the Earth Systems · doi:10.1175/aies-d-22-0039.1 · Cited by 17
Topological spaces of persistence modules and their properties
Bubenik, P., & Vergili, T. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-018-0022-4 · Cited by 32
Topological Fidelity and Image Thresholding: A Persistent Homology Approach
Chung, Y.-M., & Day, S. (2018) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-018-0802-4 · Cited by 17
“, [13, 19], persistent homology has also been used in a number of image processing settings.”
BackgroundAutomatic Detection of Image Morphing by Topology-based Analysis
Jassim, S., & Asaad, A. (2018) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco.2018.8553317 · Cited by 17
“In the literature, this is known as persistent homology based analysis, (e.g. [15], [19]).”
BackgroundCharacterizing the Shape of Activation Space in Deep Neural Networks
Gebhart, T., Schrater, P., & Hylton, A. (2019) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2019.00254 · Cited by 22
“Background on persistent homology can be found in the Supplemental Materials as well as [6] and [4].”
BackgroundHierarchical phenomena in multicomponent liquids: simulation methods, analysis, chemistry
Servis, M. J., Martínez-Baez, E., & Clark, A. E. (2020) · Physical Chemistry Chemical Physics · doi:10.1039/d0cp00164c · Cited by 18
Topological signal processing and learning: Recent advances and future challenges
Isufi, E., Leus, G. J. T., Beferull‐Lozano, B., Barbarossa, S., & Lorenzo, P. D. (2025) · Signal Processing · doi:10.1016/j.sigpro.2025.109930 · Cited by 20
Learning Topology: Bridging Computational Topology and Machine Learning
Moroni, D., & Pascali, M. A. (2021) · Pattern Recognition and Image Analysis · doi:10.1134/s1054661821030184 · Cited by 16
“They also introduced barcodes , a combinatorial invariant; a complete introduction to persistence homology and its application from the perspective of barcodes is provided by Ghrist in [37].”
BackgroundDuan, S., Wu, X., Zou, Y., et al. (2022) · Structural Health Monitoring · doi:10.1177/14759217221119790 · Cited by 16
Machine learning techniques on homological persistence features for prostate cancer diagnosis
Rammal, A., Assaf, R., Goupil, A., et al. (2022) · BMC Bioinformatics · doi:10.1186/s12859-022-04992-5 · Cited by 16
“Barcode representation has been proposed for the visualization of topological persistence [27], in which various horizontal line segments or bars are utilized to represent the persistence of the topological features.”
BackgroundProbing center vortices and deconfinement in SU(2) lattice gauge theory with persistent homology
Sale, N., Biagio Lucini, A., & Giansiracusa, J. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.107.034501 · Cited by 16
Curiosity as filling, compressing, and reconfiguring knowledge networks
Patankar, S. P., Zhou, D., Lynn, C. W., Kim, J. Z., Ouellet, M., Ju, H., Zurn, P., Lydon‐Staley, D. M., & Bassett, D. S. (2023) · Collective Intelligence · doi:10.1177/26339137231207633 · Cited by 20
“…treating gaps in networks as topological cavities, and by tracking their evolution using techniques from applied algebraic topology (Bianconi, 2021; Ghrist, 2007; Hatcher, 2002) ( In contrast, compression progress theory posits that humans subtract or discard information (Lynn and Bassett, 2021;…”
BackgroundMethodGovc, D., & Škraba, P. (2017) · Foundations of Computational Mathematics · doi:10.1007/s10208-017-9368-6 · Cited by 16
“A powerful tool in TDA is persistent homology, which studies the homology of a filtration rather than a single space.”
BackgroundCluster-based Data Reduction for Persistent Homology
Moitra, A., Malott, N. O., & Wilsey, P. A. (2018) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata.2018.8622440 · Cited by 16
“The ability of TDA to identify shapes despite certain deformations in the space renders it immune to noise and leads to discovering properties of data that are not discernible by conventional methods of data analysis [1], [2].”
BackgroundChung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610 · Cited by 18
Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector
Yesilli, M. C., Tymochko, S., Khasawneh, F. A., et al. (2019) · IEEE International Conference on Machine Learning and Applications (ICMLA) · doi:10.1109/icmla.2019.00200 · Cited by 16
“D persistent homology, but we defer a more thorough treatment to references in the literature such as [30]–[34].”
BackgroundMeasuring human activity spaces from GPS data with density ranking and summary curves
Chen, Y., & Dobra, A. (2020) · The Annals of Applied Statistics · doi:10.1214/19-aoas1311 · Cited by 20
“The Betti number curve is closely related to the barcode plot in topological data analysis (Ghrist (2008), Wasserman (2016)).”
Hausmann, M., Neitzel, C., Bobkova, E., Nagel, D., Hofmann, A., Chramko, T., Smirnova, E., Kopečná, O., Pagáčová, E., Boreyko, A. V., et al. (2020) · Frontiers in Physics · doi:10.3389/fphy.2020.578662 · Cited by 16
Complexity of Supersymmetric Systems and the Cohomology Problem
Cade, C., & Crichigno, P. M. (2024) · Quantum · doi:10.22331/q-2024-04-30-1325 · Cited by 39
“This solves a much harder problem than estimating normalized Betti numbers, does not rely on any assumption regarding the clique-density of the input graph, and might even find application in persistent homology, where one must track holes across multiple complexes [76].”
BackgroundDiaconis, P. W., Holmes, S., & Janson, S. (2012) · Annals of Combinatorics · doi:10.1007/s00026-012-0175-0 · Cited by 16
“For an application of rectangle intersections, see Rim and Nakajima [36] and for sphere intersections see Ghrist [16].”
MethodMedians of populations of persistence diagrams
Turner, K. (2019) · Homology Homotopy and Applications · doi:10.4310/hha.2020.v22.n1.a15 · Cited by 15
Pritchard, Y., Sharma, A., Clarkin, C. E., Ogden, H. E., Mahajan, S., & Sánchez-García, R. J. (2023) · Scientific Reports · doi:10.1038/s41598-023-28985-3 · Cited by 15
Cosmological Probes of Structure Growth and Tests of Gravity
Hou, J., Bautista, J., Berti, M., Cuesta-Lazaro, C., Hernández‐Aguayo, C., Tröster, T., & Zheng, J. (2023) · Universe · doi:10.3390/universe9070302 · Cited by 15
“, [284] and [285,286]) refers to a collection of statistical methods that extract information from the structure in data.”
Topological change of soil microbiota networks for forest resilience under global warming
Gong, H., Wang, H., Wang, Y., Zhang, S., Liu, X., Che, J., Wu, S., Wu, J., Sun, X., Zhang, S., et al. (2024) · Physics of Life Reviews · doi:10.1016/j.plrev.2024.08.001 · Cited by 16
Ramachandran, R. K., Wilson, S., & Berman, S. (2017) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2016.2647641 · Cited by 15
“TDA can be used to compute prominent topological features of a point cloud, which can be presented graphically in a compact fashion with persistence diagrams [24] and barcode diagrams [25].”
MethodData Analysis from Empirical Moments and the Christoffel Function
Pauwels, E., Putinar, M., & Lasserre, J. B. (2020) · Foundations of Computational Mathematics · doi:10.1007/s10208-020-09451-2 · Cited by 15
Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics
Stojisavljević, V., & Zhang, J. (2021) · International Journal of Mathematics · doi:10.1142/s0129167x21500403 · Cited by 15
“For a detailed exposition see [22, 27, 14, 63, 11, 21, 41, 15, 45].”
Guo, G., Zhao, Y., Liu, C., Fu, Y., Xi, X., Jin, L., Shi, D., Wang, L., Duan, Y., Huang, J., et al. (2022) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2022.1024205 · Cited by 16
“During the filtering process, changes in the network topology are visualized using barcodes or persistence diagrams (Carlsson et al., 2005; Ghrist, 2008).”
MethodGhahremani, M., Yoo, J., Chung, S. J., Yoo, K., Ye, J. C., & Jeong, Y. (2018) · Journal of Movement Disorders · doi:10.14802/jmd.17061 · Cited by 15
Lupo, U., Medina-Mardones, A. M., & Tauzin, G. (2022) · Journal of Applied and Computational Topology · doi:10.1007/s41468-022-00093-7 · Cited by 15
Computational Topology Techniques for Characterizing Time-Series Data
Sanderson, N., Shugerman, E., Molnar, S., Meiss, J. D., & Bradley, E. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-68765-0_24 · Cited by 20
“There are many other ways to build simplicial complexes, including the α-complex [8], the Vietoris-Rips complex [12], or even building a complex based on a cubical grid [13].”
BackgroundXia, K., & Wei, G. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1612.01735 · Cited by 22
“The importance of retaining geometric information in topological analysis has been recognized,(30) and topology has been advocated as a new approach for tackling big datasets.(28,34,44,111,119) Most recently, persistent homology has been developed as a powerful tool for analyzing biomolecular topological fingerprints,…”
MethodQuantum Topological Data Analysis with Linear Depth and Exponential SpeedupPreprint
Ubaru, S., Akhalwaya, I. Y., Squillante, M. S., Clarkson, K. L., & Lior Horesh, C. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2108.02811 · Cited by 44
“The factor ( 1) l produces the so-called oriented [14] sum of boundary simplices, which keeps track of neighbouring simplices so that ∂ k 1 ∂ k | s k = 0, given that the boundary of the boundary is empty.”
BackgroundMethodInfluentialChulián, S., Stolz, B. J., Martínez-Rubio, Á., Goñi, C. B., Gutiérrez, J. F. R., Caballero‐Velázquez, T., Quintana, Á. M., Ramírez, M. R., Robleda, A. C., Soler, J. L. F., et al. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011329 · Cited by 14
“A more detailed summary of TDA can be found in the S1 Text, as well as in further references [26, 54, 70].”
BackgroundKulkarni, S., Pharasi, H. K., Vijayaraghavan, S., Kumar, S., Chakraborti, A., & Samal, A. (2024) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2024.129785 · Cited by 18
“We use two primary tools to visualize the persistent homology of our filtration, namely the persistence barcode [89] and persistence diagram [88].”
BackgroundMethodInfluentialPersistent Homology: a Step-by-step Introduction for Newcomers
Fugacci, U., Scaramuccia, S., Iuricich, F., & Floriani, L. D. (2016) · PORTO@Iris · doi:10.2312/stag.20161358 · Cited by 21
“The interval approach has led to the current notion of barcode , widely surveyed in [Ghr08] and first called by this name in [CZCG05].”
BackgroundAssignments to sheaves of pseudometric spaces
Robinson, M. (2020) · Compositionality · doi:10.32408/compositionality-2-2 · Cited by 14
“s of topological data analysis is persistence: one should not select a single threshold, but instead consider the ensemble of all thresholds. This idea led to the persistent homology of a point cloud [8, 35, 6, 7], an idea that has since become quite popular because of its broad applicability and straightforward usage…”
BackgroundTopological data analysis for true step detection in periodic piecewise constant signals
Khasawneh, F. A., & Munch, E. (2018) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2018.0027 · Cited by 23
“ve mathematical guarantees for the resulting change detection which enables accurate identication and counting of the true pulses. The approach we utilize is based on Topological Data Analysis (TDA) [15,16,17,18,19,20] and it is outlined in Section3. The prevalence of Topological Data Analysis (TDA) has exploded in re…”
MethodInfluentialHomological Algebra of Modules over Posets
Miller, E. (2025) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1516361 · Cited by 15
“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…”
Photonic band structure design using persistent homology
Leykam, D., & Angelakis, D. G. (2021) · APL Photonics · doi:10.1063/5.0041084 · Cited by 14
“The characterization of topological properties in the presence of finite resolution or noise can be carried out using persistent homology, which is a tool from the field of topological data analysis (TDA) [15–17].”
BackgroundMethodCircular and unified analysis in network neuroscience
Rubinov, M. (2023) · eLife · doi:10.7554/elife.79559 · Cited by 16
“Persistent-homology barcodes ( Ghrist, 2008 ).”
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
Alexander, Z., Bradley, E., Meiss, J. D., & Sanderson, N. (2015) · SIAM Journal on Applied Dynamical Systems · doi:10.1137/140971415 · Cited by 14
“A large body of research in the field of computational topology has revolved around the concept of persistence [DE95, EH08, Ghr08, Rob99].”
BackgroundCluster Analysis of Haze Episodes Based on Topological Features
Zulkepli, N. F. S., Noorani, M. S. M., Razak, F. A., et al. (2020) · Sustainability · doi:10.3390/su12103985 · Cited by 14
Testing contractibility in planar rips complexes
Chambers, E. W., Erickson, J., & Worah, P. (2008) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1377676.1377721 · Cited by 12
“Applications of these algorithms include problems in graph drawing [20], map simpli.cation [5], simpli.cation and pa rameterization of surface meshes [33, 52], and approximation algorithms [17] and .xed-parameter tractable algorithms [38] for generalizations of planar graphs.”
MethodUsing persistent homology to reveal hidden information in neural dataPreprint
Spreemann, G., Dunn, B. A., Botnan, M. B., & Baas, N. A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1510.06629 · Cited by 20
“The unfamiliar reader may want to read the wordier survey [12].”
BackgroundCategory-specific topological learning of metal–organic frameworks
Chen, D., Chen, C., & Wei, G.-W. (2025) · Journal of Materials Chemistry A · doi:10.1039/d4ta08877h · Cited by 14
Morimoto, J., Conceição, P., Mirth, C. K., & Lihoreau, M. (2022) · The American Naturalist · doi:10.1086/723599 · Cited by 13
Higher-order connection Laplacians for directed simplicial complexes
Gong, X., Higham, D. J., Zygalakis, K. C., & Bianconi, G. (2024) · Journal of Physics Complexity · doi:10.1088/2632-072x/ad353b · Cited by 12
Structural Analysis of Multivariate Point Clouds Using Simplicial Chains
Rieck, B., & Leitte, H. (2014) · Computer Graphics Forum · doi:10.1111/cgf.12398 · Cited by 12
Graph induced complex on point data
Dey, T. K., Fan, F., & Wang, Y. (2015) · Computational Geometry · doi:10.1016/j.comgeo.2015.04.003 · Cited by 13
Weighted-Persistent-Homology-based Machine Learning for RNA Flexibility Analysis
Pun, C. S., Yong, B. Y. S., & Xia, K. (2019) · PLoS ONE · doi:10.1371/journal.pone.0237747 · Cited by 13
Persistent homology analysis of a generalized Aubry-André-Harper model
Yu, H., Xia, S., Angelakis, D. G., Song, D., Chen, Z., & Leykam, D. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.054210 · Cited by 18
CONNECTIVITY PROPERTIES OF SIERPIŃSKI RELATIVES
TAYLOR, T. D. (2011) · Fractals · doi:10.1142/s0218348x11005531 · Cited by 13
Chowdhury, S., Dai, B., & Mémoli, F. (2018) · PLoS ONE · doi:10.1371/journal.pone.0202561 · Cited by 14
Gene Coexpression Network Comparison via Persistent Homology
Duman, A. N., & Pirim, H. (2018) · International Journal of Genomics · doi:10.1155/2018/7329576 · Cited by 13
Dong, Z., Hu, C., Zhou, C., & Lin, H. (2020) · Computers & Graphics · doi:10.1016/j.cag.2020.05.029 · Cited by 13
“A persistence barcode [6, 7], shown in Figure 1 as an instance, is ordered linear segments obtained by embedding each pair into 2D Euclidean space.”
Multidimensional persistent homology is stable
Cerri, A., Fabio, B. D., Ferri, M., Frosini, P., & Landi, C. (2009) · AMS Acta (University of Bologna) · doi:10.6092/unibo/amsacta/2603 · Cited by 14
“Recent surveys on the topic include [13, 19, 28].”
Database queries and constraints via lifting problems
Spivak, D. I. (2013) · Mathematical Structures in Computer Science · doi:10.1017/s0960129513000479 · Cited by 26
The Topological 'Shape' of BrexitPreprint
Stolz, B. J., Harrington, H. A., & Porter, M. A. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2843662 · Cited by 20
“2 Vietoris–Rips Filtration We apply a VR filtration [10] to our point-cloud data.”
MethodInterleaving and Gromov-Hausdorff distancePreprint
Bubenik, P., de Silva, V., & Scott, J. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.06288 · Cited by 13
“Introduction Persistent homology [26, 46] has been a highly successful tool in applied topology [27, 15].”
MethodPersistent Homology for The Quantitative Prediction of Fullerene StabilityPreprint
Xia, K., Feng, X., Tong, Y., et al. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1412.2369 · Cited by 13
A multi-parameter persistence framework for mathematical morphology
Chung, Y.-M., Day, S., & Hu, C.-S. (2022) · Scientific Reports · doi:10.1038/s41598-022-09464-7 · Cited by 16
“Computational topology and the field of topological data analysis offer powerful tools for analyzing structure in data [1, 2, 3, 4, 5].”
MethodArfi, B. (2023) · Quality & Quantity · doi:10.1007/s11135-023-01708-6 · Cited by 14
Triadic percolation induces dynamical topological patterns in higher-order networks
Millán, A. P., Sun, H., Torres, J. J., & Bianconi, G. (2024) · PNAS Nexus · doi:10.1093/pnasnexus/pgae270 · Cited by 13
“The TDA of a set of points proceeds by first generating a simplicial complex that represents the data at different values of a filtration or threshold parameter f s , and then evaluating the homology classes of the simplicial complex as function of f s .”
BackgroundInfluentialComputing persistent homology within Coq/SSReflect
Heras, J., Coquand, T., Mörtberg, A., & Siles, V. (2013) · ACM Transactions on Computational Logic · doi:10.1145/2528929 · Cited by 16
Santarcangelo, E. L., & Scattina, E. (2019) · International Journal of Clinical and Experimental Hypnosis · doi:10.1080/00207144.2019.1649539 · Cited by 12
“Topological data analysis (TDA) (Ghrist, 2008) has shown that the EEG patterns of activation in highs maintaining the actual and imaginatively rotated position of the head is less different than lows’ (Petri, Ibanez-Marcelo, Campioni, Phingyomark, & Santarcangelo, 2018).”
MethodFast Computation of Persistent Homology with Data Reduction and Data Partitioning
Malott, N. O., & Wilsey, P. A. (2019) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata47090.2019.9006572 · Cited by 13
“conventional methods of data analysis [3], [4].”
Persistent homology of Z2 gauge theories
Sehayek, D., & Melko, R. G. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.085111 · Cited by 19
From calcium imaging to graph topology
Blevins, A. S., Bassett, D. S., Scott, E. K., & Vanwalleghem, G. C. (2022) · Network Neuroscience · doi:10.1162/netn_a_00262 · Cited by 11
“Methods from topological data analysis (TDA) aim to provide a quantitative and qualitative description of how the pieces of the system combine to form the larger whole (Blevins & Bassett, 2020; Geniesse et al., 2019; Ghrist, 2007; Munch, 2017).”
MethodDey, A. K., Young, S. J., & Gel, Y. R. (2023) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1093/jrsssa/qnad066 · Cited by 12
A New Approximation Algorithm for the Matching Distance in Multidimensional Persistence
Cerri, A., & Frosini, P. (2020) · Journal of Computational Mathematics · doi:10.4208/jcm.1809-m2018-0043 · Cited by 11
Exploring generalized shape analysis by topological representations
Zhou, Z., Huang, Y., Wang, L., & Tan, T. (2016) · Pattern Recognition Letters · doi:10.1016/j.patrec.2016.04.002 · Cited by 12
Studying the utility preservation in social network anonymization via persistent homology
Gao, T., & Li, F. (2018) · Computers & Security · doi:10.1016/j.cose.2018.04.003 · Cited by 12
“homology [6] is the homology of a filtration.”
BackgroundA Topological Approach to Representational Data Models
Purvine, E., Aksoy, S. G., Joslyn, C., Nowak, K., Praggastis, B., & Robinson, M. (2018) · Lecture notes in computer science · doi:10.1007/978-3-319-92043-6_8 · Cited by 12
“For a rigorous discussion of the persistent homology methodology, the interested reader is directed to [36–38].”
MethodTuning the performance of a computational persistent homology package
Hylton, A., Henselman‐Petrusek, G., Sang, J., & Short, R. (2019) · Software Practice and Experience · doi:10.1002/spe.2678 · Cited by 12
“The topological data produced by using persistent homology can be visualized through a barcode (25).”
BackgroundModuli spaces of morse functions for persistence
Catanzaro, M. J., Curry, J. M., Fasy, B. T., Lazovskis, J., Malen, G., Riess, H., Wang, B., & Zabka, M. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00055-x · Cited by 20
“f Morse theory, as it studies homology groups of sublevel sets connected by inclusion maps, Ma→−Mb(for a≤b). The evolution of shape is captured by what is known as the persistence diagram or barcode [24, 30]. Barcodes enjoy properties such as simplicity, as a barcode is simply a collection of intervals in the real lin…”
BackgroundTopological Data Analysis of Vascular Disease: A Theoretical Framework
Nicponski, J., & Jung, J.-H. (2020) · Frontiers in Applied Mathematics and Statistics · doi:10.3389/fams.2020.00034 · Cited by 12
Bot Detection on Social Networks Using Persistent Homology
Nguyen, M., Aktas, M. E., & Akbaş, E. (2020) · Mathematical and Computational Applications · doi:10.3390/mca25030058 · Cited by 12
“The basic idea is giving shape to data by replacing it with a family of simplicial complexes, which can be considered as a higher dimensional generalization of graphs, and then studying the topology of the resulted shapes to gain knowledge about it [5].”
Background
Metadata
- Authors
- Ghrist, R.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Ghrist, Robert
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2007-10-26
- Volume
- 45
- Issue
- 1
- Pages
- 61-75
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2071391326
- Semantic Scholar
- cad0bff789bb0e626245b21a47791ef76a9cd5c2
- Cited by
- 1,236
- References
- 32
- Citations by year
- 2026: 78 · 2025: 98 · 2024: 81 · 2023: 105 · 2022: 93 · 2021: 111 · 2020: 121 · 2019: 121 · 2018: 86 · 2017: 66 · 2016: 76 · 2015: 46 · 2014: 41 · 2013: 37 · 2012: 18
- FWCI
- 16.70
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- ScholarlyCommons (University of Pennsylvania) (submittedVersion)
- Scholarly Commons (University of Pennsylvania) (submittedVersion)
- Funders
- Defense Advanced Research Projects Agency (HR0011-07-1-0002)
- University of Illinois at Urbana-Champaign
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistent homology, barcodes, computational algebraic topology, feature detection, shape recognition, high-dimensional data, point clouds, natural image classification
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