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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Differential Private Social Network Publication and Persistent Homology Preservation
Gao, T., & Li, F. (2021) · IEEE Transactions on Network Science and Engineering · doi:10.1109/tnse.2021.3106673 · Cited by 12
A Survey on the High Performance Computation of Persistent Homology
Malott, N. O., Chen, S., & Wilsey, P. A. (2022) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2022.3147070 · Cited by 12
“Data analysis with topology has demonstrated excellent results in several fields [1], [2], [3], [4], [5].”
BackgroundMethodEXAMPLES OF USING BINARY CANTOR SETS TO STUDY THE CONNECTIVITY OF SIERPIŃSKI RELATIVES
TAYLOR, T. D., Hudson, C., & Anderson, A. (2012) · Fractals · doi:10.1142/s0218348x12500065 · Cited by 12
Algorithms and theory of computation handbook: special topics and techniques
Atallah, M. J., & Blanton, M. (2010) · Chapman & Hall/CRC eBooks · Cited by 12
Stabilizing the unstable output of persistent homology computations
Bendich, P. L., Bubenik, P., & Wagner, A. (2019) · Journal of Applied and Computational Topology · doi:10.1007/s41468-019-00044-9 · Cited by 11
Topological Neuroscience: Linking Circuits to Function
Curto, C., & Sanderson, N. (2025) · Annual Review of Neuroscience · doi:10.1146/annurev-neuro-112723-034315 · Cited by 17
Transient cell assembly networks encode stable spatial memories
Babichev, A., & Dabaghian, Y. A. (2017) · Scientific Reports · doi:10.1038/s41598-017-03423-3 · Cited by 10
“The list of the Betti numbers of a space X is known as its topological barcode, , which in many cases captures the topological identity of topological spaces 37 .”
BackgroundThe persistent homotopy type distance
Landi, C. (2019) · IRIS UNIMORE (University of Modena and Reggio Emilia) · doi:10.4310/hha.2019.v21.n2.a13 · Cited by 11
Grain analysis of atomic force microscopy images via persistent homology
Duman, A. N. (2020) · Ultramicroscopy · doi:10.1016/j.ultramic.2020.113176 · Cited by 11
Leave-One-Out Kernel Density Estimates for Outlier Detection
Kandanaarachchi, S., & Hyndman, R. J. (2021) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2021.2000425 · Cited by 11
“Readers are referred to Ghrist (2008) and Carlsson (2009) for an overview and Wasserman (2018) for a statistical viewpoint on the subject.”
BackgroundMethodHomology of homologous knotted proteins
Benjamin, K. S., Mukta, L., Moryoussef, G., et al. (2023) · Journal of The Royal Society Interface · doi:10.1098/rsif.2022.0727 · Cited by 11
Persistent Homology over Directed Acyclic Graphs
Chambers, E. W., & Letscher, D. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_2 · Cited by 14
“See any of the recent books or surveys on topological persistence for full coverage of this broad topic and its applications [10, 11, 13, 21].”
BackgroundSynchronization among Groups of Spectators for Highlight Detection in Movies
Muszyński, M., Κωστούλας, Θ., Lombardo, P., Pun, T., & Chanel, G. (2016) · ACM International Conference on Multimedia (ACM MM) · doi:10.1145/2964284.2967229 · Cited by 11
“Finally, we introduce the basics of persistent homology: filtrations and persistence diagrams [17], [18], [8] plotted in Figure 2.”
MethodUsing persistent homology to reveal hidden covariates in systems governed by the kinetic Ising model
Spreemann, G., Dunn, B. A., Botnan, M. B., & Baas, N. A. (2018) · Physical review. E · doi:10.1103/physreve.97.032313 · Cited by 11
Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence
Makarenko, I., Bushby, P., Fletcher, A., et al. (2018) · Journal of Plasma Physics · doi:10.1017/s0022377818000752 · Cited by 11
“tions can be found, for example, in Morozov (2008); Adler et al. (2010); Edelsbrunner & Harer (2010); Adler & Taylor (2011); Edelsbrunner (2014); Toth et al. (2017), while Park et al. (2013), Ghrist (2008), Pranav et al. (2017) and Motta (2018) also provide useful and less formal expositions. 2.1. Topological…”
BackgroundInfluentialComputing discrete Morse complexes from simplicial complexes
Fugacci, U., Iuricich, F., & Floriani, L. D. (2019) · Graphical Models · doi:10.1016/j.gmod.2019.101023 · Cited by 11
“Persistent homology [28, 29, 30] aims at overcoming intrinsic limitations of standard homology by allowing for a multi-scale approach defined through a filtration.”
BackgroundPersistence Cycles for Visual Exploration of Persistent Homology
Iuricich, F. (2021) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2021.3110663 · Cited by 11
“Persistent homology [33], [38], [84] provides a multiscale description of homology by means of a filtration.”
BackgroundNetwork science: Ising states of matter
Sun, H., Panda, R. K., Verdel, R., Rodríguez, Á., Dalmonte, M., & Ginestra Bianconi, A. (2024) · Physical review. E · doi:10.1103/physreve.109.054305 · Cited by 14
Topological Characterization of Haze Episodes Using Persistent Homology
Zulkepli, N. F. S., Noorani, M. S. M., Razak, F. A., Ismail, M., & Alias, M. A. (2019) · Aerosol and Air Quality Research · doi:10.4209/aaqr.2018.08.0315 · Cited by 12
“One can recognize any noise in the barcode by looking at the short lines, while the significant features are signified by the long lines (Ghrist, 2008).”
A Notion of Harmonic Clustering in Simplicial Complexes
Ebli, S., & Spreemann, G. (2019) · IEEE International Conference on Machine Learning and Applications (ICMLA) · doi:10.1109/icmla.2019.00182 · Cited by 11
“topological data analysis (TDA) as a field [8], [9] (see also [10]– [13] for modern introductions and surveys), have confirmed the usefulness of viewing data through a higher-dimensional analog of graphs [14], [15].”
BackgroundPersistence Images: An Alternative Persistent Homology Representation.Preprint
Chepushtanova, S., Emerson, T., Hanson, E., Kirby, M. J. L., Motta, F. C., Neville, R., Peterson, C., Shipman, P. D., & Ziegelmeier, L. (2015) · arXiv (Cornell University) · Cited by 11
Bazaikin, Y., Байков, В. А., Taimanov, I. A., & Yakovlev, A. A. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1302.6885 · Cited by 10
“A detailed account of persistent homology and applications can be found in [8, 9, 10, 11, 12, 13, 14, 15, 16].”
BackgroundPersistence modules: Algebra and algorithmsPreprint
Škraba, P., & Vejdemo‐Johansson, M. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1302.2015 · Cited by 13
“For surveys or books on the subject, we recommend: [6, 18, 23, 36].”
BackgroundKnowledge gaps in the early growth of semantic networksPreprint
Sizemore, A. E., Karuza, E. A., Giusti, C., & Bassett, D. S. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1709.00133 · Cited by 9
“We refer the interested reader to [11, 62, 22] and the Appendix for more details.”
MethodPersistent Homology of Geospatial Data: A Case Study with VotingPreprint
Feng, M., & Porter, M. A. (2019) · doi:10.31235/osf.io/9mbdn · Cited by 11
“r which f b x;d x (x) = 0. If f b x;j(x) 6= 0 for all b x<jl, then xlives forever and we associate the interval [b x;1) to it. The collection of half-open intervals is known as the \barcode" [17] of X, and we use it to visualize the mth persistent homology. Generators with longer associated half-open intervals…”
MethodDATABASE QUERIES AND CONSTRAINTS VIA LIFTING PROBLEMSPreprint
Spivak, D. I. (2016) · Cited by 14
A topological data analysis based classifier
Kindelan, R., Frías, J., Cerda, M., et al. (2023) · Advances in Data Analysis and Classification · doi:10.1007/s11634-023-00548-4 · Cited by 10
Detecting Carbon Nanotube Orientation with Topological Analysis of Scanning Electron Micrographs
Dong, L., Hang, H., Park, J. G., Mio, W., & Liang, R. (2022) · Nanomaterials · doi:10.3390/nano12081251 · Cited by 10
“explanation for this convention may be found in [32,33].”
BackgroundMethodMitochondrial networks through the lens of mathematics
Lewis, G. R., & Marshall, W. F. (2023) · Physical Biology · doi:10.1088/1478-3975/acdcdb · Cited by 11
Maslarova, A., Shin, J. N., Navas-Olivé, A., Vöröslakos, M., Hamer, H. M., Doerfler, A., Henin, S. E., Buzsáki, G., & Liu, A. A. (2025) · Nature Communications · doi:10.1038/s41467-025-66562-6 · Cited by 13
“Persistent homology diagrams 91 were computed adapting the code from previously published works 92 .”
Homological analysis of multi-qubit entanglement
Pierro, A. D., Mancini, S., Memarzadeh, L., et al. (2018) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/123/30006 · Cited by 10
Ryu, H. M., Habeck, C. G., Stern, Y., & Lee, S. (2023) · Human Brain Mapping · doi:10.1002/hbm.26304 · Cited by 13
“Topological data analysis (TDA) is a relatively new and still developing branch of statistics to infer, analyze, and exploit the underlying geometric or topological features of complex data (Boissonnat et al., 2018; Chazal & Michel, 2021; Ghrist, 2008; Hatcher, 2002; Zomorodian & Carlsson, 2005).”
BackgroundA novel simplicial complex representation of social media texts: The case of Twitter
Ghahremani, Y., & Amiri, B. (2023) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2023.113642 · Cited by 10
-connectedness, finite approximations, shape theory and coarse graining in hyperspaces
Morón, M. A., Cuchillo-Ibanez, E., & Luzón, A. (2008) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2008.05.016 · Cited by 10
Chepushtanova, S., Kirby, M. J. L., Peterson, C., et al. (2015) · IEEE International Geoscience and Remote Sensing Symposium (IGARSS) · doi:10.1109/igarss.2015.7325797 · Cited by 10
“The Betti numbers, as functions of the scale ǫ, can be visualized in a distinct barcode for each dimension k [6].”
BackgroundSmooth Summaries of Persistence Diagrams and Texture Classification
Chung, Y.-M., Hull, M., & Lawson, A. (2020) · IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) · doi:10.1109/cvprw50498.2020.00428 · Cited by 10
“Topological Data Analysis (TDA) is a field of research lying at the intersection of mathematics, statistics, and computer science that is concerned with understanding data through its shape (see survey articles and references therein [8; 9; 21; 42; 13]).”
BackgroundPersistent homology of fractional Gaussian noise
Masoomy, H., Askari, B., Najafi, M. N., & Movahed, M. S. (2021) · Physical review. E · doi:10.1103/physreve.104.034116 · Cited by 10
Persistent Tor-algebra for protein–protein interaction analysis
Liu, X., Feng, H., Lü, Z., & Xia, K. (2023) · Briefings in Bioinformatics · doi:10.1093/bib/bbad046 · Cited by 11
Zinc Finger Proteins and the 3D Organization of Chromosomes
Feinauer, C., Hofmann, A., Goldt, S., Liu, L., Máté, G., & Heermann, D. W. (2013) · Advances in protein chemistry and structural biology · doi:10.1016/b978-0-12-410523-2.00003-1 · Cited by 10
ROLE OF LOCAL GEOMETRY IN ROBUSTNESS OF POWER GRID NETWORKS
Islambekov, U., Dey, A. K., Gel, Y. R., et al. (2018) · IEEE Global Conference on Signal and Information Processing · doi:10.1109/globalsip.2018.8646573 · Cited by 10
“As a result, barcodes provide an insight on the significance of various topological features and allow to distinguish persistent features from topological noise [16].”
BackgroundA Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information
Joslyn, C., Charles, L., Deperno, C. S., Gould, N. P., Nowak, K., Praggastis, B., Purvine, E., Robinson, M., Strules, J., & Whitney, P. (2020) · Sensors · doi:10.3390/s20123418 · Cited by 9
Blevins, A. S., & Bassett, D. S. (2020) · Springer eBooks · doi:10.1007/978-3-319-70658-0_87-1 · Cited by 9
Distance Functions, Critical Points, and the Topology of Random \v{C}ech ComplexesPreprint
Bobrowski, O., & Adler, R. J. (2011) · arXiv (Cornell University) · Cited by 10
The population dynamics of a canonical cognitive circuitPreprint
Chaudhuri, R., Gerçek, B., Pandey, B., Peyrache, A., & Fiete, I. (2019) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/516021 · Cited by 10
Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations
Schiff, Y., Chenthamarakshan, V., Hoffman, S. C., Ramamurthy, K. N., & Das, P. (2022) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp43922.2022.9747088 · Cited by 9
“At the same time, TDA representations are a natural fit for such generative models for several other reasons: They incorporate global topological information (such as presence of connected components, holes, and voids in molecules) [18, 19], which cannot be captured by GNNs or point cloud-based NNs.”
BackgroundNetwork modeling and topology of aging
Li, F., Yang, D., Wu, S., Xue, C., Sang, M., Liu, X., Che, J., Wu, J., Gragnoli, C., Griffin, C. H., et al. (2024) · Physics Reports · doi:10.1016/j.physrep.2024.10.006 · Cited by 9
Grouping real functions defined on 3D surfaces
Biasotti, S., Spagnuolo, M., & Falcidieno, B. (2013) · Computers & Graphics · doi:10.1016/j.cag.2013.05.007 · Cited by 9
Topological features of multivariate distributions: Dependency on the covariance matrix
Aromi, L. L., Katz, Y. A., & Vives, J. (2021) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2021.105996 · Cited by 9
Applications of persistent homology in nuclear collisions
Hamilton, G. A., Dore, T., & Plumberg, C. J. (2022) · Physical Review C · doi:10.1103/physrevc.106.064912 · Cited by 9
Dong, Z., Lin, H., Zhou, C., Zhang, B., & Li, G. (2024) · Machine Learning · doi:10.1007/s10994-023-06492-w · Cited by 11
“These multidimensional homological features are represented as a persistence diagram (PD) or a barcode [3].”
BackgroundInfluentialPersistent Memories in Transient Networks
Babichev, A., & Dabaghian, Y. A. (2017) · Springer proceedings in physics · doi:10.1007/978-3-319-47810-4_14 · Cited by 12
Spanning Trees as Approximation of Data Structures
Alcaide, D., & Aerts, J. A. (2020) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2020.2995465 · Cited by 9
“The stability of topological features is visualized in a so-called barcode [25] and analyzed to identify suitable parameterization [26], [27].”
BackgroundTopological analysis of traffic pace via persistent homology*
Carmody, D., & Sowers, R. B. (2020) · Journal of Physics Complexity · doi:10.1088/2632-072x/abc96a · Cited by 9
“Using the Vietoris–Rips complexes, persistent homology allows one to reliably reconstruct the topology of the surface, even in the presence of sampling noise, and quantify how robust that reconstruction is to the sampling noise via a barcode [Ghr08].”
MethodDigital image reduction for the analysis of topological changes in the pore space of rock matrix
Prokhorov, D. I., Lisitsa, V. V., & Bazaikin, Y. (2021) · Computers and Geotechnics · doi:10.1016/j.compgeo.2021.104171 · Cited by 9
Chung, M. K., Garcia‐Ramos, C., de Paiva, F. B., Mathis, J. R., Prabhakaran, V., Nair, V. A., Meyerand, M. E., Hermann, B. P., Binder, J. R., & Struck, A. F. (2023) · NeuroImage · doi:10.1016/j.neuroimage.2023.120436 · Cited by 15
“The Rips complex 𝜖 is a simplicial complex, where 𝑘 -simplices are formed by ( 𝑘 +1) nodes that are pairwise within distance 𝜖 (Ghrist, 2008).”
Persistent extensions and analogous bars: data-induced relations between persistence barcodes
Yoon, H. R., Ghrist, R., & Giusti, C. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00115-y · Cited by 14
“A more complete development of this material can be found in, for example, [9, 20, 3, 10].”
BackgroundSelf-citationSpectral sequences, exact couples and persistent homology of filtrations
Basu, S., & Parida, L. (2016) · Expositiones Mathematicae · doi:10.1016/j.exmath.2016.06.007 · Cited by 9
“We refer the reader to survey articles [6, 11, 16] for details regarding these applications.”
BackgroundAnalyzing wireless communication network vulnerability with homological invariants
Robinson, M. (2014) · IEEE Global Conference on Signal and Information Processing · doi:10.1109/globalsip.2014.7032250 · Cited by 9
“Following standard practice [25, 26], an element of a ∈ Hk(X \ roi L•) is said to be born at Q if a ∈ Hk(X \ roi LQ) is nontrivial, and its preimage is trivial in all Hk(X \ roi LP ) with P < Q.”
BackgroundHigher Interpolation and Extension for Persistence Modules
Bubenik, P., de Silva, V., & Nanda, V. (2017) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/16m1100472 · Cited by 9
“The combination of rigorously developed foundations [6, 10], efficient computability [21, 25], and stability properties [8, 11] has resulted in the widespread adoption of topological persistence [15, 25] as a technique for the analysis of large and complex datasets [7, 16, 22].”
BackgroundTopological Data Analysis: Developments and Applications
Motta, F. C. (2017) · Springer eBooks · doi:10.1007/978-3-319-58895-7_19 · Cited by 9
Geometry and topology of the space of sonar target echos
Robinson, M., Fennell, S., DiZio, B., & Dumiak, J. (2018) · The Journal of the Acoustical Society of America · doi:10.1121/1.5027825 · Cited by 10
Kashiwara, M., & Schapira, P. (2019) · International Mathematics Research Notices · doi:10.1093/imrn/rnz145 · Cited by 9
Random geometric complexes in the thermodynamic regimePreprint
Yogeshwaran, D., Subag, E., & Adler, R. J. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1403.1164 · Cited by 9
“rest in random geometric complexes was stimulated by their connections to topological data analysis and, more broadly, applied topology. There are a number of accessible surveys on this subject (e.g. [6, 9, 12, 15, 45]), all of which share a common theme of studying topological invariants of simplicial complexes built…”
BackgroundCrackle: The Persistent Homology of NoisePreprint
Adler, R. J., Bobrowski, O., & Weinberger, S. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1301.1466 · Cited by 8
“Some recent references are [2,3,7,13,16] with two reviews, from different aspects, in [1] and [10].”
BackgroundCrawford, L. A., Monod, A., Chen, A., Mukherjee, S., & Rabadán, R. (2016) · arXiv (Cornell University) · Cited by 8
Topological Data Analysis for Object DataPreprint
Patrangenaru, V., Bubenik, P., Paige, R. L., et al. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1804.10255 · Cited by 9
Computing Wasserstein Distance for Persistence Diagrams on a Quantum ComputerPreprint
Berwald, J., Gottlieb, J. M., & Munch, E. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1809.06433 · Cited by 28
Computing Topological Persistence for Simplicial MapsPreprint
Dey, T. K., Fan, F., & Wang, Y. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1208.5018 · Cited by 9
“It is known that the persistence diagram [11] of Vietoris-Rips (Rips in short) filtrations provides avenues for topological analysis of data [1, 18, 21].”
BackgroundThe growing topology of the C. elegans connectomePreprint
Helm, A., Blevins, A. S., & Bassett, D. S. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.12.31.424985 · Cited by 8
PERCEPT: A New Online Change-Point Detection Method using Topological Data Analysis
Zheng, X., Mak, S., Xie, L., & Xie, Y. (2022) · Technometrics · doi:10.1080/00401706.2022.2124312 · Cited by 8
“Next, we introduce the bottleneck distance, a standard metric for topological distance (Ghrist, 2008).”
BackgroundMethodInfluentialRelational Persistent Homology for Multispecies Data with Application to the Tumor Microenvironment
Stolz, B. J., Dhesi, J., Bull, J. A., Harrington, H. A., Byrne, H. M., & Yoon, I. H. R. (2024) · Bulletin of Mathematical Biology · doi:10.1007/s11538-024-01353-6 · Cited by 10
“For details of PH, see Ghrist (2008); Edelsbrunner and Harer (2008); Carlsson (2009); Edelsbrunner et al (2000).”
A Taxonomy of Neuroscientific Strategies Based on Interaction Orders
Neri, M., Brovelli, A., Castro, S., Fraisopi, F., Gatica, M., Herzog, R. A., Mediano, P. A. M., Mindlin, I., Petri, G., Bor, D., et al. (2025) · European Journal of Neuroscience · doi:10.1111/ejn.16676 · Cited by 11
“One of the key tools in TDA, persistent homology (Ghrist 2007; Zomorodian and Carlsson 2004), has been successfully applied to analyse collective dynamics of neuronal populations, uncovering robust topological features that correlate with different cognitive states or behavioural tasks (Curto and…”
Ellis, S. P., & Klein, A. (2014) · Homology Homotopy and Applications · doi:10.4310/hha.2014.v16.n1.a14 · Cited by 8
“Recently there has been much interest in using persistent homology for data analysis, e.g., [Ghrist, 2008] and [Carlsson, 2009].”
MethodResting state dynamic functional connectivity: Network topology analysis
Knyazeva, I., Poyda, A. A., Orlov, V. A., Verkhlyutov, V. M., Makarenko, N., Kozlov, S., Velichkovsky, B. M., & Ushakov, V. (2017) · Biologically Inspired Cognitive Architectures · doi:10.1016/j.bica.2017.10.001 · Cited by 9
Bizhani, M., & Ardakani, O. H. (2021) · Energy & Fuels · doi:10.1021/acs.energyfuels.1c03255 · Cited by 8
Zigzag persistence for coral reef resilience using a stochastic spatial model
McDonald, R. A., Neuhausler, R., Robinson, M. J., Larsen, L. G., Harrington, H. A., & Bruna, M. (2023) · Journal of The Royal Society Interface · doi:10.1098/rsif.2023.0280 · Cited by 10
“Persistent homology (PH), a prominent tool in TDA, takes in data and outputs a multiscale topological summary of features, such as connected components, loops, and cavities [25].”
MethodDynamic topological data analysis of functional human brain networks
Chung, M. K., Das, S. R., & Ombao, H. (2023) · Foundations of Data Science · doi:10.3934/fods.2023013 · Cited by 11
“Within TDA, persistent homology based approaches have become increasingly popular as a tool for analyzing different brain imaging data because they can capture the persistences [51, 60] of different topological features that are robust under different scales [13, 34, 44, 22].”
Backgroundk-means clustering for persistent homology
Cao, Y., Leung, P., & Monod, A. (2024) · Advances in Data Analysis and Classification · doi:10.1007/s11634-023-00578-y · Cited by 8
“For each finite dimensional H • ( X, ϵ i ), the persistence module can be decomposed into rank one summands corresponding to birth and death times of homology classes (Chazal et al. , 2016) (Ghrist, 2008).”
BackgroundBeshkov, K., Fyhn, M., Hafting, T., & Einevoll, G. T. (2024) · iScience · doi:10.1016/j.isci.2024.109370 · Cited by 9
Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning
Su, Z., Tong, Y., & Wei, G.-W. (2024) · AIMS Mathematics · doi:10.3934/math.20241333 · Cited by 11
“A major feature of persistent homology is its multi-scale analysis, which creates a family of topological spaces from the original data to track the topological persistence, i.e., the lifespan of topological invariants across scales [5, 27].”
Methods of Computational Topology for the Analysis of Dynamics of Active Regions of the Sun
Makarenko, N., Malkova, D., Machin, M., et al. (2014) · Journal of Mathematical Sciences · doi:10.1007/s10958-014-2170-y · Cited by 8
Generative Design of Electromagnetic Structures Through Bayesian Learning
Patel, R., Roy, K., Choi, J., & Han, K. J. (2017) · IEEE Transactions on Magnetics · doi:10.1109/tmag.2017.2762351 · Cited by 7
“Our proposed BC L uses statistical clique embedded in the data for searching optimum design parameters [6]–[9].”
MethodPersistence barcodes and Laplace eigenfunctions on surfaces
Polterovich, I., Polterovich, L., & Stojisavljević, V. (2018) · Geometriae Dedicata · doi:10.1007/s10711-018-0383-9 · Cited by 8
“We refer the reader to articles [2,9,16,21,44] and monographs [10,15,30] for an introduction into this rapidly developing subject and further details.”
BackgroundExpert Selection in Prediction Markets With Homological Invariants
Bousoño‐Calzón, C., Molina-Bulla, H., Escudero-Garzas, J. J., & Herrera-Galvez, F. J. (2018) · IEEE Access · doi:10.1109/access.2018.2846878 · Cited by 8
“The result of the persistent homology tool for each player is a kind of topological signature for him, which can be depicted either as a barcode or as a rotated diagram [36], see Fig.”
BackgroundFrom Geometry to Topology: Inverse Theorems for Distributed Persistence
Solomon, E., Wagner, A., & Bendich, P. (2021) · Journal of computational geometry · doi:10.20382/jocg.v14i2a8 · Cited by 8
Multiscale persistent topological descriptor for porous structure retrieval
Dong, Z., Pu, J., & Lin, H. (2021) · Computer Aided Geometric Design · doi:10.1016/j.cagd.2021.102004 · Cited by 8
Lattice Paths for Persistent Diagrams
Chung, M. K., & Ombao, H. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-87444-5_8 · Cited by 9
Representation of the fermionic boundary operator
Akhalwaya, I. Y., He, Y., Lior Horesh, C., Jejjala, V., Kirby, W., Naidoo, K., & Ubaru, S. (2022) · Physical Review A · doi:10.1103/physreva.106.022407 · Cited by 8
“TDA permits representing large volumes of data using a few global and interpretable features called Betti numbers [37].”
MethodLin, B. (2022) · Algorithms · doi:10.3390/a15100371 · Cited by 10
“Figure 1 is a schematic diagram outlining how to perform a filtration process (by sweeping the ), document the “birth” and “death” of each complexes (the colored lines of various length in the chart), and generate this as a “barcode” representation [13] for the downstream analyses.”
BackgroundNetworks and Islands of Genome Nano-architecture and Their Potential Relevance for Radiation Biology
Hausmann, M., Hildenbrand, G. L., & Pilarczyk, G. (2022) · Results and problems in cell differentiation · doi:10.1007/978-3-031-06573-6_1 · Cited by 8
Adsorbate Organization Characterized by Sublevelset Persistent Homology
Kumar, N., & Clark, A. E. (2023) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.3c00090 · Cited by 8
“[29 ] Analysis of the distribution of topological features can be used to create metrics or descriptors of the image - as has been done in image of liquid crystal nanocomposites[26] or immune cells.”
$$A_\infty $$ A ∞ -persistence
Guillamon, F. B., & Murillo, A. (2014) · Applicable Algebra in Engineering Communication and Computing · doi:10.1007/s00200-014-0241-4 · Cited by 10
“When dealing with persistent homology over a field, its effective computation as well as its known geometrical meaning, are best understood through the barcode of the homology of the given filtration K [4,10], which is obtained via the classification of finite graded modules over a principal ideal domain.”
BackgroundMethodFiltrations induced by continuous functions
Fabio, B. D., & Frosini, P. (2013) · Topology and its Applications · doi:10.1016/j.topol.2013.05.013 · Cited by 8
“…introducing a nested collection of sets (usually carriers of simplicial complexes), the latter of giving a continuous function from a topological space to R or Rn (called a filtering function), whose sub-level sets represent the elements of the considered filtration (cf., e.g., [11, 15]).”
BackgroundAbduction: A categorical characterization
Tohmé, F., Caterina, G., & Gangle, R. (2015) · Journal of Applied Logic · doi:10.1016/j.jal.2014.12.004 · Cited by 8
“Heuristic tools like plotting the data, extracting statistics, running simulations or looking at barcodes [15] may help to state hypotheses about the features of the process that in fact generated the data [25].”
MethodG-invariant persistent homology
Frosini, P. (2014) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.3139 · Cited by 9
“For more details about persistent homology and its applications we refer the reader to [6, 7, 9, 14, 20].”
BackgroundMind the Gap: A Study in Global Development Through Persistent Homology
Banman, A., & Ziegelmeier, L. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_8 · Cited by 7
“For foundational material and overviews of computational homology in the setting of persistence, see [5, 6, 8, 2, 17].”
BackgroundMethodInfluentialPersistent homology-based gait recognition robust to upper body variations
León, J. L., Alonso-Baryolo, R., García-Reyes, E. B., et al. (2016) · Proceedings - International Conference on Pattern Recognition/Proceedings/International Conference on Pattern Recognition · doi:10.1109/icpr.2016.7899780 · Cited by 8
“For a detailed introduction see [11], [12].”
Duman, A. N., Yilbas, B. S., Pirim, H., et al. (2017) · Coatings · doi:10.3390/coatings7090139 · Cited by 8
Perkins, M., & Daniels, K. (2017) · International Conference on Information Visualisation (IV) · doi:10.1109/iv.2017.9 · Cited by 8
“TDA is of interest for data exploration and data mining ([17],”
BackgroundA Numerical Approach for the Filtered Generalized Čech Complex
Espinoza, J. F., Hernández-Amador, R., Hernández, H. A., & Ramonetti-Valencia, B. (2019) · Algorithms · doi:10.3390/a13010011 · Cited by 8
“dition, we show how the algorithm yields the minimal enclosing ball for a nite set of points in the plane. 1. Introduction Recently, in the study of data point clouds from a topological approach (cf. [4, 5, 11, 16, 24]) the need to develop algorithms to construct dierent simplicial structures has arisen, such as the V…”
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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