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 901–1,000 of 1,274
Lin, Z., Zulkepli, N. F. S., Kasihmuddin, M. S. M., et al. (2025) · IEEE Access · doi:10.1109/access.2025.3548558 · Cited by 1
“A filtration F = F ( K ) of a simplicial complex K is a nested sequence written with inclusion maps (continuous functions) as One of the commonly used complexes is Vietoris-Rips complex [33].”
Methods for Brain Connectivity Analysis with Applications to Rat Local Field Potential Recordings
El-Yaagoubi, A. B., Aslan, S., Gomawi, F., Redondo, P. V., Roy, S., Sultan, M. S., Talento, M. S., Tarrazona, F. T., Wu, H., Cooper, K. W., et al. (2025) · Entropy · doi:10.3390/e27040328 · Cited by 1
“PH typically summarizes this information using visual tools such as barcodes [74], persistence landscapes [75], persistence images [76], or persistence diagrams as illustrated in the left column of Figure 19, where for each di-mension, the birth–death pairs ( b , d ) of topological features are…”
Chen, Q., & Long, E. (2025) · Physics of Fluids · doi:10.1063/5.0265896 · Cited by 1
Topological Data Analysis and Graph-Based Learning for Multimodal Recommendation
Bachiri, K., Yahyaouy, A., Malek, M., & Rogovschi, N. (2025) · IEEE Access · doi:10.1109/access.2025.3582480 · Cited by 3
“By leveraging concepts such as persistent homology, TDA identifies invariant features in datasets, capturing global and local structures that persist across different resolution scales [17].”
InfluentialTopological and geometric characterization of synthetic aperture sonar collections
Robinson, M., Memon, Z., & Gualtieri, M. (2025) · The Journal of the Acoustical Society of America · doi:10.1121/10.0037085 · Cited by 2
Ferranti, G., Failla, C. R., Finocchiaro, P., Pluchino, A., Rapisarda, A., Tudisco, S. S., & Vecchio, G. (2025) · Sensors · doi:10.3390/s25154579 · Cited by 1
“Our methodology, inspired by the field of topological data analysis and persistent homology techniques [10–13], addresses this by introducing the concept of volumetric multi-parameter persistence or VM-PP.”
An improved K-means algorithm based on persistent homology
Peng, N., Gao, S., Yin, X., et al. (2025) · Journal of Computational Science · doi:10.1016/j.jocs.2025.102680 · Cited by 1
Euler Characteristic Surfaces: A Stable Multiscale Topological Summary of Time Series Data
Roy, A., Mitra, A., & Dutta, T. (2025) · La Matematica · doi:10.1007/s44007-025-00165-x · Cited by 2
“While persistent homology has computational complexity of order about O ( n 3 ) (where n is the number of simplices), the Euler Characteristic Surface has computational complexity of order about O ( n ).”
ToBaFu: Topology-based fusion model for classification of two-dimensional cancer images
Xing, Y., Chen, H., & Zheng, Q. (2025) · Neural Networks · doi:10.1016/j.neunet.2025.108117 · Cited by 1
Reinforcement Learning in Categorical Cybernetics
Hedges, J., & Sakamoto, R. R. (2025) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.429.15 · Cited by 1
“…the major pillars of machine learning, along with deep learning (neural networks and differentiable programming), unsupervised learning (statistical clustering methods, which includes topological data analysis [Ghr08]) and variational learning (Bayesian inference and related probabilistic methods).”
Shovon, A. R., Kumar, S., & Deshpande, G. (2025) · IEEE Access · doi:10.1109/access.2025.3616256 · Cited by 1
“The application of topological data analysis (TDA) in network analysis extends beyond conventional graph theory, harnessing computational topology tools to characterize network or data structure architectures with greater adapt-ability [20].”
Uncovering Lung Lesion Patterns in Computed Tomography Scans Through Topological Machine Learning
Benedictis, S. G. D., & Buono, N. D. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-11381-8_7 · Cited by 1
Analyzing Spatial Correlation Patterns of Water Quality Variables in Fluvial Networks
Pronello, N., Castiglia, S., Frontuto, V., Golini, N., Ignaccolo, R., & Ippoliti, L. (2026) · Journal of Agricultural Biological and Environmental Statistics · doi:10.1007/s13253-026-00726-9 · Cited by 1
Leveraging topological noise for dynamic state change detection using persistent homology
Antosh, B. R., Das, S., & Thyagu, N. N. (2026) · Frontiers in Physics · doi:10.3389/fphy.2026.1829651 · Cited by 1
Impact of leaky dynamics on predictive path integration accuracy in recurrent neural networks
Zhang, Y., Zhang, Y., Zheng, M., & Xu, K. (2026) · Nonlinear Dynamics · doi:10.1007/s11071-026-12740-0 · Cited by 1
Lecture 2 Fundamentals of Data, Information, and Knowledge
Holzinger, A. (2014) · Springer eBooks · doi:10.1007/978-3-319-04528-3_2 · Cited by 1
Face Recognition Using Simplicial Complexes
Zou, C., & Hancock, E. R. (2010) · Lecture notes in computer science · doi:10.1007/978-3-642-15992-3_3 · Cited by 1
Metric Reconstruction Via Optimal Transport
Adamaszek, M., Adams, H., & Frick, F. (2018) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/17m1148025 · Cited by 1
“Forschungsinstitut Oberwolfach in 2015. 1 of molecules [31, 43], branched polymers [28], and sensor networks [14], for example. Helpful expository introductions to persistent homology include [8] and [18]. Though we started with a metric space X, the Vietoris–Rips complex VR(X;r) does not come equipped with a natural…”
TOPOLOGICAL METHODS FOR THE QUANTIFICATION AND ANALYSIS OF COMPLEX PHENOTYPES
Medina, P., & Doerge, R. W. (2016) · Conference on Applied Statistics in Agriculture · doi:10.4148/2475-7772.1484 · Cited by 1
“Common methods for achieving this are simplicial complexes [5], or approximating functions [8], such as the kernel density estimator.”
BackgroundMethodGeometric goodness of fit measure to detect patterns in data point clouds
Hernández, A., & Solís, M. (2022) · Computational Statistics · doi:10.1007/s00180-022-01244-1 · Cited by 1
Sinkhorn Divergence of Topological Signature Estimates for Time Series Classification
Stephen, C. (2018) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2018.00113 · Cited by 1
“For an introduction see [9], for algorithmic aspects [10], and for examples of metric space representations on the space of PDs needed to use TDA in off-the-shelf machine learning pipelines see [11], [12].”
BackgroundStreamflow data analysis using persistent homology
Musa, S. M. S. S., Noorani, M. S. M., Razak, F. A., et al. (2019) · AIP conference proceedings · doi:10.1063/1.5111228 · Cited by 1
Cell Complexes, Filaments, Vortexes and Shapes Within a Shape
Peters, J. F. (2019) · Intelligent systems reference library · doi:10.1007/978-3-030-22192-8_2 · Cited by 1
2D+t track detection via relative persistent homology
Assaf, R., Goupil, A., Rammal, A., et al. (2020) · International Journal of Imaging Systems and Technology · doi:10.1002/ima.22503 · Cited by 1
“19,29 In a group of chains, there are two kinds of subgroups.”
BackgroundPersistent homology and application on residues 1 to 28 of amyloid beta peptide
Gao, Y., Lei, F., & Li, S. X. (2020) · Proteins Structure Function and Bioinformatics · doi:10.1002/prot.26026 · Cited by 1
“In Aβ(1-28), atoms are connected by a covalent bonds with similar lengths, independent of its conformational state.”
BackgroundMethodInfluentialAdvanced Homological Analysis for Biometric Identification Using Accelerometer
Yan, K., Zhang, L., & Wu, H. (2020) · IEEE Sensors Journal · doi:10.1109/jsen.2020.3046481 · Cited by 1
“Each point b z , d z corresponds to the lifespan of an underlying topological feature (usually chosen to be a high-dimensional hole), where b z and d z specify the associated birth time and death time , respectively [32].”
BackgroundTopological Data Analysis Approach for Weighted Networks Embedding
Knyazeva, I., & Talalaeva, O. (2021) · Lecture notes in networks and systems · doi:10.1007/978-3-030-64877-0_6 · Cited by 1
Persistent homology in ℓ∞ metric
Beltramo, G., & Škraba, P. (2021) · Computational Geometry · doi:10.1016/j.comgeo.2021.101821 · Cited by 1
Medical Image Segmentation for Skin Lesion Detection via Topological Data Analysis
Jazayeri, N., Jazayeri, F., & Sajedi, H. (2022) · International Conference on Ubiquitous Information Management and Communication (IMCOM) · doi:10.1109/imcom53663.2022.9721758 · Cited by 1
Topological Dynamics and Topological Data Analysis
(2021) · Springer proceedings in mathematics & statistics · doi:10.1007/978-981-16-0174-3 · Cited by 1
Comparing Directed and Weighted Road Maps
Bittner, A., Fasy, B. T., Grudzien, M., Hajra, S. G., Huang, J., Pelatt, K. E., Thatcher, C. M., Tumurbaatar, A., & Wenk, C. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_4 · Cited by 2
Mathematics and Art: Unifying Perspectives
Russell, H. M., & Sazdanović, R. (2021) · Springer eBooks · doi:10.1007/978-3-319-57072-3_125 · Cited by 1
Robinson, M. (2014) · Mathematical engineering · doi:10.1007/978-3-642-36104-3_6 · Cited by 1
Tierny, J. (2017) · Mathematics and visualization · doi:10.1007/978-3-319-71507-0_2 · Cited by 1
Collapsibility and homological properties of $$\mathfrak {I}$$-contractible transformations
Espinoza, J. F., Frias-Armenta, M.-E., & Hernández-Hernández, H. A. (2022) · Boletín de la Sociedad Matemática Mexicana · doi:10.1007/s40590-022-00434-7 · Cited by 1
“Such table of the persistent homology classes is usually represented graphically as a barcode or as a persistence diagram (see [16]).”
BackgroundTopological Data Analysis of Time-Series as an Input Embedding for Deep Learning Models
Byers, M., Hinkle, L. B., & Metsis, V. (2022) · IFIP advances in information and communication technology · doi:10.1007/978-3-031-08337-2_33 · Cited by 3
“More information about the construction of these complexes can be found in [8].”
Publishing Private High-dimensional Datasets: A Topological Approach
Alipourjeddi, N., & Miri, A. (2022) · International Wireless Communications and Mobile Computing Conference (IWCMC) · doi:10.1109/iwcmc55113.2022.9824981 · Cited by 1
Single-Cell Topological Simplicial Analysis Reveals Higher-Order Cellular Complexity
Lin, B. (2022) · IEEE International Conference on Bioinformatics and Biomedicine (BIBM) · doi:10.1109/bibm55620.2022.9995365 · Cited by 1
Stable Topological Feature Vectors via Hermite Function Expansion on Persistence Curves
Chung, Y.-M. (2022) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata55660.2022.10020423 · Cited by 3
“Thus, by (7), the persistence curve of this example is P (D,ψ5(b, d))(x) = 1 11χ[1,2)(x) + 2 11χ[2,4)(x) + 2 11χ[2,4)(x) + 2 11χ[3,5)(x) + 4 11χ[1,5)(x).”
An Unsupervised Object Localization Based on Topological Data Analysis
Cheng, H. (2022) · International Conference on Computational Science and Computational Intelligence (CSCI) · doi:10.1109/csci58124.2022.00277 · Cited by 1
“We modify the Algorithm 1 from [5] as to generate a bounding box of interest in an image as following:”
MethodAn enhanced topological analysis for Lamb waves based SHM methods
Lejeune, A., Hascoët, N., Rébillat, M., Monteiro, É., & Mechbal, N. (2023) · Structural Health Monitoring · doi:10.1177/14759217231196217 · Cited by 1
“The approach used a barcode persistence representation 17,18 assessing the Betti number evolution and the p-Wasserstein distance 19 to exhibit the effect of damages, in terms of manifold structures, on the data topology.”
A Topological Data Analysis-Based Approach to Object Localization: A Comparison with ViT and Yolov7
Cheng, H., & Ong, Z. Z. (2023) · Lecture notes in networks and systems · doi:10.1007/978-3-031-47451-4_10 · Cited by 1
Interpreting High-Dimensional Projections With Capacity
Zhang, Y., Liu, J.-S., Lai, C., et al. (2023) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2023.3324851 · Cited by 1
“There are similarities between our approach and the concepts of TDA [51].”
ResultCombinatorial methods for barcode analysis
Rodriguez, E. J. (2023) · Journal of Applied and Computational Topology · doi:10.1007/s41468-023-00143-8 · Cited by 1
“This theory has been applied to the study of data, where a filtration is applied to some dataset and its persistent homology groups are used to determine features in the underlying distribution [5, 10].”
BackgroundMethodTime varying analysis of dynamic resting-state functional brain network to unfold memory function
Azizi, T. (2023) · Neuroscience Informatics · doi:10.1016/j.neuri.2023.100148 · Cited by 1
Topological learning in multiclass data sets
Griffin, C. H., Karn, T., & Apple, B. (2024) · Physical review. E · doi:10.1103/physreve.109.024131 · Cited by 2
“Our approach is inspired by topological data analysis [35, 36], which provides methods for extracting topological information about a topological space T on which a data set X ⊆ R n resides.”
MethodPersistent Magnitude for the Quantitative Analysis of the Structure and Stability of Carboranes
Bi, W., Fu, X., Li, J., et al. (2024) · Journal of Computational Biophysics and Chemistry · doi:10.1142/s2737416524400027 · Cited by 1
“19 It aims to identify and track the persistence of these features across different scales.”
BackgroundA Method of Defect Characterization in Guided Wave Testing Using Persistent Homology
Duan, S.-Y., & Wu, X. (2023) · IEEE Far East NDT New Technology & Application Forum (FENDT) · doi:10.1109/fendt59575.2023.10461536 · Cited by 1
Islambekov, U., Pathirana, H., Khormali, O., Akçora, C. G., & Smirnova, E. (2024) · Foundations of Data Science · doi:10.3934/fods.2024024 · Cited by 2
From samples to persistent stratified homotopy types
Mäder, T., & Waas, L. (2024) · Journal of Applied and Computational Topology · doi:10.1007/s41468-024-00170-z · Cited by 1
The shift-dimension of multipersistence modules
Chachólski, W., Corbet, R., & Sattelberger, A.-L. (2024) · Journal of Applied and Computational Topology · doi:10.1007/s41468-024-00169-6 · Cited by 1
“The persistent homology of a one-parameter filtration is algebraically well understood; by a basic structure theorem from algebra, the persistence module is uniquely determined by its barcode [17] in the discrete setting [14, 29] as well as in the real setting [3] under mild finiteness assumptions.”
BackgroundPersistent homology with k-nearest-neighbor filtrations reveals topological convergence of PageRank
Le, M., & Taylor, D. (2024) · Foundations of Data Science · doi:10.3934/fods.2024038 · Cited by 5
Integration of LLM in Barcode Scanning for Visually Impaired People
Grumeza, T.-R., Baczur, R., & Fortiş, A.-E. (2024) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-3-031-70011-8_42 · Cited by 1
Piecewise Computation of Persistent Homology
Singh, R. P., Malott, N. O., & Wilsey, P. A. (2024) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata62323.2024.10825374 · Cited by 1
Generalized Morse theory of distance functions to surfaces for persistent homology
Song, A., Yim, K. M., & Monod, A. (2025) · Advances in Applied Mathematics · doi:10.1016/j.aam.2025.102857 · Cited by 13
“Persistent homology is the leading algebraic method in topological data analysis that measures multiscale features in data and fundamentally depends on a distance function (Edelsbrunner & Harer, 2008; Ghrist, 2008; Carlsson, 2009).”
Wang, R., Qiu, C., Hu, C.-S., Li, Z., & Wu, Y. (2025) · Physics Letters B · doi:10.1016/j.physletb.2025.139405 · Cited by 3
Empirical power of quantum encoding methods for binary classification
Luca, G. D., Vlasic, A., Vitz, M., & Pham, A. T. (2025) · Quantum Machine Intelligence · doi:10.1007/s42484-025-00298-w · Cited by 3
Toscano-Durán, V., Narteni, S., Carlevaro, A., et al. (2025) · Communications in computer and information science · doi:10.1007/978-3-032-08324-1_18 · Cited by 1
Topology of Reticulate Evolution
Emmett, K. (2016) · doi:10.7916/d8028rkg · Cited by 1
Applications of Topological Graph Theory to 2-Manifold Learning
Berry, T., & Schluchter, S. A. (2024) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-031-62166-6_20 · Cited by 1
Multimapper: Data Density Sensitive Topological Visualization
Deb, B., Sarkar, A., Kumari, N., Rupela, A., Gupta, P., & Krishnamurthy, B. (2018) · IEEE ... International Conference on Data Mining workshops · doi:10.1109/icdmw.2018.00153 · Cited by 1
“The well-known TDA method of persistence , which is usually applied directly on the point cloud [Ghrist, 2008], can be translated to the Mapper setting via Multiscale Mapper – [Dey et al., 2016] have given an algorithm that, given a tower of covers, computes the persistence diagram of the resulting…”
MethodAngarita, M. A. O., Ebroul Izquierdo, M., & Cañadas, A. M. (2019) · Cyber Security in Networking Conference (CSNet) · doi:10.1109/csnet47905.2019.9108952 · Cited by 1
“In general, a point cloud can be sampled in a n -dimensional metric space [28].”
BackgroundChung, M. K. (2019) · Cambridge University Press eBooks · doi:10.1017/9781316882610.003 · Cited by 1
Fülöp, T., Desroches, M., Cohen, A. A., Santos, F. A. N., & Rodrigues, S. (2020) · Mechanisms of Ageing and Development · doi:10.1016/j.mad.2020.111390 · Cited by 9
“4, Panel C) [Ghrist, 2008].”
Contagion Dynamics for Manifold Learning
Mahler, B. I. (2022) · Frontiers in Big Data · doi:10.3389/fdata.2022.668356 · Cited by 2
“…to these dimensionality considerations via MDS, we analyze our data topologically by computing the persistent homology of the Vietoris–Rips filtration (Ghrist, 2008) based either on the dissimilarity measure directly or on the Euclidean distances between points in the associated high-dimensional…”
MethodCognitive Fatigue Detection from EEG Signals using Topological Signal Processing
Das, A. K., Kumar, K., Gavas, R. D., Jaiswal, D., Chatterjee, D., Ramakrishnan, R. K., Chandra, M. G., & Pal, A. (2020) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco47968.2020.9287418 · Cited by 2
“This method of analyzing homology groups at different scales is known as the Rips filtration technique [18].”
MethodFlux in Tilted Potential Systems: Negative Resistance and Persistence
Baryshnikov, Y. M., & Kvalheim, M. D. (2023) · Communications in Mathematical Physics · doi:10.1007/s00220-022-04622-4 · Cited by 1
Spontaneous Symmetry Breaking in Data Visualization
Feldager, C. W., Hauberg, S., & Hansen, L. K. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-86340-1_35 · Cited by 1
“This lets us inspect the sensitivity of a chosen threshold for multiple models visually as each bar corresponds to a model [10].”
BackgroundA Brief Introduction to Multidimensional Persistent Betti Numbers
Cerri, A., & Frosini, P. (2021) · Springer proceedings in mathematics & statistics · doi:10.1007/978-981-16-0174-3_18 · Cited by 1
Polyhedral expansions of compacta associated to finite approximations
Ruiz, D. M. (2022) · Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas · doi:10.1007/s13398-022-01236-2 · Cited by 1
“Topological persistence [16, 27] uses finite sequences of polyhedra extracted from a dataset to reconstruct topological properties of the underlying space with some guarantees [21, 23, 18, 46].”
MethodAnalysis of Brain fMRI Data via Topological Data Clustering Method IoPS
Jung, T., & Ryoo, H. S. (2021) · IEEE International Conference on Bioinformatics and Biomedicine (BIBM) · doi:10.1109/bibm52615.2021.9669771 · Cited by 1
“When homology is persistent despite change of ϵ , it is the robust characteristics of data [3].”
BackgroundLearning Topology: Bridging Computational Topology and Machine Learning
Moroni, D., & Pascali, M. A. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-68821-9_20 · Cited by 1
A simplified algorithm for identifying abnormal changes in dynamic networks
Azamir, B., Bennis, D., & Michel, B. (2022) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2022.128224 · Cited by 1
ON THE APPLICATION OF TOPOLOGICAL DATA ANALYSIS: A Z24 BRIDGE CASE STUDY
Gowdridge, T., Dervilis, N., & Worden, K. (2022) · International Workshop on Structural Health Monitoring (IWSHM) · doi:10.12783/shm2021/36304 · Cited by 1
Topological Comparison of Some Dimension Reduction Methods Using Persistent Homology on EEG Data
Kwessi, E. A. (2023) · Axioms · doi:10.3390/axioms12070699 · Cited by 2
“The following definition is borrowed from [38]:”
Topological schemas of cognitive maps and spatial learning in the hippocampusPreprint
Babichev, A., Cheng, S., & Dabaghian, Y. A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1509.00171 · Cited by 1
“With methods drawn from persistent homology theory (Zomorodian and Carlsson, 2005; Ghrist, 2008) one can determine the minimal period Tmin required for removing the spurious loops, which provides a theoretical estimate of the time required to learn the environment (Dabaghian et al., 2012; Arai,…”
MethodOptimal homeomorphisms between closed curvesPreprint
Cerri, A., & Fabio, B. D. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0906.4648 · Cited by 1
“As such, a class of methods has been developed with the purpose of performing a topological exploration of the shape, according to some quantitative geometric properties provided by a real function chosen to extract shape features [1, 3, 18, 20, 25].”
MethodKnown unknowns, unknown unknowns and information flow: new concepts in decentralized controlPreprint
Belabbas, M.-A. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1110.1152 · Cited by 1
“The role of homological algebra and algebraic topology in control theory and applied sciences has been recognized in many different contexts such as feedback stabilization, computer graphics, sensor networks or data analysis [17], [18], [19], [20].”
Linear Embedding of Large-Scale Brain Networks for Twin fMRI.Preprint
Chung, M. K., Villalta‐Gil, V., Rathouz, P. J., Lahey, B. B., & Zald, D. H. (2015) · arXiv (Cornell University) · Cited by 1
Stabilizing the output of persistent homology computations.Preprint
Bendich, P. L., & Bubenik, P. (2015) · arXiv (Cornell University) · Cited by 1
Biwer, C. (2017) · Deep Blue (University of Michigan) · Cited by 1
Topological Data Analysis of DNA sequence data in human gut microbiome
Petrov, P. (2014) · ERA: Education and Research Archive (University of Alberta) · doi:10.7939/r3xw4831v · Cited by 1
“The number of intervals in the barcode that span the parameter interval is equal to the rank of the persistent homology group [12].”
BackgroundDigital Geometry, a SurveyPreprint
Chen, L., & Coeurjolly, D. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1807.02222 · Cited by 1
“In recent years, persistent analysis using profound mathematics has become a powerful method in Big Data and data science [19, 10, 41].”
BackgroundMethodTopological Inference of Manifolds with BoundaryPreprint
Wang, Y., & Wang, B. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.05759 · Cited by 1
“see [15,23,24] for introduction to persistent homology.”
Assessing biological models using topological data analysisPreprint
Ulmer, M. P., Ziegelmeier, L., & Topaz, C. M. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1811.04827 · Cited by 1
“create a simplicial complex, the Vietoris-Rips complex is a common choice as it is computationally tractable [11].”
BackgroundGeometrical correlation indices using homological constructions on manifoldsPreprint
Chacón, M. S., Alvarado, A. J. H., & Rojas, R. A. Z. (2018) · arXiv (Cornell University) · Cited by 1
Mesh Learning Using Persistent Homology on the Laplacian EigenfunctionsPreprint
Zhang, Y., Liu, H., Rosen, P., & Hajij, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.09639 · Cited by 1
“A persistence diagram can be represented equivalently by persistence barcodes [11].”
BackgroundEllis, S. P., & Klein, A. (2012) · arXiv (Cornell University) · doi:10.13140/2.1.3087.3286 · Cited by 1
Horizon Visibility Graphs and Time Series Merge Trees are DualPreprint
Stephen, C. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.08825 · Cited by 1
“Topological data analysis (TDA) for time series follows a seemingly different path [6, 10, 21, 24].”
BackgroundExplicit topological priors for deep-learning based image segmentation using persistent homologyPreprint
Clough, J. R., Öksüz, İ., Byrne, N., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1901.10244 · Cited by 1
“iagram is that it is stable in the presence of noise, and there are theoretical guarantees that small changes to the original data can only make small changes to the positions and lengths of the bars [10]. For an array of input data S, we will describe its PH by denoting each bar in the barcode diagram as H‘ d (S) = (…”
Quantales, persistence, and magnitude homologyPreprint
Cho, S. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.02905 · Cited by 1
““persistent homology”, at least as applied to metric spaces, refers to homology of the l8 (i.e. Vietoris-Rips) complex in the vast majority, if not all, of its occurrences in the literature (of which [Ghr08], [dSG07], [ZC05] are a few well-known examples), and this is the sense of “persistent homology” to which we adh…”
BackgroundMethodResultInfluentialAn Excision Theorem for Persistent HomologyPreprint
Palser, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.03348 · Cited by 1
“The maps αk−1 and βk−1 are precisely those seen in the sequence (7), which are induced by the inclusions of A ∩ B into A and B, respectively.”
BackgroundPrivacy Preserving in Online Social Network Data Sharing and Publication
Gao, T. (2019) · Purdue e-Pubs (Purdue University System) · doi:10.25394/pgs.9901862.v1 · Cited by 1
Multivariate Central Limit Theorems for Random Simplicial ComplexesPreprint
Akinwande, G., & Reitzner, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.00975 · Cited by 1
“For an introduction to this we refer to the surveys on persistent homology by Carlsson [1] and Ghrist [3].”
MethodTopological Machine Learning for Multivariate Time SeriesPreprint
Wu, C., & Hargreaves, C. A. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1911.12082 · Cited by 1
Data analysis from empirical moments and the Christoffel functionPreprint
Pauwels, E., Putinar, M., & Lasserre, J. B. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.08480 · Cited by 1
Steinhaus Filtration and Stable Paths in the MapperPreprint
Arendt, D., Broussard, M., Krishnamoorthy, B., Saul, N., & Thrall, A. (2019) · arXiv (Cornell University) · doi:10.4230/lipics.socg.2025.10 · Cited by 1
“The need to rigorously seed a solution with a notion of stability in topological data analysis (TDA) has been addressed primarily using topological persistence [4, 11, 17].”
Topological Features of Electroencephalography are Reference-InvariantPreprint
Billings, J., Tivadar, R. I., Murray, M. M., Franceschiello, B., & Petri, G. (2020) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.09.25.311829 · Cited by 1
“Because of this capacity, topological observables describe information qualitatively robust to noise, deformation and continuous transformations Zomorodian and Carlsson (2005); Ghrist (2008).”
BackgroundFunctional strong laws of large numbers for Euler characteristic processes of extreme sample cloudsPreprint
Thomas, A. M., & Owada, T. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.09110 · Cited by 1
“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.”
BackgroundMethodArgumentative Topology: Finding Loop(holes) in LogicPreprint
Tymochko, S., New, Z., Bynum, L., Purvine, E., Doster, T. J., Chaput, J., & Emerson, T. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.08952 · Cited by 6
“For a detailed description of these concepts see [3, 7].”
Background
Metadata
- Authors
- Ghrist, R.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Ghrist, Robert
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2007-10-26
- Volume
- 45
- Issue
- 1
- Pages
- 61-75
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2071391326
- Semantic Scholar
- cad0bff789bb0e626245b21a47791ef76a9cd5c2
- Cited by
- 1,236
- References
- 32
- Citations by year
- 2026: 78 · 2025: 98 · 2024: 81 · 2023: 105 · 2022: 93 · 2021: 111 · 2020: 121 · 2019: 121 · 2018: 86 · 2017: 66 · 2016: 76 · 2015: 46 · 2014: 41 · 2013: 37 · 2012: 18
- FWCI
- 16.70
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- ScholarlyCommons (University of Pennsylvania) (submittedVersion)
- Scholarly Commons (University of Pennsylvania) (submittedVersion)
- Funders
- Defense Advanced Research Projects Agency (HR0011-07-1-0002)
- University of Illinois at Urbana-Champaign
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistent homology, barcodes, computational algebraic topology, feature detection, shape recognition, high-dimensional data, point clouds, natural image classification
About these checks
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