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.”
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Topological inference of manifolds with boundary
Wang, Y., & Wang, B. (2019) · Computational Geometry · doi:10.1016/j.comgeo.2019.101606 · Cited by 6
“see [15,23,24] for introduction to persistent homology.”
Persistent Homology Metrics Reveal Quantum Fluctuations and Reactive Atoms in Path Integral Dynamics
Hu, Y., Ounkham, P., Maršálek, O., Markland, T. E., Krishmoorthy, B., & Clark, A. E. (2021) · Frontiers in Chemistry · doi:10.3389/fchem.2021.624937 · Cited by 5
Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds
Thomas, A. M., & Owada, T. (2021) · Extremes · doi:10.1007/s10687-021-00419-1 · Cited by 5
“It is one of the most widely used and robust tools in the TDA toolbox — see Adler et al. (2010) or Ghrist (2008) for an introduction to persistent homology and Edelsbrunner and Harer (2010) for a more thorough treatment.”
BackgroundMethodTopological Features of Electroencephalography are Robust to Re-referencing and Preprocessing
Billings, J., Tivadar, R. I., Murray, M. M., Franceschiello, B., & Petri, G. (2022) · Brain Topography · doi:10.1007/s10548-021-00882-w · Cited by 5
“point clouds or weighted networks, that are invariant under homeomorphic transformations, which include, deformations, rotations, contractions and any other continuous transformation (Zomorodian and Carlsson 2005; Ghrist 2008).”
BackgroundRemote sensing image segmentation by combining manifold projection and persistent homology
Li, Y., Ye, X., & Zhao, Q. (2022) · Measurement · doi:10.1016/j.measurement.2022.111414 · Cited by 5
Polytopal Complex Construction and Use in Persistent Homology
Singh, R. P., & Wilsey, P. A. (2022) · IEEE ... International Conference on Data Mining workshops · doi:10.1109/icdmw58026.2022.00087 · Cited by 5
“The output of PH is a set of Persistence Intervals (PIs) of the form 〈dim, birth, death〉, that characterize the dimension of the feature, the connectivity distance (birth) when the feature first forms, and the connectivity distance (death) when the feature disappears [13], [14].”
BackgroundPersistent Homology for RNA Data Analysis
Xia, K., Liu, X., & Wee, J. (2023) · Methods in molecular biology · doi:10.1007/978-1-0716-2974-1_12 · Cited by 5
MorphMatrix: A Toolkit Facilitating Shape-Changing Interface Design
Dai, S., Ullmer, B., & Newman, W. E. (2024) · doi:10.1145/3623509.3633383 · Cited by 5
Tracking the topology of neural manifolds across populations
Yoon, I. H. R., Henselman‐Petrusek, G., Yu, Y., Ghrist, R., Smith, S. L., & Giusti, C. (2024) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2407997121 · Cited by 6
Self-citationSuay‐García, B., Climent, J., Pérez‐Gracia, M. T., & Falcó, A. (2025) · Expert Opinion on Drug Discovery · doi:10.1080/17460441.2025.2477625 · Cited by 6
“The term was first introduced by Cang and Wei [44].”
Persistent Stanley–Reisner theory
Suwayyid, F., & Wei, G. (2025) · Foundations of Data Science · doi:10.3934/fods.2025009 · Cited by 9
“Filtration-based methods (e.g., the scale parameter in Vietoris-Rips complexes) identify these features and encode their evolution in persistence modules, whose persistence barcodes or persistence diagrams capture the lifespan of each feature [4, 5].”
Peterson, J. K. (2015) · Computational Cognitive Science · doi:10.1186/s40469-015-0003-z · Cited by 5
“The best tantalizing glimpse into this is in (Ghrist 2007).”
BackgroundA Randomized Subdivision Algorithm for Determining the Topology of Nodal Sets
Cochran, G. S., Wanner, T., & Dłotko, P. (2013) · SIAM Journal on Scientific Computing · doi:10.1137/120903154 · Cited by 5
“In recent years, computational topology has increasingly been used as a tool for understanding high-dimensional and complex data generated both from experiment and from numerical simulations [8, 9, 10, 11, 12, 14, 15, 18, 27, 29, 30].”
MethodTowards A Topological Framework for Integrating Semantic Information Sources
Joslyn, C., Hogan, E., & Robinson, M. (2014) · Cited by 5
On the concept of space in neuroscience
Baas, N. A. (2016) · Current Opinion in Systems Biology · doi:10.1016/j.coisb.2016.12.002 · Cited by 5
The Topological Data Analysis of Time Series Failure Data in Software Evolution
Costa, J. P., & Grbac, T. G. (2017) · doi:10.1145/3053600.3053604 · Cited by 5
“Figure 2: Encoding persistent topological features of a pointcloud (on the left) into a persistence diagram (on the right) by approximating the space through simplicial complexes (in the middle) [11].”
BackgroundUsing Persistent Homology to Represent Online Social Network Graphs
Lee, C. S. G., Gao, T., & Li, F. (2017) · IEEE International Conference on Mobile Ad-Hoc and Sensor Systems (MASS) · doi:10.1109/mass.2017.98 · Cited by 5
Bump hunting by topological data analysis
Sommerfeld, M., Heo, G., Kim, P., Rush, S. T. A., & Marron, J. S. (2017) · Stat · doi:10.1002/sta4.167 · Cited by 5
“We give a brief review of the main ideas here and refer to the growing literature for details (Bubenik & Kim, 2007; Ghrist, 2008; Carlsson & Zomorodian, 2009; Carlsson, 2009; Chazal et al., 2011).”
BackgroundGraph based analysis for gene segment organization In a scrambled genome
Hajij, M., Jonoska, N., Kukushkin, D., et al. (2020) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2020.110215 · Cited by 5
Forman-Ricci curvature and persistent homology of unweighted complex networks
Roy, I., Vijayaraghavan, S., Ramaia, S. J., & Samal, A. (2020) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2020.110260 · Cited by 5
Approximate and discrete Euclidean vector bundles
Scoccola, L., & Perea, J. A. (2023) · Forum of Mathematics Sigma · doi:10.1017/fms.2023.16 · Cited by 5
“We make free use of basic tools from persistence theory such as Vietoris–Rips complexes, persistent cohomology and persistence diagrams; see, for example, [20, 50].”
MethodStatistical and topological summaries aid disease detection for segmented retinal vascular images
Nardini, J. T., Pugh, C. W. J., & Byrne, H. M. (2023) · Microcirculation · doi:10.1111/micc.12799 · Cited by 5
Functional neuroimaging in the era of Big Data and Open Science: A modern overview
Lazar, N. A. (2023) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1609 · Cited by 6
“Topological data analysis (TDA; Carlsson, 2009; Chazal & Michel, 2021; Ghrist, 2008) is one such direction which is gaining popularity in the neuroimaging field.”
BackgroundPersistent homological cell tracking technology
Oda, H., Tonami, K., Nakata, Y., Takubo, N., & Kurihara, H. (2023) · Scientific Reports · doi:10.1038/s41598-023-37760-3 · Cited by 6
“This technology is supported by the persistent homological figure detection technology 4 , which detects over-lapping disk-like figures using the death points of the persistent barcodes 5,6 .”
MethodGeometric Quantification of Cost Uncertainty Propagation: A Case Study
Schwabe, O., Shehab, E., & Erkoyuncu, J. A. (2015) · Procedia CIRP · doi:10.1016/j.procir.2015.08.078 · Cited by 5
Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2016) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2016.1260472 · Cited by 5
“…Letscher, and Zomorodian (2002); Cohen-Steiner, Edelsbrunner, and Harer (2005); Bubenik and Kim (2007); Chazal and Oudot (2008); Edelsbrunner and Harer (2008); Ghrist (2008); Carlsson (2009); Carlsson and Zomorodian (2009); Edelsbrunner and Harer (2010); Chazal et al. (2011); Turner et al. (2012).”
BackgroundEremeev, S., Kuptsov, K., & Romanov, S. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-73013-4_16 · Cited by 5
Computational Topology to Monitor Human Occupancy
Barsocchi, P., Cassará, P., Giorgi, D., Moroni, D., & Pascali, M. A. (2018) · International Workshop on Computational Intelligence for Multimedia Understanding (IWCIM) · doi:10.3390/proceedings2020099 · Cited by 5
“Basing on the theory of Persistent Homology, we evaluate the relevance of topological features as their persistence (a topological event’s lifespan) in the so-called barcodes [10].”
MethodNovel Modulation Recognizer for Frequency-Hopping Signals Based on Persistence Diagram
Huang, X., Zhang, L., Yan, K., et al. (2020) · IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB) · doi:10.1109/bmsb49480.2020.9379548 · Cited by 5
“…be represented by a set of points Θ μ on a two-dimensional birth-death plane such that Each point ( b zμ , d zμ ) corresponds to the lifespan of a topological feature (usually a higher-dimensional hole), where b zμ and d zμ specify the corresponding birth time and death time , respectively [16].”
BackgroundTopology-guided sampling of nonhomogeneous random processes
Mischaikow, K., & Wanner, T. (2010) · The Annals of Applied Probability · doi:10.1214/09-aap652 · Cited by 5
“Given the current activity surrounding the ideas of using topological methods for data analysis and remote sensing [8, 9, 15], we believe the importance of this perspective will grow.”
BackgroundA point calculus for interlevel set homology
Bendich, P. L., Cabello, S., & Edelsbrunner, H. (2011) · Pattern Recognition Letters · doi:10.1016/j.patrec.2011.10.007 · Cited by 5
“For intuition, one might imagine a0-dimensional class as a connected component of X, a 1-dimensional class as a loop going around a tunnel, and2-dimensional class as a closed surface enclosing a void insideX.”
BackgroundThe Complexity of Some Topological Inference Problems
Weinberger, S. (2013) · Foundations of Computational Mathematics · doi:10.1007/s10208-013-9152-1 · Cited by 5
ADVANCES IN MULTIDIMENSIONAL SIZE THEORY
Cerri, A., & Frosini, P. (2011) · Image Analysis & Stereology · doi:10.5566/ias.v29.p19-26 · Cited by 4
“These observations had led to pay close attention to the study of Topological Persistence in a multidimensional setting (Frosini and Mulazzani, 1999; Biasotti et al., 2007; Carlsson and Zomorodian, 2007; Biasotti et al., 2008a; Edelsbrunner and Harer, 2008; Ghrist, 2008; Carlsson, 2009).”
BackgroundChen, L. M. (2015) · Springer eBooks · doi:10.1007/978-3-319-25127-1_6 · Cited by 5
Topological Signals of Singularities in Ricci Flow
Alsing, P. M., Blair, H. A., Corne, M., Jones, G., Miller, W. A., Mischaikow, K., & Nanda, V. (2017) · Axioms · doi:10.3390/axioms6030024 · Cited by 5
“Since our treatment of these topics is cursory, we refer the interested reader to [18,19,21,29] for overviews of persistent homology and its applications and to [17,30] for various technical details pertaining to Ricci flow and its discrete analogues.”
Topological Signatures of Singularities in Simplicial Ricci FlowPreprint
Alsing, P. M., Blair, H. A., Corne, M., Jones, G., Miller, W. A., Mischaikow, K., & Nanda, V. (2015) · arXiv (Cornell University) · Cited by 5
Topological mapping of space in bat hippocampusPreprint
Hoffman, K., Babichev, A., & Dabaghian, Y. A. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1601.04253 · Cited by 5
“…mapping of space in bat hippocampus Kentaro Hoffman1, Andrey Babichev1,2 and Yuri Dabaghian1,2 1Jan and Dan Duncan Neurological Research Institute, Baylor College of Medicine, Houston, TX 77030, 2Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005…”
BackgroundCrawford, L. A., Monod, A., Chen, A., Mukherjee, S., & Rabadán, R. (2016) · arXiv (Cornell University) · Cited by 5
Spectral Sparsification of Simplicial Complexes for Clustering and Label PropagationPreprint
Osting, B., Palande, S., & Wang, B. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.08436 · Cited by 5
“…activity for the development of data analysis methods that focus on simplicial complexes, including methods based on computational topology [8, 20, 23, 27], higher-order random processes [5, 26], generalized Cheeger inequalities [28, 54], isoperimetric inequalities [45], high-dimensional…”
MethodMeasuring Human Activity Spaces from GPS Data with Density Ranking and Summary CurvesPreprint
Chen, Y., & Dobra, A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.05017 · Cited by 5
Weighted persistent homology for biomolecular data analysisPreprint
Meng, Z., Anand, D. V., Lu, Y., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1903.02890 · Cited by 5
Universality Theorems for Generative ModelsPreprint
Khrulkov, V., & Oseledets, I. V. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.11520 · Cited by 5
“For many datasets used in practice, it seems very unlikely that the data comes from manifolds with very simple topological properties, as even basic visual patterns may possess quite non-trivial topological structure (Ghrist, 2008).”
One-Dimensional Reduction of Multidimensional Persistent HomologyPreprint
Cagliari, F., Fabio, B. D., & Ferri, M. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0702713 · Cited by 4
Serrano, D. H., & Gómez, D. S. (2019) · arXiv (Cornell University) · Cited by 5
Centrality measures in simplicial complexes: applications of TDA to Network SciencePreprint
Serrano, D. H., & Gómez, D. S. (2019) · arXiv (Cornell University) · Cited by 4
Entropy and Diversity: The Axiomatic ApproachPreprint
Leinster, T. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.02113 · Cited by 24
“(For expository accounts of persistent homology, see Ghrist [113] or Carlsson [59].”
On shapes recognition in topological data analysis
Janes, F., Mwanzalima, M., & Mpimbo, M. (2025) · Research in Mathematics · doi:10.1080/27684830.2025.2573577 · Cited by 4
Algorithm for Interpretable Graph Features via Motivic Persistent Cohomology
Maruyama, Y. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-15621-1_8 · Cited by 4
Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability
Maruyama, Y. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-15621-1_7 · Cited by 4
Mashatola, L., Kader, Z., Abdulla, N., & Kaur, M. (2024) · International Journal of Data Science and Analytics · doi:10.1007/s41060-024-00534-9 · Cited by 4
Kodama, T., Arimura, H., Tokuda, T., Tanaka, K., Yabuuchi, H., Gowdh, N. F. M., Liam, C. K., Chai, C.-S., & Ng, K. H. (2024) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2024.109519 · Cited by 4
Introduction to the special session on Topological Data Analysis, ICASSP 2016
Chintakunta, H., Robinson, M., & Krim, H. (2016) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2016.7472911 · Cited by 4
“The reader is directed to two surveys in [10], [7].”
BackgroundCurvature Sets Over Persistence Diagrams
Gómez, M., & Mémoli, F. (2024) · Discrete & Computational Geometry · doi:10.1007/s00454-024-00634-0 · Cited by 10
Topological information retrieval with dilation-invariant bottleneck comparative measures
Cao, Y., Monod, A., Vlontzos, A., Schmidtke, L., & Kainz, B. (2023) · Information and Inference A Journal of the IMA · doi:10.1093/imaiai/iaad022 · Cited by 4
“We now briefly formalize these technicalities; a complete discussion with full details can be found in the literature on applied and computational topology (e.g., Carlsson (2009); Ghrist (2008); Edelsbrunner and Harer (2008)).”
BackgroundPersistent Homology on Musical Bars
Callet, V. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-07015-0_29 · Cited by 4
Persistent Homology of Semialgebraic Sets
Basu, S., & Karisani, N. (2023) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1494415 · Cited by 6
Barcodes distinguishing morphology of neuronal tauopathy
Beers, D., Goniotaki, D., Hanger, D. P., Goriely, A., & Harrington, H. A. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.043006 · Cited by 7
Resting-State f MRI Advances for Functional Brain Dynamics
Larrivee, D. C. (2023) · IntechOpen eBooks · doi:10.5772/intechopen.113802 · Cited by 3
Persistent homotopy groups of metric spaces
Mémoli, F., & Zhou, L. (2024) · Journal of Topology and Analysis · doi:10.1142/s1793525324500018 · Cited by 4
On Vietoris–Rips complexes of finite metric spaces with scale 2
Feng, Z., & Nukala, N. C. P. (2024) · Journal of Homotopy and Related Structures · doi:10.1007/s40062-024-00340-x · Cited by 4
“Along with the development of topological data analysis [10, 6], determining the homotopy types of Vietoris-Rips complex of finite metric spaces has become crucial in applied topology.”
BackgroundSchrader, P. (2024) · doi:10.1117/12.3009160 · Cited by 4
“…the intrinsic topological structure of raw sensor data through their various representations (e.g., persistence diagrams (PD, [10]), bar codes(BC, [11]), persistence landscapes (PL, [12]), persistence silhouettes (Sil, [13]), persistence images (PI, [14]), Betti curves(Bet, [15]), persistence…”
BackgroundAbid, M., & Saqlain, M. (2024) · Management Science Advances. · doi:10.31181/msa1120241 · Cited by 4
Topology Unveiled: A New Horizon for Economic and Financial Modeling
Wei, Y., Watada, J., & Wang, Z. (2025) · Mathematics · doi:10.3390/math13020325 · Cited by 5
Zia, M., Jones, B., Feng, H., & Wei, G. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.5c00074 · Cited by 6
“When studying the 3D structure of biomolecules, such as proteins, we consider 𝑁 atoms, each represented by their position as { x 𝑖 | x 𝑖 ∈ R 3 , 𝑖 = 1 , 2 , . . ., 𝑁 } .”
Persistent homology for structural characterization in disordered systems
Wang, A., & Zou, L. (2025) · Physical review. E · doi:10.1103/physreve.111.045306 · Cited by 9
“…[55] is a sequence of nested complexes ∅ = K 0 ⊆ K 1 ⊆ . . . ⊆ K n = VR( P, ϵ n ), with ϵ 0 ϵ 1 . . . ϵ n , and the homology groups satisfy [71] H k is born at some K i and dies at some K j , forming birth-death pairs ( ϵ i , ϵ j ) or equivalent birth-persistence pairs ( ϵ i , ϵ j − ϵ i ).”
Lee, J., Mun, J., Choo, M., & Park, S. (2025) · npj Digital Medicine · doi:10.1038/s41746-025-01635-w · Cited by 5
“The dimension of holes was sub-classi fi ed into three different homology groups 78,79 : 0-dimensional groupH 0 (noholes; ‘ point ’ );1-dimensionalgroupH 1 (aholemadebysingle line; ‘ ring ’ ), and 2-dimensional group H 2 (a hallow space enclosed by single surface; including examplesof ‘ hallowsphere…”
Malinowski, J., Kostrzewa, M., Balcerek, M., Tomczuk, W., & Szwabiński, J. (2025) · Journal of Physics Photonics · doi:10.1088/2515-7647/add826 · Cited by 5
“Since the models in the second challenge differ from the previous ones, we decided to further enrich the feature set by adding characteristics derived from TDA [66, 67].”
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
Frosini, P. (2012) · Acta Applicandae Mathematicae · doi:10.1007/s10440-012-9769-0 · Cited by 4
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies
Chepushtanova, S., Kirby, M., Peterson, C., & Ziegelmeier, L. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-39441-1_21 · Cited by 4
“For a detailed discussion of homology, see [16], and for further discussions of persistent homology, see [3, 14,15].”
BackgroundNetwork Topology vs. Geometry: From persistent Homology to Curvature
Saucan, E., & Jost, J. (2017) · Cited by 5
A Simplified Topological Representation of Text for Local and Global Context
Sami, I. R., & Farrahi, K. (2017) · ACM International Conference on Multimedia (ACM MM) · doi:10.1145/3123266.3123330 · Cited by 4
“Cech Complex or Rips Complex theory can be used to build a family of simplicial complexes [4].”
BackgroundInfluentialPerformance enhancement of a computational persistent homology package
Hylton, A., Henselman‐Petrusek, G., Sang, J., & Short, R. (2017) · IEEE International Performance, Computing, and Communications Conference · doi:10.1109/pccc.2017.8280468 · Cited by 4
“The topological data produced by using persistent homology can be visualized through a barcode [14].”
BackgroundHaze detection using persistent homology
Zulkepli, N. F. S., Noorani, M. S. M., Razak, F. A., et al. (2019) · AIP conference proceedings · doi:10.1063/1.5111219 · Cited by 4
Calcina, S. G. S., & Gameiro, M. (2021) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2021.01.013 · Cited by 4
Persistent Topology of Protein Space
Hamilton, W., Borgert, J. E., Hamelryck, T., & Marron, J. S. (2022) · Association for Women in Mathematics series · doi:10.1007/978-3-030-95519-9_10 · Cited by 5
Persistent homology and the shape of evolutionary games
Stenseke, J. (2021) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2021.110903 · Cited by 4
“For a more in-depth introduction to PH, barcodes, and the computation of PH, see (Zomorodian and Carlsson, 2005; Ghrist, 2008; Edelsbrunner and Morozov, 2014; Otter et al., 2017).”
BackgroundExtremal lifetimes of persistent cycles
Chenavier, N., & Hirsch, C. (2021) · Extremes · doi:10.1007/s10687-021-00430-6 · Cited by 4
“However, since the minimal lifetimes of features tend to 0 in increasing sampling windows, there is no contradiction with classical approximation results such as (Ghrist 2008, Lemma 2.1.)”
ResultInfluentialVittorietti, M., Giambalvo, O., Genova, V. G., & Aiello, F. (2022) · Statistical Methods & Applications · doi:10.1007/s10260-022-00666-x · Cited by 4
Topological Data Analysis in Automotive Industry
Beutenmüller, F., Dierolf, B., Keckeisen, M., Pausinger, F., & Vaudrevange, P. K. S. (2023) · Proceedings · doi:10.1007/978-3-658-42048-2_4 · Cited by 4
A topological classifier to characterize brain states: When shape matters more than variance
Ferrà, A., Cecchini, G., Fisas, F.-P. N., Casacuberta, C., & Cos, I. (2023) · PLoS ONE · doi:10.1371/journal.pone.0292049 · Cited by 4
“The Vietoris–Rips filtration [43, 44] associated with X is a nested family of abstract simplicial complexes V t ( X ) for t � 0, where V t ( X ) has a 0-simplex (that is, a vertex) for each point in X and a k -simplex with k � 1 for each collection of points v 0 , . . . , v k in X such that k v i −…”
Computing Geometric Feature Sizes for Algebraic Manifolds
Rocco, S. D., Edwards, P. B., Eklund, D., Gäfvert, O., & Hauenstein, J. D. (2023) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1522656 · Cited by 4
Persistent homological 3D cell detection
Oda, H. (2024) · JSIAM Letters · doi:10.14495/jsiaml.16.25 · Cited by 4
Topological state-space estimation of functional human brain networks
Chung, M. K., Huang, S.-G., Carroll, I. C., Calhoun, V. D., & Goldsmith, H. H. (2024) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011869 · Cited by 5
“…edge weight w = ( w ij ), the binary network X � ¼ ð V ; w � Þ is a graph consisting of the node set V and the binary edge weights w � = ( w � , ij ) given by ( Note w � is the adjacency matrix of X � , which is a simplicial complex consisting of 0-simpli-ces (nodes) and 1-simplices (edges) [60].”
BackgroundBiwer, C., Rothberg, A. E., IglayReger, H. B., Derksen, H., Burant, C., & Najarian, K. (2017) · PLoS ONE · doi:10.1371/journal.pone.0177696 · Cited by 4
Topological data analysis and applications
Costa, J. P. (2017) · International Convention on Information and Communication Technology, Electronics and Microelectronics · doi:10.23919/mipro.2017.7973488 · Cited by 4
“Persistence allows us to compute the homology at all scales, thereby giving us the ability to find ranges of scales where the structure of the space is stable [12].”
BackgroundWubie, B. A., Andrès, A., Greiner, R., Hoehn, B., Montaño‐Loza, A. J., Kneteman, N. M., & Heo, G. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_9 · Cited by 4
General framework for testing Poisson‐Voronoi assumption for real microstructures
Vittorietti, M., Kok, P., Sietsma, J., et al. (2020) · Applied Stochastic Models in Business and Industry · doi:10.1002/asmb.2517 · Cited by 4
Estimating Betti Numbers Using Deep Learning
Paul, R., & Chalup, S. K. (2019) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.2019.8852277 · Cited by 4
“Then the associated Betti numbers can be calculated for each complex of the filtration and be plotted in a barcode diagram [15].”
MethodDiscrete Morse theory, persistent homology and Forman–Ricci curvature
Saucan, E. (2021) · Mathematics Computation and Geometry of Data · doi:10.4310/mcgd.2021.v1.n2.a1 · Cited by 4
ToFU: Topology functional units for deep learning
Oballe, C., Boothe, D. L., Franaszczuk, P. J., & Maroulas, V. (2021) · Foundations of Data Science · doi:10.3934/fods.2021021 · Cited by 6
Graph Embedding Based on Euclidean Distance Matrix and its Applications
Liu, Z., Li, H., Li, R., Zeng, Y., & Ma, J. (2021) · ACM International Conference on Information and Knowledge Management (CIKM) · doi:10.1145/3459637.3482261 · Cited by 5
“To capture the topological properties of the graph, we create a metric space from a distance matrix of the graph and compute its barcode [20] by Javaplex 1 .”
MethodKerboua-Benlarbi, S., Mziou-Sallami, M., & Doufene, A. (2022) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-3-030-90618-4_16 · Cited by 4
Metric Structures on Datasets: Stability and Classification of Algorithms
Mémoli, F. (2011) · Lecture notes in computer science · doi:10.1007/978-3-642-23678-5_1 · Cited by 4
“linkage based methods, spectral clustering, k-means, etc); – obtaining a dendrogram: the result of applying a hierarchical clustering algorithm to the data; – computing the average distance to the barycenter of the dataset (assumed to be embedded in Euclidean space); – computing the average distance between all pairs…”
MethodCone Fields and Topological Sampling in Manifolds with Bounded Curvature
Turner, K. (2013) · Foundations of Computational Mathematics · doi:10.1007/s10208-013-9176-6 · Cited by 4
Invariance properties of the Multidimensional matching distance in Persistent Topology and HomologyPreprint
Cerri, A., & Frosini, P. (2010) · arXiv (Cornell University) · doi:10.6092/unibo/amsacta/2765 · Cited by 5
“In order to compare them, the concept of persistent homology group has been defined, and the scientific community has become more and more interested in this subject, both from the theoretical [3, 7, 21, 27] and the applicative point of view [4, 5, 9, 11, 14, 19, 20, 28, 31].”
BackgroundGeometry in the space of persistence modules
de Silva, V., & Nanda, V. (2013) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/2462356.2462402 · Cited by 4
“The theory of persistence has been at the vanguard of such analysis [4, 11, 9]: for instance, the persistent homology of a point cloud provides a global multi-scale description of the appearance and disappearance of topological features as the points are thickened into balls”
BackgroundSpatiotemporal Barcodes for Image Sequence Analysis
González-Dı́az, R., Jiménez, M.-J., & Medrano, B. (2015) · Lecture notes in computer science · doi:10.1007/978-3-319-26145-4_5 · Cited by 4
“A p–barcode [7] is a graphical representation of the p–th persistent homology groups as a collection of horizontal line segments (bars) in a plane.”
BackgroundTopological tracking of connected components in image sequences
González-Dı́az, R., Jiménez, M.-J., & Medrano, B. (2018) · Journal of Computer and System Sciences · doi:10.1016/j.jcss.2017.12.005 · Cited by 4
“A persistence d-barcode [7] is a graphical representation of the d-th persistent homology groups as a collection of horizontal line segments (bars) in a plane.”
BackgroundA Probabilistic Topological Approach to Feature Identification Using a Stochastic Robotic Swarm
Ramachandran, R. K., Wilson, S., & Berman, S. (2018) · Springer proceedings in advanced robotics · doi:10.1007/978-3-319-73008-0_1 · Cited by 4
Topological Data Analysis for Image Forgery Detection
Al-Jaberi, A. K. (2020) · Indian Journal of Forensic Medicine & Toxicology · doi:10.37506/ijfmt.v14i3.10668 · Cited by 4
“Instead of computing aforementioned topological invariants at a single distance threshold, TDA depends on calculating the persistency of these invariants across an increasing series of distance thresholds using what is known as persistent homology (9), (10).”
BackgroundMethod
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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