The DOI matches its Crossref record; 32 of the 32 references it cites were found in a registry or an index; 8 of the 28 sources its claims cite were checked for support.
Barcodes: The persistent topology of data
Open accessAbstract
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— persistent homology —and a novel representation of this algebraic characterization— barcodes . We sketch an application of these techniques to the classification of natural images.
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Claims the paper makes
17 Own interpretation · 13 Sources not found · 12 No basis given · 5 Supported · 3 Partly supported · 2 Not checked yet · 2 Own result · 1 No full text of the sources · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“R ↩→ C √ 2 ↩→ R √ 2 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.1. Persistence.
What the sources say
- Supportedcites [10] DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358. source
Theorem 2.5 (§2.3; page not provided) gives a chain of inclusions from a Rips complex through a Čech complex to a Rips complex at the larger parameter. Its required scale ratio is √(2d/(d+1)), which is at most √2, so the claimed √2-scaled chain follows for a positive parameter. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
What the sources say
- Supportedcites [7] CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271. source
The Main Theorem (Section 3.1, p. 7) says persistence diagrams are stable under small-amplitude perturbations of a function. The Introduction (p. 2) says its assumptions are satisfied by Morse functions, and Applications (p. 16) identifies barcodes as persistence diagrams represented by intervals. (LLM, conf=0.92)
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“H * (C; F ) ∼ = i x t i • F [x] ⊕ ⎛ ⎝ j x r j • (F [x]/(x s j • F [x])) ⎞ ⎠ .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
What the sources say
- Supportedcites [22] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
Section 3.2 (p. 12) says that finite-type persistence modules over a field correspond to finitely generated graded modules over F[t] and describes their decomposition into free modules and quotients by powers of t. Section 2.1 (p. 6) identifies these as the free and torsional portions of the decomposition. (LLM, conf=0.91)
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“However, the Čech complex and various topologically equivalent subcomplexes (e.g., the alpha complex of [13]) are delicate objects to compute, relying on the precise distances between the nodes in E n .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
What the sources say
- Supportedcites [13] EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72. source
The citation identifies the alpha complex as an example of the subcomplexes being discussed. Section 3.3 defines alpha-complex membership using circumsphere radii, and Section 6 (Implementation) describes the need for exact arithmetic and symbolic perturbation to ensure correct computation. The paper need not independently establish the sentence’s broader claim about Čech complexes for this example citation to be supported. (LLM, conf=0.88)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The full data set consists of roughly 8,000,000 points in E 9 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Partly supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. sourceExperimental — accuracy not yet measured
Section 2 (page unavailable) reports about 4.2 million optical patches and 0.79 million range patches—roughly 5 million total, not 8 million. Section 3 (page unavailable) represents each original 3 × 3 patch as a vector in nine-dimensional space, although the preprocessed data lie on a seven-dimensional sphere. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The witness complex of Carlsson and de Silva [8,9,14] is one solution to this problem.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
What the sources say
- Partly supportedcites [8] D V. de Silva, “A weak definition of Delaunay triangulation”, preprint (2003). sourceExperimental — accuracy not yet measured
Section 6 (p. 8) describes using a subsample as the vertices of a topology-approximating simplicial complex and using weak witnesses to construct it. That supports the witness-complex alternative, but our copy does not discuss Rips complexes or say that their unmanageable size is the problem this construction solves. (LLM, conf=0.82)
- Not checked — no full text of the sourcecites [9] DC V. de Silva and G. Carlsson, “Topological estimation using witness complexes”, in SPBG’04 Symposium on Point-Based Graphics (2004), 157-166. source
- Not matchedcites [14] GO L. Guibas and S. Oudot, “Reconstruction using witness complexes”, in Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms (2007).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M on a topological seven-sphere S 7 ⊂ E 8 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. source
Section 3 (Preprocessing; page unavailable) says the authors subtract each patch’s mean, normalize its contrast, and transform the resulting points onto a seven-dimensional sphere in R⁸. The Introduction (page unavailable) says the study focuses on high-contrast data, including regions far from the origin. More precisely, preprocessing uses the logarithms of intensity values. (LLM, conf=0.92)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The Čech theorem (or, equivalently, the “nerve theorem”) states that C has the homotopy type of the union of closed radius /2 balls about the point set {x α }.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As with β k , the barcode for H k does not give any information about the finer structure of the homology, but merely a continuously parameterized rank.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The torsional elements correspond to those homology generators which appear at parameter r j and disappear at parameter r j + s j .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Probably not a substantial claim. 10
- It may state two separate claims as one. 16
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3. Example: natural images
What the sources say
- Partly supportedcites [20] HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366. sourceExperimental — accuracy not yet measured
The Images section describes a collection photographed with a digital camera, but gives its size as 4,212 images, not 4,167. It says the images came from woods, open landscapes and urban areas, often in consecutive series, rather than describing them as random outdoor scenes. The paper was published in 1998, but the passage does not specify when the photographs were taken. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Partly supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, in Figure 4, one sees (from a very coarse sampling) that the point cloud likely represents a connected object with one or two significant 'holes' as measured by H 1 and no significant higher homology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the figure, it certainly appears as though an ideal choice of , if it exists, is rare: by the time is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
hedgeddirectSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“So coarse a reduction of natural images (into three-by-three squares of greyscale intensities) still leads to a point cloud of too high a dimension to visualize.Worse still, what structure there is is blurred and foggy: points appear at first to be distributed over the entire S 7 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.2. "Hover through the fog".
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The resulting homology H * (C) retains the structure of an R[x]-module, but, unlike the chain module, is not necessarily free.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the combinatorics of the 1-skeleton completely determines the complex, and the Rips complex can be stored as a graph and reconstituted instead of storing the entire boundary operator needed for a Čech complex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The reason is that the Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For sufficiently small, the complex is a discrete set; for sufficiently large, the complex is a single highdimensional simplex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Nor is it easily classified: the Artin-Rees theory from commutative algebra implies that the problem of classifying (finite-type) persistence modules such as H * (C) is equivalent to classifying finitely generated nonnegatively graded R[x]-modules.This is known to be very difficult in, say, Z[x].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In general, it is neither a subcomplex of E n nor does it necessarily behave like an n-dimensional space at all (Figure 2).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“From a computational point of view, the Rips complex is less expensive that the corresponding Čech complex, even though the Rips complex has more simplices (in general).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Despite being both computable and insightful, the homology of a complex associated to a point cloud at a particular is insufficient: it is a mistake to ask which value of is optimal.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2. Algebraic topology for data
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of the data point corresponding to the primary circle reveals a pattern of 3-by-3 patches with one light region and one dark region separated by a linear transition.This nodal curve between light and dark is linear and appears in a circular family parameterized by the angle of the nodal line, as shown in Figure 5.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This necessitates efficient sampling or reduction of the complex with accurate topology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These persistent generators, upon close examination, do correspond to meaningful structures in the data, inspiring a sensible parametrization of the global structure of the data set.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of these three circles reveals that each intersects the highk primary twice, yet the two secondary circles are disjoint.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“[3] reports that a repetition of the experiment with a camera held at a constant angle π/4 yields a data set whose secondary persistent H 1 generators exhibit a bias towards true vertical and true horizontal: the axis of pixellation appears less relevant than the axis of gravity in natural image data.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is worth noting that for chain filtrations arising from realistic data sets, the Rips complexes are of an unmanageable size.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The computation of persistent relative homology is more subtle, since the ensuing parameterized chain complex C is no longer free as an F [x]-module.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Following this analogy, the H 1 primary generator fills the role of a next term in the expansion of the homotopy type of the data set, collating the nodal curve between two contrasting patches.The secondary circles, interpolating between single and dual nodal curves, act as higher-order terms in the expansion, in which horizontal and vertical biases arise.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“However, there are fundamental algebraic difficulties in constructing an analogous theory of persistence modules in this setting [24].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
What the sources say
- Not checked — no full text of the sourcecites [24] ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193. source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As noted in [3], each secondary circle regulates images with three contrasting regions and interpolates between these states and the primary circle.The difference between the two secondary circles lies in their bias for horizontal and vertical stratification respectively.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
The authors state this as an assumption.
Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
“One assumes a finite-type condition that each C i * is finitely generated as an R[x]-module and that the sequence stabilizes in i (in the case of an infinite sequence of chain complexes).”
Citations
Cites
Density Estimation for Statistics and Data Analysis
Silverman, B. W. (2018) · Routledge eBooks · doi:10.1201/9781315140919 · Cited by 15,938
Density Estimation for Statistics and Data Analysis
(1986) · Springer eBooks · doi:10.1007/978-1-4899-3324-9 · Cited by 8,569
Density Estimation for Statistics and Data Analysis
Dehnad, K., & Silverman, B. W. (1987) · Technometrics · doi:10.2307/1269475 · Cited by 5,479
(2001) · Cited by 5,374
Three-dimensional alpha shapes
Edelsbrunner, H., & Mücke, E. P. (1994) · ACM Transactions on Graphics · doi:10.1145/174462.156635 · Cited by 2,488
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Stability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280
Independent component filters of natural images compared with simple cells in primary visual cortex
van Hateren, J. H., & van der Schaaf, A. (1998) · Proceedings of the Royal Society of London. Series B: Biological Sciences · doi:10.1098/rspb.1998.0303 · Cited by 1,106 · cited 1×
“A collection of 4167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
BackgroundTopological persistence and simplification
Edelsbrunner, H., Letscher, D., & Zomorodian, A. J. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.2000.892133 · Cited by 716 · cited 2×
“Persistence, as introduced by Edelsbrunner, Letscher, and Zomo-rodian [12] and refined by Carlsson and Zomorodian [22], is a rigorous response to this problem.”
Method
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Items 801–900 of 1,274
Persistence Based on LBP Scale Space
Janusch, I., & Kropatsch, W. G. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-39441-1_22 · Cited by 2
“Short-lived features are treated as noise [5].”
BackgroundPersistent homology of graph-like digital images
Ege, Ö., & Karaca, İ. (2020) · Annali di Matematica Pura ed Applicata (1923 -) · doi:10.1007/s10231-020-00962-x · Cited by 2
“For detailed knowledge related to persistent homology, see [16] and more recent papers include different applications, see [15, 17, 24].”
BackgroundOn the local behavior of spaces of range image patches
Li, J., & Xia, S. (2020) · Multimedia Tools and Applications · doi:10.1007/s11042-020-09913-3 · Cited by 2
“The normal topological structures of homotopy and homology do not provide any identification [10].”
BackgroundA density-based approach to feature detection in persistence diagrams for firn data
Lawson, A., Hoffman, T., Chung, Y.-M., et al. (2021) · Foundations of Data Science · doi:10.3934/fods.2021012 · Cited by 2
“Topological data analysis (TDA) is a rising field at the intersection of Mathematics, Statistics, and Machine Learning [14, 4, 25, 6].”
BackgroundMultiparameter Persistent Homology via Generalized Morse Theory
Bubenik, P., & Catanzaro, M. J. (2024) · Fields Institute communications · doi:10.1007/978-3-031-57204-3_4 · Cited by 3
Distributed Computation of Persistent Homology from Partitioned Big Data
Malott, N. O., Verma, R. R., Singh, R. P., & Wilsey, P. A. (2021) · IEEE International Conference on Cluster Computing (CLUSTER) · doi:10.1109/cluster48925.2021.00050 · Cited by 2
Topological data analysis for the energy and stability of endohedral metallofullerenes
Zhao, Y., Wang, Y., Ding, Y., & Han, H. (2021) · Journal of Mathematical Chemistry · doi:10.1007/s10910-021-01309-4 · Cited by 2
Measuring population-level plant gene flow with topological data analysis
Migdałek, G., & Żelawski, M. (2022) · Ecological Informatics · doi:10.1016/j.ecoinf.2022.101740 · Cited by 2
Homology groups of embedded fractional Brownian motion
Masoomy, H., Tajik, S., & Movahed, M. S. (2022) · Physical review. E · doi:10.1103/physreve.106.064115 · Cited by 2
Feature-Based Complexity Measure for Multinomial Classification Datasets
Erwin, K. N., & Engelbrecht, A. P. (2023) · Entropy · doi:10.3390/e25071000 · Cited by 4
“Topological data analysis (TDA) measures the topological features in data, and the relationships between them, to assess complexity [5,6].”
BackgroundCluster Persistence for Weighted Graphs
Bobrowski, O., & Škraba, P. (2023) · Entropy · doi:10.3390/e25121587 · Cited by 4
“Persistent homology [3,4] is a powerful topological tool that provides multi-scale structural information about data and networks [5–8].”
BackgroundAdvantages of Persistent Cohomology in Estimating Animal Location From Grid Cell Population Activity
Kawahara, D., & Fujisawa, S. (2024) · Neural Computation · doi:10.1162/neco_a_01645 · Cited by 3
Spatiotemporal analysis of global grain trade multilayer networks considering topological clustering
Tu, Y., Shu, Z., Wu, W., et al. (2024) · Transactions in GIS · doi:10.1111/tgis.13149 · Cited by 2
“TDA is an emerging methodology that combines algebraic topology and data science (Chazal & Michel, 2021; Ghrist, 2008) and was first proposed by American scientist Gunnar Carlsson in 2009 (Carlsson, 2009).”
MethodEncoding Group Interests With Persistent Homology for Personalized Search
Meng, Y., Li, R.-H., Qin, H., Wu, X., Duan, H., Lu, Y., & Wang, G. (2024) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3410029 · Cited by 3
“The topological changes of the Rips filtration can be represented by a set of barcode [7], [31], which is constructed by plotting the changing topological features over different filtration values.”
BackgroundProvable Quantum Speedups for Computing Persistence in Topological Data Analysis
Gyurik, C., Schmidhuber, A., King, R., Dunjko, V., & Hayakawa, R. (2026) · PRX Quantum · doi:10.1103/gvys-hl8h · Cited by 23
“Specifically, the k th ( i, j )-persistent Betti number counts the number of k -dimensional holes that are present in the i th simplicial complex and persist until the j th (which are typically summarized visually in a so-called barcode [9]).”
Homotopy types of Vietoris–Rips complexes of hypercube graphs
Feng, Z. (2025) · Journal of Topology and Analysis · doi:10.1142/s1793525325500062 · Cited by 2
Identifying public transit deserts: A travel demand-independent persistent homology-based method
Carmody, D., Zhu, O., He, Z., et al. (2025) · Transportation Safety and Environment · doi:10.1093/tse/tdaf015 · Cited by 3
Sykes, J., Nicuşan, A. L., Werner, D., et al. (2025) · Scientific Reports · doi:10.1038/s41598-025-97175-0 · Cited by 2
“The K¨onigsberg bridge problem, solved by Euler, involved determining whether it was possible to cross every bridge in the city once without repeating a crossing [4].”
Falk, M., Hausmann, M., Schäfer, M., Falková, I., Toufar, J., & Toufarová, L. (2025) · Genetics. · doi:10.5772/intechopen.1010246 · Cited by 3
Topological Time Series Analysis of Market Crashes: A Persistence Homology Approach
Zhang, F., & Wu, Y. (2025) · International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC) · doi:10.1145/3745533.3745634 · Cited by 2
“Topological data analysis is focus on extracting information by topological reconstructing from noisy data, and crutial step is the computation of topological invariants, i.e., topological features [21].”
Node persistence from topological data analysis reveals changes in brain functional connectivity
Mondal, M., Yadav, Y., Jost, J., & Samal, A. (2025) · Patterns · doi:10.1016/j.patter.2025.101427 · Cited by 2
Persistent Homology applied to location problems
Bennis, D., Gharib, F., & Lebbar, G. (2018) · MATEC Web of Conferences · doi:10.1051/matecconf/201820000003 · Cited by 2
“In this section, we give a short presentation on some basic notions concerning simplicial complexes, simplicial homology and persistent homology (more details can be found in [5,7,20]).”
BackgroundMethodInfluentialPendekatan Baharu untuk Mengelompok Stesen Pengawasan Kualiti Udara menggunakan Homologi Gigih
Zulkepli, N. F. S., Noorani, M. S. M., Razak, F. A., et al. (2020) · Sains Malaysiana · doi:10.17576/jsm-2020-4904-25 · Cited by 2
Topological Data Analysis on Simple English Wikipedia Articles
Wright, M., & Zheng, X. (2020) · The PUMP Journal of Undergraduate Research · doi:10.46787/pump.v3i0.2410 · Cited by 2
“For a well-illustrated introduction to persistent homology, see [3].”
BackgroundComparing Vector Networks via Frequency Domain Persistent Homology
Yue, Z., Cassidy, B., & Solo, V. (2021) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc45484.2021.9683493 · Cited by 2
Short Topological Decompositions of Non-orientable Surfaces
Fuladi, N., Hubard, A., & de Mesmay, A. (2023) · Discrete & Computational Geometry · doi:10.1007/s00454-023-00580-3 · Cited by 2
“Due to their extra difficulty, non-orientable surfaces have been often somehow neglected in computational topology, but there are many reasons to want to correct this: natural models of random surfaces yield non-orientable surfaces with overwhelming probability, they appear naturally as configuration spaces in diverse…”
Vehicle Plate Recognition Using Improved Neocognitron Neural Network
Kangin, D., Kolev, G., & Angelov, P. P. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-40728-4_78 · Cited by 2
“So, these classes should measure a distance between these classes regarding geometrical information as well as topological in order to classify patterns robustly, as it is done for persistent homology [16].”
BackgroundA topological approach to spectral clustering
Rieser, A. (2021) · Foundations of Data Science · doi:10.3934/fods.2021005 · Cited by 2
“We encounter a more topologically-oriented approach in [3], in which persistent homology [2, 7, 13] is used to help a statistician identify high-density regions of a distribution function.”
MethodGinzburg, V. L., & Ṗasechnik, D. V. (2017) · Arnold Mathematical Journal · doi:10.1007/s40598-016-0062-6 · Cited by 2
“This filtration is extremely important and, in particular, it allows one to treat Floer theory in the context of persistent homology and topological data analysis; see Carlsson (2009); Ghrist (2008).”
BackgroundShape Classification According to LBP Persistence of Critical Points
Janusch, I., & Kropatsch, W. G. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-32360-2_13 · Cited by 2
“Short-lived features are treated as noise [5].”
Topological Path Planning in GPS Trajectory Data
Corcoran, P. (2016) · Sensors · doi:10.3390/s16122203 · Cited by 2
“The Čech complex Cr(X) for r ≥ 0, as defined in Equation (A2), is the abstract simplicial complex whose k-simplices correspond to unordered (k+1)-tuples of points {xα} 0 whose closed r/2 neighbourhoods intersect [32].”
BackgroundSpatial objects classification algorithm on the basis of topological features of a form
Eremeev, S. V., & Kuptsov, K. V. (2017) · Russian-Pacific Conference on Computer Technology and Applications (RPC) · doi:10.1109/rpc.2017.8168065 · Cited by 2
Another look at recovering local homology from samples of stratified sets
Mileyko, Y. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00062-y · Cited by 2
Statistical Methods in Topological Data Analysis for Complex, High-Dimensional Data
Medina, P., & Doerge, R. W. (2015) · Conference on Applied Statistics in Agriculture · doi:10.4148/2475-7772.1130 · Cited by 4
“Reviews of the Topological Data Analysis workflow may be found in Carlsson [1], Ghrist [6], and Nanda and Sazadanovi´c [7].”
BackgroundKarimova, L., Terekhov, A., Makarenko, N., et al. (2020) · Cogent Engineering · doi:10.1080/23311916.2020.1808340 · Cited by 2
“Lifetimes are usually depicted by a set of horizontal segments—barcodes (Ghrist, 2008).”
BackgroundTemporal and Spatial Analyses of TB Granulomas to Predict Long-Term Outcomes
Joslyn, L. R., Renardy, M., Weissman, C. R., et al. (2020) · Springer eBooks · doi:10.1007/978-3-030-56510-7_15 · Cited by 2
Computational Topology and its Applications in Geometric Design
Lin, H., Dong, Z., & Chen, J. (2021) · Recent Patents on Engineering · doi:10.2174/1872212115666210901124742 · Cited by 2
Data-driven and Automatic Surface Texture Analysis Using Persistent Homology
Yesilli, M. C., & Khasawneh, F. A. (2021) · IEEE International Conference on Machine Learning and Applications (ICMLA) · doi:10.1109/icmla52953.2021.00219 · Cited by 2
Point Cloud Object Segmentation Using Multi Elevation-Layer 2D Bounding-Boxes
Brodeur, T., Aliakbarpour, H., & Suddarth, S. (2021) · IEEE/CVF International Conference on Computer Vision Workshops (ICCVW) · doi:10.1109/iccvw54120.2021.00438 · Cited by 2
Homology Preserving Graph Compression
Aktas, M. E., Nguyen, T. D., & Akbaş, E. (2021) · International Conference on Machine Learning and Applications · doi:10.1109/icmla52953.2021.00153 · Cited by 2
Extremal event graphs: A (stable) tool for analyzing noisy time series data
Belton, R., Cummins, B., Gedeon, T., & Fasy, B. T. (2022) · Foundations of Data Science · doi:10.3934/fods.2022019 · Cited by 3
Comparison of Word Embeddings of Unaligned Audio and Text Data Using Persistent Homology
Yessenbayev, Z., & Kozhirbayev, Z. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-20980-2_59 · Cited by 2
Research Papers in Statistical Inference for Time Series and Related Models
Liu, Y., Hirukawa, J., & Kakizawa, Y. (2023) · Springer eBooks · doi:10.1007/978-981-99-0803-5 · Cited by 2
“Over the past twenty years, topological data analysis (TDA) has witnessed significant advances that aim to study and understand various patterns in the data, thanks to the pioneering work of [11], [13], [14] and [19].”
BackgroundTopological Data Analysis for Directed Dependence Networks of Multivariate Time Series Data
Yaagoubi, A. E., & Ombao, H. (2023) · Springer eBooks · doi:10.1007/978-981-99-0803-5_17 · Cited by 2
Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics
Kim, W., Mémoli, F., & Stefanou, A. (2023) · Order · doi:10.1007/s11083-023-09643-9 · Cited by 2
“This enables us to define an interleaving distance between objects in the category, which encompasses many distances in TDA and facilitates a uniform treatment of those distances.”
InfluentialCombinatorial Exploration of Morse–Smale Functions on the Sphere via Interactive Visualization
Zhou, Y., Lazovskis, J., Catanzaro, M. J., et al. (2023) · doi:10.1109/topoinvis60193.2023.00012 · Cited by 2
“• MSF Designer computes and visualizes the persistent homology bar-code [11] of a Morse–Smale function to offer a global summary of its features.”
BackgroundA Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams
Kindelan, R., Petrache, M., Cerda, M., & Hitschfeld, N. (2024) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v38i12.29220 · Cited by 2
Persistence intervals of fractals
Máté, G., & Heermann, D. W. (2014) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2014.03.037 · Cited by 2
SOBRE EL ANÁLISIS DE LA FORMA DE LOS DATOS: UN NUEVO PARADIGMA EN CIENCIA DE DATOS
Fierro, J. F. E., Moya, Y. D. G., & Hernández-Amador, R. (2019) · Revista Ciencia UANL · doi:10.29105/cienciauanl22.96-4 · Cited by 2
Miao, R., Yang, Z., & Gavrishchaka, V. (2020) · International Conference on Information and Computer Technologies (ICICT) · doi:10.1109/icict50521.2020.00025 · Cited by 2
“For example, persistent homology (PH), one of computational topology tools, offers noise-tolerant topological summary of any data set [20-25].”
BackgroundMerkt, W., Ivan, V., Dinev, T., et al. (2021) · IEEE Transactions on Robotics · doi:10.1109/tro.2021.3069132 · Cited by 2
“Alternatively, these features can be visualized using a barcode diagram [34].”
MethodJoint Geometric and Topological Analysis of Hierarchical Datasets
Aloni, L., Bobrowski, O., & Talmon, R. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-86523-8_29 · Cited by 2
“7: Persistence barcode, taken from [14].”
Topological Data Analysis For Evaluating PDE-based Denoising Models
Al-Jaberi, A. K., & Hameed, E. M. (2021) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1897/1/012006 · Cited by 2
“TDA approach relies on computing the persistency of these invariants across an increasing series of thresholds using persistent homology [16] and [17].”
BackgroundMethodWaliszewski, P., & Banaszak, G. (2021) · International Conference on Control Systems and Computer Science · doi:10.1109/cscs52396.2021.00042 · Cited by 2
The application of topological data analysis to human motion recognition
Żelawski, M., & Hachaj, T. (2021) · Technical Transactions · doi:10.37705/techtrans/e2021011 · Cited by 2
“In a filtration process, a sequence of increasing subcomplexes is created and the number of holes and their lifetime is calculated and visualised as a so-called persistence diagram or barcode (Edelsbrunner & Harer, 2010; Ghrist, 2008).”
MethodBlevins, A. S., & Bassett, D. S. (2021) · Springer eBooks · doi:10.1007/978-3-319-57072-3_87 · Cited by 2
Persistent homology as a probe for center vortices and deconfinement in SU(2) lattice gauge theory
Sale, N., Lucini, B., & Giansiracusa, J. (2022) · International Symposium on Lattice Field Theory (LATTICE) · doi:10.22323/1.430.0387 · Cited by 2
Necessary conditions for discontinuities of multidimensional size functions
Cerri, A., & Frosini, P. (2009) · AMS Acta (University of Bologna) · doi:10.6092/unibo/amsacta/2625 · Cited by 2
Interleaved equivalence of categories of persistence modulesPreprint
Vejdemo‐Johansson, M. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1210.7913 · Cited by 2
Min Morse: Approximability & ApplicationsPreprint
Rathore, A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1503.03170 · Cited by 2
Comparing Metrics on Arbitrary Spaces using Topological Data AnalysisPreprint
Balchin, S., & Pillin, E. J. A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1503.04619 · Cited by 2
“This is mainly due to the fact that the main application of topological data analysis lies in computer vision, where the data is naturally given the Euclidean metric (see [5] for an overview of the current state of the theory).”
BackgroundPrediction in Projection: A new paradigm in delay-coordinate reconstructionPreprint
Garland, J. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.07360 · Cited by 2
“Since these two points are neighbors in E[2, 45] and not neighbors in E[3, 45], they are false near(est) neighbors at m = 2.”
BackgroundInfluentialWeighted Persistent HomologyPreprint
Ren, S., Wu, C., & Wu, J. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.06722 · Cited by 2
“First, we give a review of persistence [14, 25], with generalizations to the weighted case.”
Robust spatial memory maps encoded in networks with transient connectionsPreprint
Babichev, A., Morozov, D., & Dabaghian, Y. A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1710.02623 · Cited by 2
“ally equivalent to the shape of the navigated environment. Mathematically, the topological structure of a steadily growing coactivity complex can be described using persistent homology theory methods [14, 15]. In particular, this theory allows detecting topological loops in T(M E) (i.e., closed chains of simplexes ide…”
BackgroundEvolutionary homology on coupled dynamical systemsPreprint
Cang, Z., Munch, E., & Wei, G. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.04677 · Cited by 1
“Persistent homology is often visualized by barcodes (Carlsson et al. 2005; Ghrist 2008) where horizontal line segments called bars represent homology generators that survive over different filtration scales.”
BackgroundConvergence of Persistence Diagrams for Topological CracklePreprint
Owada, T., & Bobrowski, O. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.01602 · Cited by 2
“We suggest [13, 21] as a good introductory reading, while a more rigorous coverage of algebraic topology is in [23].”
MethodPersistence modules, symplectic Banach-Mazur distance and Riemannian metricsPreprint
Stojisavljević, V., & Zhang, J. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.11151 · Cited by 2
“For a detailed exposition see [22, 27, 14, 63, 11, 21, 41, 15, 45].”
Don, A. P. H., & Peters, J. F. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.10256 · Cited by 2
“nds of redundant frames. A Ghrist barcode is a topology-of-data pictograph that represents that appearance and disappearance of consecutive sequences of video frames having a particular feature value [7], [8, §5.13, pp. 104-106]. In this paper, the Betti number of a triangulated video frame vortex nerve is represented…”
InfluentialPersistent memories in transient networksPreprint
Babichev, A., & Dabaghian, Y. A. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1602.00681 · Cited by 2
Persistence Terrace for Topological Inference of Point Cloud Data
Moon, C., Giansiracusa, N., & Lazar, N. A. (2018) · Figshare · doi:10.6084/m9.figshare.5758542 · Cited by 2
Computing Persistent Homology within Coq/SSReflectPreprint
Heras, J., Coquand, T., Mörtberg, A., & Siles, V. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1209.1905 · Cited by 2
Anand, D. V., Xia, K., & Mu, Y. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.06171 · Cited by 2
Topological inference from spontaneous activity structures in FMRI videos with peristence barcodesPreprint
Don, A. P. H., Peters, J. F., Ramanna, S., et al. (2019) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/809293 · Cited by 2
“Each one-dimensional Betti number in a persistent barcode [19] represents the number of connected cycles covering an activation area of the brain.”
BackgroundIntrinsic Topological Transforms via the Distance Kernel EmbeddingPreprint
Maria, C., Oudot, S., & Solomon, E. (2019) · arXiv (Cornell University) · doi:10.4230/lipics.socg.2020.56 · Cited by 2
Dissecting Ethereum Blockchain Analytics: What We Learn from Topology and Geometry of Ethereum GraphPreprint
Li, Y., Islambekov, U., Akçora, C. G., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.10105 · Cited by 2
“The most widely used topological summaries of persistent features are the Betti numbers, barcode plots, persistent diagrams, and persistent landscapes [19].”
BackgroundPersistent Homotopy Groups of Metric SpacesPreprint
Mémoli, F., & Zhou, L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.12399 · Cited by 2
Topological Methods for Pattern Detection in Climate Data
Muszyński, G., Kurlin, V. A., Morozov, D., et al. (2022) · Special publications · doi:10.1002/9781119467557.ch13 · Cited by 2
Topological Machine Learning for Mixed Numeric and Categorical DataPreprint
Wu, C., & Hargreaves, C. A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2003.04584 · Cited by 2
Topology in cyber researchPreprint
Huntsman, S., Palladino, J., & Robinson, M. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.03299 · Cited by 2
“Perhaps the most widely recognized and utilized tool in TDA is persistent homology [70, 26, 9, 23, 27, 57].”
MethodA Fast and Robust Method for Global Topological Functional OptimizationPreprint
Solomon, E., Wagner, A., & Bendich, P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.08496 · Cited by 2
Quiver Signal Processing (QSP)Preprint
Parada-Mayorga, A., Riess, H., Ribeiro, A., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.11525 · Cited by 2
“The observation that the homology of a filtration is in fact a quiver representation has lead to a guaranteed compact description of a filtration called a barcode [20].”
BackgroundSelf-citationMultidimensional Persistence Module Classification via Lattice-Theoretic ConvolutionsPreprint
Riess, H., Hansen, J., & Ghrist, R. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.14057 · Cited by 2
“Persistent homology has the ability to discern both the global topology [10] and local geometry [5] of finite metric spaces (e.g. embedded weighted graphs, point clouds in R d ) making it a befitting feature for the purposes of training a neural network.”
BackgroundSelf-citationClassification based on Topological Data AnalysisPreprint
Kindelan, R., Frías, J., Cerda, M., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.03709 · Cited by 2
Reviews: Topological Distances and Losses for Brain NetworksPreprint
Chung, M. K., Smith, A. D. S., & Shiu, G. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.08623 · Cited by 2
“In the metric space X = ( V , w ), the Rips complex R (cid:15) ( X ) is a simplicial complex whose ( p − 1)-simplices correspond to unordered p -tuples of points that satisfy w ij ≤ in a pairwise fashion (Ghrist, 2008).”
BackgroundInfluentialBuilding Information Filtering Networks with Topological Constraints: Algorithms and Applications
Massara, G. P. (2020) · UCL Discovery (University College London) · Cited by 2
PHoS: Persistent Homology for Virtual ScreeningPreprint
Keller, B., Lesnick, M., & Willke, T. L. (2018) · ChemRxiv · doi:10.26434/chemrxiv.6969260.v1 · Cited by 2
Claudi, F., Chandra, S., & Fiete, I. (2025) · eLife · doi:10.7554/elife.107224.1 · Cited by 1
Takahashi, K. (2026) · Zenodo (CERN European Organization for Nuclear Research) · doi:10.5281/zenodo.18845339 · Cited by 2
Unsupervised topological learning for identification of atomic structures
Becker, S., Devijver, É., Molinier, R., & Jakse, N. (2022) · Physical review. E · doi:10.1103/physreve.105.045304 · Cited by 1
Ciocanel, M.-V., Juenemann, R., Dawes, A. T., & McKinley, S. A. (2021) · Bulletin of Mathematical Biology · doi:10.1007/s11538-020-00847-3 · Cited by 13
“(2015), we choose to use the Vietoris-Rips complex given that it is more computationally tractable than the corresponding C̆ech complex (Ghrist 2008).”
BackgroundMethodFinding the homology of manifolds using ellipsoids
Kališnik, S., & Lešnik, D. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00145-6 · Cited by 2
“s are independent of the choice of the tangent-normal coordinate system; they depend only on the submanifold itself. The value pin the denition of ellipsoids serves as a \persistence parameter" [20], [6], [7], [17], [4]. We purposefully do not take ellipsoids which are similar at all p(which would mean that the r…”
Lee, D., Bresten, C., Youm, K., et al. (2020) · International Journal of Modern Physics C · doi:10.1142/s012918312050117x · Cited by 1
Topological Estimation of 2D Image Data via Subsampling and Application to Firn
Belotti, V., Jester, M., Keegan, K., et al. (2021) · IEEE Access · doi:10.1109/access.2021.3050377 · Cited by 1
“It has proven successful in many scientific applications (see [5] that includes a list of applications and references therein [1]–[4]).”
BackgroundHarmonic Persistent Homology (extended abstract)
Basu, S., & Cox, N. (2022) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs52979.2021.00110 · Cited by 1
Bayesian Estimation of Topological Features of Persistence Diagrams
Martínez, A. F. (2022) · Bayesian Analysis · doi:10.1214/22-ba1341 · Cited by 1
“The method was introduced by [17] and its theory has been further considered in, for example, [8], [16], [26], [46] and [64].”
MethodClassification of Hand Movements Based on EMG Signals using Topological Features
Li, J., Yang, L., He, Y., & Fukuda, O. (2023) · International Journal of Advanced Computer Science and Applications · doi:10.14569/ijacsa.2023.0140405 · Cited by 1
“The most common constructions of the filtration of simplicial complexes are the Vietoris-Rips complex and the ˇ Cech complex [15], [16].”
BackgroundAn Algorithm Based on Topological Data Analysis for Solving Unsupervised Machine Learning Problems
Kafashan, M., & Faro, W. L. (2023) · IEEE International Workshop on Machine Learning for Signal Processing (MLSP) · doi:10.1109/mlsp55844.2023.10286003 · Cited by 1
“Pi-oneering works on TDA were started by the work of [1], [2], and [3] that mainly motivated by the idea that topology and geometry provide a powerful approach to infer robust qualitative information about data structure.”
BackgroundNew Advances in Magnetic Resonance Imaging
Larrivee, D. C. (2023) · IntechOpen eBooks · doi:10.5772/intechopen.111259 · Cited by 2
Statistical inference for dependence networks in topological data analysis
El-Yaagoubi, A. B., Chung, M. K., & Ombao, H. (2023) · Frontiers in Artificial Intelligence · doi:10.3389/frai.2023.1293504 · Cited by 1
“The development of TDA tools such as barcodes and persistence diagrams (Ghrist, 2008; Bubenik, 2015; Adams et al., 2017) have opened many new perspectives for analyzing various types of data (Umeda, 2017; Gholizadeh and Zadrozny, 2018; Motta, 2018; Xu et al., 2021; Leykam and Angelakis, 2023).”
MethodTopological data analysis in air traffic management: The shape of big flight data sets
Cuerno, M., Guijarro, L., Valdés, R. M. A., & Comendador, F. G. (2025) · PLoS ONE · doi:10.1371/journal.pone.0318108 · Cited by 2
“For graphical intuition about these constructions, we recommend the beautiful figures made by R. Ghrist in the following paper [29].”
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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