The DOI matches its Crossref record; 32 of the 32 references it cites were found in a registry or an index; 8 of the 28 sources its claims cite were checked for support.
Barcodes: The persistent topology of data
Open accessAbstract
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— persistent homology —and a novel representation of this algebraic characterization— barcodes . We sketch an application of these techniques to the classification of natural images.
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Claims the paper makes
17 Own interpretation · 13 Sources not found · 12 No basis given · 5 Supported · 3 Partly supported · 2 Not checked yet · 2 Own result · 1 No full text of the sources · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“R ↩→ C √ 2 ↩→ R √ 2 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.1. Persistence.
What the sources say
- Supportedcites [10] DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358. source
Theorem 2.5 (§2.3; page not provided) gives a chain of inclusions from a Rips complex through a Čech complex to a Rips complex at the larger parameter. Its required scale ratio is √(2d/(d+1)), which is at most √2, so the claimed √2-scaled chain follows for a positive parameter. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
What the sources say
- Supportedcites [7] CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271. source
The Main Theorem (Section 3.1, p. 7) says persistence diagrams are stable under small-amplitude perturbations of a function. The Introduction (p. 2) says its assumptions are satisfied by Morse functions, and Applications (p. 16) identifies barcodes as persistence diagrams represented by intervals. (LLM, conf=0.92)
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“H * (C; F ) ∼ = i x t i • F [x] ⊕ ⎛ ⎝ j x r j • (F [x]/(x s j • F [x])) ⎞ ⎠ .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
What the sources say
- Supportedcites [22] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
Section 3.2 (p. 12) says that finite-type persistence modules over a field correspond to finitely generated graded modules over F[t] and describes their decomposition into free modules and quotients by powers of t. Section 2.1 (p. 6) identifies these as the free and torsional portions of the decomposition. (LLM, conf=0.91)
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“However, the Čech complex and various topologically equivalent subcomplexes (e.g., the alpha complex of [13]) are delicate objects to compute, relying on the precise distances between the nodes in E n .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
What the sources say
- Supportedcites [13] EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72. source
The citation identifies the alpha complex as an example of the subcomplexes being discussed. Section 3.3 defines alpha-complex membership using circumsphere radii, and Section 6 (Implementation) describes the need for exact arithmetic and symbolic perturbation to ensure correct computation. The paper need not independently establish the sentence’s broader claim about Čech complexes for this example citation to be supported. (LLM, conf=0.88)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The full data set consists of roughly 8,000,000 points in E 9 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Partly supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. sourceExperimental — accuracy not yet measured
Section 2 (page unavailable) reports about 4.2 million optical patches and 0.79 million range patches—roughly 5 million total, not 8 million. Section 3 (page unavailable) represents each original 3 × 3 patch as a vector in nine-dimensional space, although the preprocessed data lie on a seven-dimensional sphere. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The witness complex of Carlsson and de Silva [8,9,14] is one solution to this problem.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
What the sources say
- Partly supportedcites [8] D V. de Silva, “A weak definition of Delaunay triangulation”, preprint (2003). sourceExperimental — accuracy not yet measured
Section 6 (p. 8) describes using a subsample as the vertices of a topology-approximating simplicial complex and using weak witnesses to construct it. That supports the witness-complex alternative, but our copy does not discuss Rips complexes or say that their unmanageable size is the problem this construction solves. (LLM, conf=0.82)
- Not checked — no full text of the sourcecites [9] DC V. de Silva and G. Carlsson, “Topological estimation using witness complexes”, in SPBG’04 Symposium on Point-Based Graphics (2004), 157-166. source
- Not matchedcites [14] GO L. Guibas and S. Oudot, “Reconstruction using witness complexes”, in Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms (2007).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M on a topological seven-sphere S 7 ⊂ E 8 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. source
Section 3 (Preprocessing; page unavailable) says the authors subtract each patch’s mean, normalize its contrast, and transform the resulting points onto a seven-dimensional sphere in R⁸. The Introduction (page unavailable) says the study focuses on high-contrast data, including regions far from the origin. More precisely, preprocessing uses the logarithms of intensity values. (LLM, conf=0.92)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The Čech theorem (or, equivalently, the “nerve theorem”) states that C has the homotopy type of the union of closed radius /2 balls about the point set {x α }.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As with β k , the barcode for H k does not give any information about the finer structure of the homology, but merely a continuously parameterized rank.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The torsional elements correspond to those homology generators which appear at parameter r j and disappear at parameter r j + s j .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Probably not a substantial claim. 10
- It may state two separate claims as one. 16
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3. Example: natural images
What the sources say
- Partly supportedcites [20] HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366. sourceExperimental — accuracy not yet measured
The Images section describes a collection photographed with a digital camera, but gives its size as 4,212 images, not 4,167. It says the images came from woods, open landscapes and urban areas, often in consecutive series, rather than describing them as random outdoor scenes. The paper was published in 1998, but the passage does not specify when the photographs were taken. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Partly supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, in Figure 4, one sees (from a very coarse sampling) that the point cloud likely represents a connected object with one or two significant 'holes' as measured by H 1 and no significant higher homology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the figure, it certainly appears as though an ideal choice of , if it exists, is rare: by the time is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
hedgeddirectSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“So coarse a reduction of natural images (into three-by-three squares of greyscale intensities) still leads to a point cloud of too high a dimension to visualize.Worse still, what structure there is is blurred and foggy: points appear at first to be distributed over the entire S 7 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.2. "Hover through the fog".
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The resulting homology H * (C) retains the structure of an R[x]-module, but, unlike the chain module, is not necessarily free.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the combinatorics of the 1-skeleton completely determines the complex, and the Rips complex can be stored as a graph and reconstituted instead of storing the entire boundary operator needed for a Čech complex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The reason is that the Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For sufficiently small, the complex is a discrete set; for sufficiently large, the complex is a single highdimensional simplex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Nor is it easily classified: the Artin-Rees theory from commutative algebra implies that the problem of classifying (finite-type) persistence modules such as H * (C) is equivalent to classifying finitely generated nonnegatively graded R[x]-modules.This is known to be very difficult in, say, Z[x].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In general, it is neither a subcomplex of E n nor does it necessarily behave like an n-dimensional space at all (Figure 2).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“From a computational point of view, the Rips complex is less expensive that the corresponding Čech complex, even though the Rips complex has more simplices (in general).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Despite being both computable and insightful, the homology of a complex associated to a point cloud at a particular is insufficient: it is a mistake to ask which value of is optimal.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2. Algebraic topology for data
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of the data point corresponding to the primary circle reveals a pattern of 3-by-3 patches with one light region and one dark region separated by a linear transition.This nodal curve between light and dark is linear and appears in a circular family parameterized by the angle of the nodal line, as shown in Figure 5.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This necessitates efficient sampling or reduction of the complex with accurate topology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These persistent generators, upon close examination, do correspond to meaningful structures in the data, inspiring a sensible parametrization of the global structure of the data set.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of these three circles reveals that each intersects the highk primary twice, yet the two secondary circles are disjoint.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“[3] reports that a repetition of the experiment with a camera held at a constant angle π/4 yields a data set whose secondary persistent H 1 generators exhibit a bias towards true vertical and true horizontal: the axis of pixellation appears less relevant than the axis of gravity in natural image data.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is worth noting that for chain filtrations arising from realistic data sets, the Rips complexes are of an unmanageable size.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The computation of persistent relative homology is more subtle, since the ensuing parameterized chain complex C is no longer free as an F [x]-module.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Following this analogy, the H 1 primary generator fills the role of a next term in the expansion of the homotopy type of the data set, collating the nodal curve between two contrasting patches.The secondary circles, interpolating between single and dual nodal curves, act as higher-order terms in the expansion, in which horizontal and vertical biases arise.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“However, there are fundamental algebraic difficulties in constructing an analogous theory of persistence modules in this setting [24].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
What the sources say
- Not checked — no full text of the sourcecites [24] ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193. source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As noted in [3], each secondary circle regulates images with three contrasting regions and interpolates between these states and the primary circle.The difference between the two secondary circles lies in their bias for horizontal and vertical stratification respectively.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
The authors state this as an assumption.
Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
“One assumes a finite-type condition that each C i * is finitely generated as an R[x]-module and that the sequence stabilizes in i (in the case of an infinite sequence of chain complexes).”
Citations
Cites
Density Estimation for Statistics and Data Analysis
Silverman, B. W. (2018) · Routledge eBooks · doi:10.1201/9781315140919 · Cited by 15,938
Density Estimation for Statistics and Data Analysis
(1986) · Springer eBooks · doi:10.1007/978-1-4899-3324-9 · Cited by 8,569
Density Estimation for Statistics and Data Analysis
Dehnad, K., & Silverman, B. W. (1987) · Technometrics · doi:10.2307/1269475 · Cited by 5,479
(2001) · Cited by 5,374
Three-dimensional alpha shapes
Edelsbrunner, H., & Mücke, E. P. (1994) · ACM Transactions on Graphics · doi:10.1145/174462.156635 · Cited by 2,488
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Stability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280
Independent component filters of natural images compared with simple cells in primary visual cortex
van Hateren, J. H., & van der Schaaf, A. (1998) · Proceedings of the Royal Society of London. Series B: Biological Sciences · doi:10.1098/rspb.1998.0303 · Cited by 1,106 · cited 1×
“A collection of 4167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
BackgroundTopological persistence and simplification
Edelsbrunner, H., Letscher, D., & Zomorodian, A. J. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.2000.892133 · Cited by 716 · cited 2×
“Persistence, as introduced by Edelsbrunner, Letscher, and Zomo-rodian [12] and refined by Carlsson and Zomorodian [22], is a rigorous response to this problem.”
Method
Cited by
Sorted by citations
Networks beyond pairwise interactions: Structure and dynamics
Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G., & Petri, G. (2020) · Physics Reports · doi:10.1016/j.physrep.2020.05.004 · Cited by 1,879
“Recently the concept has been extended to weighted and growing simplicial complexes [118].”
BackgroundNetworks beyond pairwise interactions: Structure and dynamics
Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J., & Petri, G. (2020) · UCL Discovery (University College London) · Cited by 1,179
A roadmap for the computation of persistent homology
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P., & Harrington, H. A. (2017) · EPJ Data Science · doi:10.1140/epjds/s13688-017-0109-5 · Cited by 769
“If S is a subset of Euclidean space, one can consider a “thickening” S (cid:15) of S given by the union of balls of a certain fixed radius (cid:15) around its points and then compute the ˇCech complex.”
MethodHomological scaffolds of brain functional networks
Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P. J., & Vaccarino, F. (2014) · Journal of The Royal Society Interface · doi:10.1098/rsif.2014.0873 · Cited by 675
“Technical details about the theory of persistent homology and how the computation is performed can be found in the works of Carlsson, Zomorodian and Edelsbrunner [28,31–35].”
Statistical topological data analysis using persistence landscapesPreprint
Bubenik, P. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1207.6437 · Cited by 628
“Of particular interest are the homological features that persist as the resolution changes.”
BackgroundVisualizing high-dimensional data: advances in the past decade
Liu, S., Maljovec, D., Wang, B., Bremer, P.-T., & Pascucci, V. (2017) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2016.2640960 · Cited by 427
What are higher-order networks?
Bick, C., Gross, E., Harrington, H. A., & Schaub, M. T. (2023) · SIAM Review · doi:10.1137/21m1414024 · Cited by 420
“Thus, for an appropriate choice of basis [282] of the homology groups Hn, a barcode summarizes the information carried by the homology groups and the maps f i,j [51, 109], its persistent homology.”
BackgroundChaudhuri, R., Gerçek, B., Pandey, B., Peyrache, A., & Fiete, I. (2019) · Nature Neuroscience · doi:10.1038/s41593-019-0460-x · Cited by 388
Cliques and cavities in the human connectome
Sizemore, A. E., Giusti, C., Kahn, A. E., Vettel, J. M., Betzel, R. F., & Bassett, D. S. (2017) · Journal of Computational Neuroscience · doi:10.1007/s10827-017-0672-6 · Cited by 384
“In order to quantitatively characterize this macroscale structure, we employ an enrichment of networks that comes from the field of algebraic topology [18], developed precisely to understand the interplay between these weak and strong connections in systems [19].”
MethodCang, Z., & Wei, G.-W. (2017) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1005690 · Cited by 330
Attractor and integrator networks in the brain
Khona, M., & Fiete, I. (2022) · Nature reviews. Neuroscience · doi:10.1038/s41583-022-00642-0 · Cited by 356
“rings, torii), especially those of dimension 3, topological data analysis methods become important[112, 113, 114, 25, 115, 116, 106].”
BackgroundThe Why, How, and When of Representations for Complex Systems
Torres, L., Blevins, A. S., Bassett, D. S., & Eliassi‐Rad, T. (2021) · SIAM Review · doi:10.1137/20m1355896 · Cited by 340
“erhaps most often, we study weighted simplicial complexes through the lens of persistent homology, which returns the organization of topological cavities housed within the weighted simplicial complex [220, 35, 79, 142] (see a few recent uses in [184, 82, 153, 193]). Lastly, in hypergraphs we can naturally weight hyper…”
BackgroundThe theory of multidimensional persistence
Carlsson, G., & Zomorodian, A. (2007) · Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07 · doi:10.1145/1247069.1247105 · Cited by 299
Persistent Brain Network Homology From the Perspective of Dendrogram
Lee, H.-K., Kang, H., Chung, M. K., Kim, B.-N., & Lee, D. S. (2012) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2012.2219590 · Cited by 261
“is a simplicial complex whose -simplices correspond to unordered -tuples of points which are pairwise within distance [38].”
BackgroundMethodThe Theory of Multidimensional Persistence
Carlsson, G., & Zomorodian, A. (2009) · Discrete & Computational Geometry · doi:10.1007/s00454-009-9176-0 · Cited by 270
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
Mischaikow, K., & Nanda, V. (2013) · Discrete & Computational Geometry · doi:10.1007/s00454-013-9529-6 · Cited by 277
“The use of topological methods for data analysis is rapidly growing and persistent homology is proving to be one of the more successful techniques [1, 7, 8, 11].”
MethodPersistent homology analysis of protein structure, flexibility, and folding
Xia, K., & Wei, G. (2014) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.2655 · Cited by 254
“Usually, the persistent diagram is visualized through barcodes [47], in which various horizontal line segments or bars are the homology generators lasted over filtration scales.”
BackgroundMethodCang, Z., Mu, L., & Wei, G.-W. (2018) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1005929 · Cited by 257
“One-dimensional (1D) topological invariants generated from persistent homology is often visualized by persistence barcodes [53, 54] and persistence diagrams [55].”
BackgroundRevealing key structural features hidden in liquids and glasses
Tanaka, H., Tong, H., Shi, R., & Russo, J. (2019) · Nature Reviews Physics · doi:10.1038/s42254-019-0053-3 · Cited by 260
Patania, A., Petri, G., & Vaccarino, F. (2017) · EPJ Data Science · doi:10.1140/epjds/s13688-017-0114-8 · Cited by 255
Confidence sets for persistence diagrams
Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., & Singh, A. (2014) · The Annals of Statistics · doi:10.1214/14-aos1252 · Cited by 251
“The reach of A—denoted by reach(A)— is the largest ε ≥ 0 such that each point in A ⊕ ε has a unique projection onto A (Federer (1959)).”
BackgroundQuantum algorithms for topological and geometric analysis of data
Lloyd, S., Garnerone, S., & Zanardi, P. (2016) · Nature Communications · doi:10.1038/ncomms10138 · Cited by 241
“The primary difference between the topological and geometrical algorithms presented here, and algorithms for, for example, constructing clusters 19 , principal components 20 , and support vector machines 21 , is that our topological algorithms require only a small qRAM of size O ( n 2 ).”
BackgroundQaiser, T., Tsang, Y., Taniyama, D., Sakamoto, N., Nakane, K., Epstein, D. B. A., & Rajpoot, N. (2019) · Medical Image Analysis · doi:10.1016/j.media.2019.03.014 · Cited by 221
“Readers are referred to Edelsbrunner and Harer (2008); Carlsson (2009); Ghrist (2008) for a description of the general theory.”
BackgroundPersistent homology of complex networks
Horak, D., Maletić, S., & Rajković, M. (2009) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2009/03/p03034 · Cited by 226
Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian
Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., & Jadbabaie, A. (2020) · SIAM Review · doi:10.1137/18m1201019 · Cited by 268
“m walks on graphs and the graph Laplacian to develop a random walk on simplicial complexes that relates to the Hodge Laplacian. Furthermore, the connection of simplicial complexes to applied topology [21,35] has already led to success in analyzing biological data [23,55], structural and functional connectivity in the…”
BackgroundTopological Strata of Weighted Complex Networks
Petri, G., Scolamiero, M., Donato, I., & Vaccarino, F. (2013) · PLoS ONE · doi:10.1371/journal.pone.0066506 · Cited by 214
Persistence-Based Clustering in Riemannian Manifolds
Chazal, F., Guibas, L. J., Oudot, S., & Škraba, P. (2013) · Journal of the ACM · doi:10.1145/2535927 · Cited by 217
Decomposition of pointwise finite-dimensional persistence modules
Crawley-Boevey, W. (2014) · Journal of Algebra and Its Applications · doi:10.1142/s0219498815500668 · Cited by 211
“) Assuming that this persistence module is a direct sum of interval modules, it is found that the collection of such intervals, the barcode, is a useful invariant of the data; see for example [1, 6].”
BackgroundBianconi, G. (2021) · doi:10.1017/9781108770996 · Cited by 223
Topological Data Analysis of Biological Aggregation Models
Topaz, C. M., Ziegelmeier, L., & Halverson, T. (2015) · PLoS ONE · doi:10.1371/journal.pone.0126383 · Cited by 187
“Foundational papers on this methodology include [35, 37, 38].”
Persistence homology of networks: methods and applications
Aktas, M. E., Akbaş, E., & Fatmaoui, A. E. (2019) · Applied Network Science · doi:10.1007/s41109-019-0179-3 · Cited by 210
Cang, Z., & Wei, G. (2018) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.2914 · Cited by 172
Towards persistence-based reconstruction in euclidean spaces
Chazal, F., & Oudot, S. (2008) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1377676.1377719 · Cited by 175
“[18] V. de Silva and R. Ghrist.”
MethodAnalysis of Big Data in Gait Biomechanics: Current Trends and Future Directions
Phinyomark, A., Petri, G., Ibáñez‐Marcelo, E., Osis, S. T., & Ferber, R. (2017) · Journal of Medical and Biological Engineering · doi:10.1007/s40846-017-0297-2 · Cited by 175
“In this paper, two major TDA techniques are presented, which can be considered as potential big data analytics for running gait analysis: (1) topological simplification, as exemplified by the Mapper algorithm [78–80], and (2) persistent homology [81, 82].”
MethodA persistence landscapes toolbox for topological statistics
Bubenik, P., & Dłotko, P. (2016) · Journal of Symbolic Computation · doi:10.1016/j.jsc.2016.03.009 · Cited by 175
Topological data analysis in biomedicine: A review
Skaf, Y., & Laubenbacher, R. (2022) · Journal of Biomedical Informatics · doi:10.1016/j.jbi.2022.104082 · Cited by 192
Persistence Images: A Stable Vector Representation of Persistent HomologyPreprint
Adams, H., 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) · doi:10.48550/arxiv.1507.06217 · Cited by 517
“The idea of persistent homology is to compute homology at many scales and observe which topological features persist across those scales (Ghrist, 2008; Carlsson, 2009; Edelsbrunner and Harer, 2008).”
BackgroundA User’s Guide to Topological Data Analysis
Munch, E. (2017) · Journal of Learning Analytics · doi:10.18608/jla.2017.42.6 · Cited by 163
“The interested reader should look to the many excellent surveys and books available on the subject (Ghrist, 2008; Carlsson, 2009; Edelsbrunner & Harer, 2010; Vejdemo-Johansson & Skraba, 2016; Ghrist, 2014).”
BackgroundQuantitative Toxicity Prediction Using Topology Based Multitask Deep Neural Networks
Wu, K., & Wei, G.-W. (2018) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.7b00558 · Cited by 160
Simplicial complexes and complex systems
Salnikov, V., Cassese, D., & Lambiotte, R. (2018) · European Journal of Physics · doi:10.1088/1361-6404/aae790 · Cited by 164
Persistent Homology: Theory and Practice
Edelsbrunner, H., & Morozov, D. (2013) · doi:10.4171/120-1/3 · Cited by 161
“Motivated by its significant applications, persistent homology has found repeated exposure in the popular mathematics literature [5, 27, 45] and features prominently in a recent text on computational topology [20].”
Betti numbers in multidimensional persistent homology are stable functions
Cerri, A., Fabio, B. D., Ferri, M., Frosini, P., & Landi, C. (2013) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.2704 · Cited by 138
“Recent surveys on the topic include [19, 20, 26, 35].”
Topology of random simplicial complexes: a survey
Kahle, M. (2014) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/620/12367 · Cited by 144
“1. Motivation. 1.1.1. Randomnessmodelsthenaturalworld. (1)Thefieldoftopologicaldataanalysishasreceivedalotofattentionover thepastseveralyears—see,forexample,thesurveyarticlesofCarlsson [16] and Ghrist [30]. In order to quantify the statistical significance of topologicalfeaturesdetectedwiththesemethods,itwillbehelpfulto…”
BackgroundInfluentialChatter detection in turning using persistent homology
Khasawneh, F. A., & Munch, E. (2015) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2015.09.046 · Cited by 141
Revealing neural correlates of behavior without behavioral measurements
Rubin, A., Sheintuch, L., Brande-Eilat, N., Pinchasof, O., Rechavi, Y., Geva, N., & Ziv, Y. (2019) · Nature Communications · doi:10.1038/s41467-019-12724-2 · Cited by 141
Embedding and approximation theorems for echo state networks
Hart, A., Hook, J., & Dawes, J. (2020) · Neural Networks · doi:10.1016/j.neunet.2020.05.013 · Cited by 122
“For the reader unfamiliar with persistent homology Ghrist (2008) offers an excellent primer.”
BackgroundPersistent homology of time-dependent functional networks constructed from coupled time series
Stolz, B. J., Harrington, H. A., & Porter, M. A. (2017) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4978997 · Cited by 124
Persistent-homology-based machine learning: a survey and a comparative study
Pun, C. S., Lee, S. X., & Xia, K. (2022) · Artificial Intelligence Review · doi:10.1007/s10462-022-10146-z · Cited by 244
“For simplification, we also define the set of k -th dimensional barcodes as, The topological persistence is usually represented in two ways, i.e., persistent barcode (PB) (Ghrist 2008b) and persistent diagram (PD) (Mischaikow and Nanda 2013).”
BackgroundTownsend, J., Micucci, C. P., Hymel, J. H., Maroulas, V., & Vogiatzis, K. D. (2020) · Nature Communications · doi:10.1038/s41467-020-17035-5 · Cited by 133
What can topology tell us about the neural code?
Curto, C. (2016) · Bulletin of the American Mathematical Society · doi:10.1090/bull/1554 · Cited by 117
“applied algebraic topology, a relatively new development in applied mathematics that was highlighted in the Bulletin of the American Mathematical Society nearly a decade ago [17].”
BackgroundCategorification of Persistent HomologyPreprint
Bubenik, P., Jonathan, & Scott, A. (2016) · Cited by 124
Persistent homology for the quantitative prediction of fullerene stability
Xia, K., Feng, X., Tong, Y., & Wei, G. W. (2014) · Journal of Computational Chemistry · doi:10.1002/jcc.23816 · Cited by 96
Sheaves, Cosheaves and ApplicationsPreprint
Curry, J. M. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1303.3255 · Cited by 205
“42) (22)For an excellent survey article see Robert Ghrist’s Chauvenet prize-winning survey article “Barcodes: The Persistent Topology of Data” [35].”
BackgroundInfluentialEvolution of network architecture in a granular material under compression
Papadopoulos, L., Puckett, J. G., Daniels, K. E., & Bassett, D. S. (2016) · Physical review. E · doi:10.1103/physreve.94.032908 · Cited by 103
Patania, A., Vaccarino, F., & Petri, G. (2017) · EPJ Data Science · doi:10.1140/epjds/s13688-017-0104-x · Cited by 106
Computing Topological Persistence for Simplicial Maps
Dey, T. K., Fan, F., & Wang, Y. (2014) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/2582112.2582165 · Cited by 118
“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].”
BackgroundPersistent Homology for Path Planning in Uncertain Environments
Bhattacharya, S., Ghrist, R., & Kumar, V. (2015) · IEEE Transactions on Robotics · doi:10.1109/tro.2015.2412051 · Cited by 105
“Although we give a quick overview of some of the required concepts in the next section, for more details the reader may refer to the standard texts, [18] and [5], on these respective subjects.”
BackgroundSelf-citationPersistent homology of complex networks for dynamic state detection
Myers, A., Munch, E., & Khasawneh, F. A. (2019) · Physical review. E · doi:10.1103/physreve.100.022314 · Cited by 105
An entropy-based persistence barcode
Chintakunta, H., Gentimis, T., González-Dı́az, R., Jiménez, M.-J., & Krim, H. (2014) · Pattern Recognition · doi:10.1016/j.patcog.2014.06.023 · Cited by 104
“In recent years persistent homology has successfully been used to characterize topological properties of given sets and structures [10].”
BackgroundCang, Z., & Wei, G. (2017) · Computer applications in the biosciences · doi:10.1093/bioinformatics/btx460 · Cited by 95
“As a result, the birth, death, and persistence of topological invariants over the filtration give rise to the barcode representation of a given data set.(29) When persistent homology is used to analyze three dimensional (3D) protein structures, one-dimensional (1D) persistent homology barcodes are obtained as topologi…”
BackgroundPersistent Homology of Collaboration Networks
Carstens, C. J., & Horadam, K. J. (2013) · Mathematical Problems in Engineering · doi:10.1155/2013/815035 · Cited by 101
“These changes can be depicted as a barcode [11, 12], where the intervals [b k , d k ] correspond to filtration values of the birth and death of an element in the ith homology group.”
BackgroundKnowledge gaps in the early growth of semantic feature networks
Sizemore, A. E., Karuza, E. A., Giusti, C., & Bassett, D. S. (2018) · Nature Human Behaviour · doi:10.1038/s41562-018-0422-4 · Cited by 98
Wu, K., Zhao, Z., Wang, R., & Wei, G. (2018) · Journal of Computational Chemistry · doi:10.1002/jcc.25213 · Cited by 101
Higher-order organization of multivariate time series
Santoro, A., Battiston, F., Petri, G., & Amico, E. (2023) · Nature Physics · doi:10.1038/s41567-022-01852-0 · Cited by 106
“In particular, we rely on persistent homology, which is a recent technique in computational topology that has been largely used for the analysis of high dimensional datasets [64, 112] and in disparate applications [113–115].”
MethodPersistent Cohomology and Circular Coordinates
de Silva, V., Morozov, D., & Vejdemo‐Johansson, M. (2011) · Discrete & Computational Geometry · doi:10.1007/s00454-011-9344-x · Cited by 98
Categorification of Persistent Homology
Bubenik, P., & Scott, J. A. (2014) · Discrete & Computational Geometry · doi:10.1007/s00454-014-9573-x · Cited by 139
“For surveys and books on this subject, see [3, 11, 12, 15, 20].”
BackgroundA topological approach for protein classification
Cang, Z., Mu, L., Wu, K., Opron, K., Xia, K., & Wei, G.-W. (2015) · Computational and Mathematical Biophysics · doi:10.1515/mlbmb-2015-0009 · Cited by 93
“The importance of retaining geometric information in topological analysis has been recognized [10], and topology has been advocated as a new approach for tackling big data sets [9, 11, 14, 54, 56].”
BackgroundPersistent homology for random fields and complexes
Adler, R. J., Bobrowski, O., Borman, M. S., Subag, E., & Weinberger, S. (2010) · Institute of Mathematical Statistics Collections · doi:10.1214/10-imscoll609 · Cited by 91
“Theorem 4.1 (Baryshnikov and Ghrist, [4]).”
BackgroundInfluentialPersistent-Homology-Based Machine Learning and Its Applications -- A SurveyPreprint
Pun, C. S., Xia, K., & Lee, S. X. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3275996 · Cited by 89
“For simplification, we also define the set of k -th dimensional barcodes as, The topological persistence is usually represented in two ways, i.e., persistent barcode (PB) (Ghrist, 2008a) and persistent diagram (PD) (Mischaikow and Nanda, 2013).”
BackgroundQiu, Y., & Wei, G. (2023) · Briefings in Bioinformatics · doi:10.1093/bib/bbad289 · Cited by 90
“Topological invariants based on Betti numbers are expressed in terms of persistence barcodes [78] (Figure 2D), persistence diagrams [79],persistence landscapes [80],or persistence images [81].”
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
Harker, S., Mischaikow, K., Mrożek, M., et al. (2013) · Foundations of Computational Mathematics · doi:10.1007/s10208-013-9145-0 · Cited by 88
“The last two decades have witnessed a rapid growth in the range of applications of algebraic topology including sensor networks, image analysis, data analysis, material science and nonlinear dynamics [5, 9, 10, 12, 13, 15, 16, 20, 28, 29].”
BackgroundKofler, A., Dewey, M., Schaeffter, T., Wald, C., & Kolbitsch, C. (2019) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2019.2930318 · Cited by 95
“Persistent homology is a mathematical tool that can be used for analysing datasets X ⊂ Rn [31].”
BackgroundMethodInfluentialMultidimensional persistence in biomolecular data
Xia, K., & Wei, G. (2015) · Journal of Computational Chemistry · doi:10.1002/jcc.23953 · Cited by 74
Functional summaries of persistence diagrams
Berry, E., Chen, Y., Cisewski-Kehe, J., & Fasy, B. T. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00048-w · Cited by 87
“Persistent homology has drawn interest in applied mathematics (Edelsbrunner et al. 2002, Scopiagno & Zorin 2004, Zomorodian 2005, Ghrist 2008, Carlsson 2009), computer science and machine learning (Adams et al. 2017), statistics (Worsley 1996, Adler et al. 2010, Chazal et al. 2014, Fasy et al.…”
BackgroundPersistent homology analysis of phase transitions
Donato, I., Gori, M., Pettini, M., Petri, G., de Nigris, S., Franzosi, R., & Vaccarino, F. (2016) · Physical review. E · doi:10.1103/physreve.93.052138 · Cited by 81
“Persistent homology [6–8], a particular sampling-based technique from algebraic topology, was originally introduced in 2002 [9] by Edelsbrunner et al. with the aim of extracting coarse topological information from high-dimensional data sets [5].”
Persistent Cohomology and Circular CoordinatesPreprint
de Silva, V., Morozov, D., & Vejdemo‐Johansson, M. (2009) · Cited by 83
Topological data analysis: Concepts, computation, and applications in chemical engineering
Smith, A. D. S., Dłotko, P., & Zavala, V. M. (2020) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2020.107202 · Cited by 81
“The homology of data provides deep insight into the structure of the data and provides quantification capabilities of their geometrical features [26, 19].”
BackgroundMethodYan, L., Masood, T. B., Sridharamurthy, R., Rasheed, F., Natarajan, V., Hotz, I., & Wang, B. (2021) · Computer Graphics Forum · doi:10.1111/cgf.14331 · Cited by 101
“…scalar field data, considered in this survey, can be classified into three categories: set-based such as persistence diagrams [ELZ02] and barcodes [Ghr08, CZCG04]; graph-based such as merge trees [BYM ∗ 14], contour trees [CSA03], and Reeb graphs [Ree46]; and complex-based such as Morse and…”
Random geometric complexes in the thermodynamic regime
Yogeshwaran, D., Subag, E., & Adler, R. J. (2015) · Probability Theory and Related Fields · doi:10.1007/s00440-015-0678-9 · Cited by 80
“[6,9,12,15,45]), all of which share a common theme of studying topological invariants of simplicial complexes built on point sets.”
BackgroundSantoro, A., Battiston, F., Lucas, M., Petri, G., & Amico, E. (2024) · Nature Communications · doi:10.1038/s41467-024-54472-y · Cited by 100
“We rely on a recent topological method 41 that combines topological data analysis 56,57 and time series analysis to reveal instantaneous higher-order patterns in fMRI data.”
Persistent Homology in Sparse Regression and Its Application to Brain Morphometry
Chung, M. K., Hanson, J. L., Ye, J., Davidson, R. J., & Pollak, S. D. (2014) · Cited by 77
Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors
Vipond, O., Bull, J. A., Macklin, P. S., Tillmann, U., Pugh, C. W., Byrne, H. M., & Harrington, H. A. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2102166118 · Cited by 76
A Survey of Vectorization Methods in Topological Data Analysis
Ali, D., Asaad, A., Jiménez, M.-J., Nanda, V., Paluzo-Hidalgo, E., & Soriano-Trigueros, M. (2023) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2023.3308391 · Cited by 111
“extracting insights from unstructured data [1], [2], [3], [4].”
BackgroundMethodPersistent topological features of dynamical systems
Maletić, S., Zhao, Y., & Rajković, M. (2016) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4949472 · Cited by 79
“̇?”
Navigating features: a topologically informed chart of electromyographic features space
Phinyomark, A., Khushaba, R. N., Ibáñez‐Marcelo, E., Patania, A., Scheme, E., & Petri, G. (2017) · Journal of The Royal Society Interface · doi:10.1098/rsif.2017.0734 · Cited by 77
“Functional groups of electromyographic features Topological data analysis (TDA) is an approach that focuses on extracting and understanding the ‘shape’ of data using techniques from topology (for an introduction and a survey, see [47,48]).”
BackgroundIsmail, M. S., Noorani, M. S. M., Ismail, M., Razak, F. A., & Alias, M. A. (2020) · Applied Soft Computing · doi:10.1016/j.asoc.2020.106422 · Cited by 74
Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules
Polterovich, L., & Shelukhin, E. (2015) · Selecta Mathematica · doi:10.1007/s00029-015-0201-2 · Cited by 74
“8 is based on the theory of oneparametric persistence modules (see [13,28] for a survey).”
BackgroundIbáñez‐Marcelo, E., Campioni, L., Phinyomark, A., Petri, G., & Santarcangelo, E. L. (2019) · NeuroImage · doi:10.1016/j.neuroimage.2019.06.044 · Cited by 76
Weighted persistent homology for biomolecular data analysis
Meng, Z., Anand, D. V., Lu, Y., Wu, J., & Xia, K. (2020) · Scientific Reports · doi:10.1038/s41598-019-55660-3 · Cited by 74
Classification of weighted networks through mesoscale homological features
Sizemore, A. E., Giusti, C., & Bassett, D. S. (2016) · Journal of Complex Networks · doi:10.1093/comnet/cnw013 · Cited by 74
“For a mathematical treatment and discussion of computational issues, we point the reader to the following useful references [21, 20, 57, 12].”
BackgroundLi, M., An, H., Angelovici, R., Bagaza, C., Batushansky, A., Clark, L. G., Coneva, V., Donoghue, M. J., Edwards, E. J., Fajardo, D. A., et al. (2018) · Frontiers in Plant Science · doi:10.3389/fpls.2018.00553 · Cited by 69
“Persistent homology is a flexible method to quantify branching structures (Edelsbrunner and Harer, 2008; Weinberger, 2011; Li et al., 2017), point clouds (Ghrist, 2008), two-dimensional and three-dimensional shapes (Gamble and Heo, 2010), and textures (Mander et al., 2013, 2017).”
MethodSingle-cell analysis reveals inflammatory interactions driving macular degeneration
Kuchroo, M. R., DiStasio, M., Song, E., Calapkulu, E., Zhang, L., Ige, M., Sheth, A. H., Majdoubi, A., Menon, M., Tong, A., et al. (2023) · Nature Communications · doi:10.1038/s41467-023-37025-7 · Cited by 78
Explicit Topological Priors for Deep-Learning Based Image Segmentation Using Persistent Homology
Clough, J. R., Öksüz, İ., Byrne, N., Schnabel, J. A., & King, A. P. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-20351-1_2 · Cited by 77
“iagram is that it is stable in the presence of noise, and there are theoretical guarantees that small changes to the original data can only make small changes to the positions and lengths of the bars [10]. For an array of input data S, we will describe its PH by denoting each bar in the barcode diagram as H‘ d (S) = (…”
Inference of Ancestral Recombination Graphs through Topological Data Analysis
Cámara, P. G., Levine, A. J., & Rabadán, R. (2016) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1005071 · Cited by 71
“Persistent homology can be represented using barcodes [43].”
BackgroundPersistent Homology in Sparse Regression and Its Application to Brain Morphometry
Chung, M. K., Hanson, J. L., Ye, J., Davidson, R. J., & Pollak, S. D. (2015) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2015.2416271 · Cited by 87
“The binary network is a simplicial complex consisting of 0-simplices (nodes) and 1-simplices (edges), a special case of the Rips complex [32].”
BackgroundPersistent homology and Floer–Novikov theory
Usher, M., & Zhang, J. (2016) · Geometry & Topology · doi:10.2140/gt.2016.20.3333 · Cited by 69
Persistence Paths and Signature Features in Topological Data Analysis
Chevyrev, I., Nanda, V., & Oberhauser, H. (2018) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2018.2885516 · Cited by 67
“ALGEBRAIC topology provides a promising framework for extracting nonlinear features from finite metric spaces via the theory of persistent homology [1], [2], [3].”
BackgroundMuszyński, G., Kashinath, K., Kurlin, V. A., Wehner, M. F., & Prabhat (2019) · Geoscientific model development · doi:10.5194/gmd-12-613-2019 · Cited by 74
“This approach is a type of20 TDA that is inspired by persistence, which is a concept in applied topology that summarizes topological variations across all values of the scalar field under consideration(Ghrist, 2008; Edelsbrunner and Morozov, 2012; Carlsson, 2009, 2014).”
BackgroundMethodSimplicial complexes and complex systems
Salnikov, V., Cassese, D., & Lambiotte, R. (2019) · Apollo (University of Cambridge) · doi:10.17863/cam.56775 · Cited by 63
Exploring uses of persistent homology for statistical analysis of landmark-based shape data
Gamble, J., & Heo, G. (2010) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2010.04.016 · Cited by 68
“More recent surveys [10,12,4] include applications and future directions for the subject.”
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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