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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Sorted by citations
Items 501–600 of 1,274
Optimally fast incremental Manhattan plane embedding and planar tight span construction
Eppstein, D. (2009) · Journal of computational geometry · doi:10.20382/v2i1a8 · Cited by 8
From Topological Analyses to Functional Modeling: The Case of Hippocampus
Dabaghian, Y. A. (2021) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2020.593166 · Cited by 8
“…Article 593166 the Betti numbers of a space or a complex X is known as its topological barcode, b(T ) = (b0(T ), b1(T ), b2(T ), . . .), which captures the topological identity of T (Zomorodian, 2005; Zomorodian and Carlsson, 2005; Ghrist, 2008; Carlsson, 2009, 2013; Edelsbrunner and Harer, 2010).”
BackgroundStable topological signatures for metric trees through graph approximations
Vandaele, R., Rieck, B., Saeys, Y., & Bie, T. D. (2021) · Pattern Recognition Letters · doi:10.1016/j.patrec.2021.03.035 · Cited by 8
Persistence Terrace for Topological Inference of Point Cloud Data
Moon, C., Giansiracusa, N., & Lazar, N. A. (2018) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2017.1422432 · Cited by 8
“ngs and give directions for future work. 3 2 Persistent Homology in TDA This section summarizes the theoretical background of (persistent) homology (Hatcher, 2002; Edelsbrunner and Harer, 2008, 2010; Ghrist, 2008; Zomorodian, 2012) and discusses some statistical shortcomings when applied to data analysis. 2.1 Homology…”
BackgroundInfluentialMeans and medians of sets of persistence diagramsPreprint
Turner, K. (2013) · arXiv (Cornell University) · Cited by 8
Defining and computing persistent Z-homology in the general casePreprint
Romero, A., Heras, J., Rubio, J., et al. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1403.7086 · Cited by 8
“Persistent homology is an algebraic method for measuring topological features of shapes and functions, with many recent applications such as point cloud data [5], sensor networks [22], optical character recognition [9] and”
MethodMeasuring and Localizing Homology ClassesPreprint
Freedman, D. Z., & Chen, C. (2008) · arXiv (Cornell University) · Cited by 6
Evaluating the Disentanglement of Deep Generative Models through Manifold TopologyPreprint
Zhou, S., Zelikman, E., Lu, F., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.03680 · Cited by 8
“RLTs use the notion of increasing proximity over time to construct a discrete distribution—known as a persistence barcode (Carlsson, 2019; Zomorodian and Carlsson, 2005; Ghrist, 2008)—over the duration of each k-dimensional hole as it appears and disappears, or their lifetime relative to other holes.”
BackgroundFülöp, T., Desroches, M., Santos, F. A. N., & Rodrigues, S. (2020) · arXiv (Cornell University) · Cited by 8
Bull, J. A., Mulholland, E. J., Moore, J. W., Bosque, J. J., Stolz, B. J., Boen, J., Eggington, H. R., Belnoue-Davis, H. L., Jones, H. J. S., Gatenbee, C., et al. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.06.02.597010 · Cited by 11
Experimental Observations of the Topology of Convolutional Neural Network Activations
Purvine, E., Brown, D., Jefferson, B. A., Joslyn, C., Praggastis, B., Rathore, A., Shapiro, M. R., Wang, B., & Zhou, Y. (2023) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v37i8.26134 · Cited by 12
Du, G., Zhou, L., Huang, Y., Wu, Y., & Hong, Z. (2024) · Acta Materialia · doi:10.1016/j.actamat.2024.120467 · Cited by 8
Using topological data analysis and machine learning to predict customer churn
Sagming, M., Heymann, R., & Visaya, M. V. (2024) · Journal Of Big Data · doi:10.1186/s40537-024-01020-6 · Cited by 7
Rodrigues, J. A. (2025) · International Journal of Topology · doi:10.3390/ijt2020008 · Cited by 7
Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis
Myers, A., Chumley, M. M., & Khasawneh, F. A. (2024) · La Matematica · doi:10.1007/s44007-024-00110-4 · Cited by 7
“More specifics can be found in [7, 8, 11, 20].”
BackgroundEnergy-based clustering: Fast and robust clustering of data with known likelihood functions.
Thürlemann, M., & Riniker, S. (2023) · Journal of Chemical Physics · doi:10.1063/5.0148735 · Cited by 5
Siva, N. K., Singh, Y., Hathaway, Q. A., Sengupta, P. P., & Yanamala, N. V. K. (2024) · Scientific Reports · doi:10.1038/s41598-024-61201-4 · Cited by 7
“The topological data analysis technique utilized in this study was persistent homology (PH), which is a TDA tool that describes the shape of data by extracting its topological invariants; the mathematical basis of PH is shown in works by Zomorodian and Carlsson 33 , Edelsbrunner et al. 34 , Ghirst et al. 35 , Bubenik…”
MethodMulti-Cover Persistence (MCP)-based machine learning for polymer property prediction
Zhang, Y., Shen, C., & Xia, K. (2024) · Briefings in Bioinformatics · doi:10.1093/bib/bbae465 · Cited by 7
“As we change these parameters, certain structures in Multi-cover appear and vanish, which is recorded using the persistent homology barcode [60,61].”
Schäfer, M., Hildenbrand, G. L., & Hausmann, M. (2024) · International Journal of Molecular Sciences · doi:10.3390/ijms252312843 · Cited by 9
Multivariate central limit theorems for random simplicial complexes
Akinwande, G., & Reitzner, M. (2020) · Advances in Applied Mathematics · doi:10.1016/j.aam.2020.102076 · Cited by 7
“For an introduction to this we refer to the surveys on persistent homology by Carlsson [1] and Ghrist [3].”
MethodMining Topological Structure in Graphs through Forest Representations
Vandaele, R., Saeys, Y., & Bie, T. D. (2020) · Ghent University Academic Bibliography (Ghent University) · Cited by 8
Randomized Self-Organizing Map
Rougier, N. P., & Detorakis, G. (2021) · Neural Computation · doi:10.1162/neco_a_01406 · Cited by 9
“For more thorough reading we refer the reader to [10, 18, 48].”
BackgroundPersistent homology for hypergraphs and computational tools — A survey for users
Ren, S. (2020) · Journal of Knot Theory and Its Ramifications · doi:10.1142/s0218216520430075 · Cited by 7
Signatures, Lipschitz-Free Spaces, and Paths of Persistence Diagrams
Giusti, C., & Lee, D. (2023) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1528471 · Cited by 9
𝓗1 persistent features of the resting-state connectome in healthy subjects
Martínez, D., Gonzalez, F. A., & Gómez, F. (2022) · Network Neuroscience · doi:10.1162/netn_a_00280 · Cited by 7
“According to Ghrist (2008, p. 3), “ the more obvious way to convert a collection of points { x α } in a metric space into a global object is to use the point cloud as vertices of a combinatorial graph whose edges are determined by proximity (the vertices within some specified distance (cid:1) ).”
BackgroundMethodResultInfluentialBiomolecular Topology: Modelling and Analysis
Liu, J., Xia, K., Wu, J., Yau, S. S. ‐., & Wei, G. (2022) · Acta Mathematica Sinica English Series · doi:10.1007/s10114-022-2326-5 · Cited by 9
“} in the Euclidean space R, the Vietoris–Rips complex [82] R is the abstract simplicial complex whose simplices are the set of points in V which are pairwise within distance .”
BackgroundConvergence of persistence diagram in the sparse regime
Owada, T. (2022) · The Annals of Applied Probability · doi:10.1214/22-aap1800 · Cited by 8
“A more rigorous description of persistent homology can be found in [16], [49], and [15], while [1] and [18] provide good introductory reading.”
BackgroundFetal Cortex Segmentation with Topology and Thickness Loss Constraints
Li, L., Ma, Q., Li, Z., et al. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-23223-7_11 · Cited by 7
Ohki, T., Kunii, N., & Chao, Z. C. (2023) · Reviews in the Neurosciences · doi:10.1515/revneuro-2022-0137 · Cited by 7
Topological data analysis reveals differences between simulated galaxies and dark matter haloes
Ouellette, A., Holder, G. P., & Kerman, E. (2023) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/stad1765 · Cited by 8
“Persistent homology (Edelsbrunner et al. 2002; Carlsson et al. 2005; Ghrist 2007) is a formalism to study the homology of a topological space as a function of a single parameter, such as scale.”
BackgroundKumar, S., Shovon, A. R., & Deshpande, G. (2023) · Human Brain Mapping · doi:10.1002/hbm.26403 · Cited by 10
“axis depicts that the bars are from 0-dimensional homology groups (H0 ð Þ; Ghrist, 2008).”
BackgroundMethodInfluentialGeneric families of finite metric spaces with identical or trivial 1-dimensional persistence
Smith, P. A., & Kurlin, V. A. (2024) · Journal of Applied and Computational Topology · doi:10.1007/s41468-024-00177-6 · Cited by 7
“The key papers of Gunnar Carlsson [5], Robert Ghrist [6], and Shmuel Weinberger [7] were followed by substantial developments of Fred Chazal [8] and others.”
Topological Features for Monitoring Human Activities at Distance
León, J. L., Alonso, R., García-Reyes, E. B., & González-Dı́az, R. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-13323-2_4 · Cited by 8
“2 [7]) is shown an example of barcode representation, where the start of a horizontal bar represents the birth of a homology class and the end its death.”
BackgroundTopological bar-codes of fractals: a new characterization of symmetric binary fractal trees
Taylor, T. D. (2009) · Banach Center Publications · doi:10.4064/bc84-0-12 · Cited by 7
“Ghrist [6] provides a wonderful survey of the current work in this area.”
BackgroundAn algebraic topological perspective to privacy
Alberto Speranzon, B., & Bopardikar, S. D. (2016) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2016.7525226 · Cited by 7
“[22-25] [47602-47678] [22-25] [47602-47678] [22-25] [47602-47678] [38-52] [47905-47909] [38-52] [47905-47909] [38-52] [47905-47909] [32-47] [47605-47603] [32-47] [47605-47603] [32-47] [47605-47603]”
BackgroundMethodMatroids And Canonical Forms: Theory And Applications
Henselman, G. (2017) · ScholarlyCommons (University of Pennsylvania) · Cited by 7
Sampling Real Algebraic Varieties for Topological Data Analysis
Dufresne, E., Edwards, P. B., Harrington, H. A., & Hauenstein, J. D. (2019) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2019.00253 · Cited by 10
“We apply the persistent homology pipeline popularized by Carlsson in [16] and summarized by Ghrist in [31].”
MethodTopological data analysis to improve exemplar-based inpainting
Jassim, S., Al-Jaberi, A. K., Asaad, A., et al. (2018) · doi:10.1117/12.2309931 · Cited by 7
“The topological features that are not surviving when changing the distance threshold will be treated as noise [20], [18].”
BackgroundHou, J., Qian, J., Wang, Y., Li, X., Du, H., & Chen, L. (2019) · International Workshop on Quality of Service · doi:10.1145/3326285.3329042 · Cited by 9
“The precise definition of homology is defined in [11, 26].”
BackgroundConvergence of persistence diagrams for topological crackle
Owada, T., & Bobrowski, O. (2020) · Bernoulli · doi:10.3150/20-bej1193 · Cited by 8
“We suggest [13, 21] as a good introductory reading, while a more rigorous coverage of algebraic topology is in [23].”
MethodNecessary conditions for discontinuities of multidimensional persistent Betti numbers
Cerri, A., & Frosini, P. (2014) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.3093 · Cited by 7
“Here the term multidimensional, or equivalently k-dimensional, refers to the fact that data properties are described through functions taking values in R [9,28,35].”
BackgroundSchwabe, O., Shehab, E., & Erkoyuncu, J. A. (2016) · Progress in Aerospace Sciences · doi:10.1016/j.paerosci.2016.05.001 · Cited by 7
“Indeed, if the amount of regressible data available for cost estimation does not admit the use of techniques relying on the Central Limit Theorem then no alternatives appear available [17, 18, 19, 20].”
MethodHomological Shape Analysis Through Discrete Morse Theory
Floriani, L. D., Fugacci, U., & Iuricich, F. (2016) · Mathematics and visualization · doi:10.1007/978-3-319-24726-7_9 · Cited by 7
Topology Preserving Data Reduction for Computing Persistent Homology
Malott, N. O., Sens, A. M., & Wilsey, P. A. (2020) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata50022.2020.9378216 · Cited by 8
Persistent homology and microlocal sheaf theoryPreprint
Kashiwara, M., & Schapira, P. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1705.00955 · Cited by 7
“See in particular the survey papers (Edelsbrunner and Harer 2008; Ghrist 2008; Oudot 2015).”
Sampling random graph homomorphisms and applications to network data analysisPreprint
Lyu, H., Mémoli, F., & Sivakoff, D. J. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.09483 · Cited by 7
“Then by computing the homology of the filtration (or the persistent homology of X ), one can associate X with a topological invariant f (X ) called the persistence diagram [25] (or barcodes [29]).”
MethodUnsupervised Space-Time Clustering using Persistent HomologyPreprint
Islambekov, U., & Gel, Y. R. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.11525 · Cited by 7
“ng out TDA related computations in C++, Python and R. Perseus is another noteworthy persistent homology software [20]. For more details on TDA, the reader may consult survey papers such as [21], [7], [22], [23]. 3 Clustering using Betti Numbers (CBN) Algorithm Let X be a point cloud consisting of npoints belonging to…”
MethodPLLay: Efficient Topological Layer based on Persistence LandscapesPreprint
Kim, K., Kim, J., Zaheer, M., Kim, J. S., Chazal, F., & Wasserman, L. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2002.02778 · Cited by 11
“The power of topology lies in its capacity which differentiates sets in topological space in a robust and meaningful geometric way Carlsson (2009); Ghrist (2008).”
BackgroundTopology-guided sampling of nonhomogeneous random processesPreprint
Mischaikow, K., & Wanner, T. (2016) · Cited by 6
Pérez, J. B., Hauke, S., Lupo, U., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2107.05412 · Cited by 7
“In recent years, persistent homology (PH) (see e.g. [24, 19, 21, 34, 11, 35, 9, 31] for surveys) has been a key driving force behind the ever-increasing adoption of topological approaches in a wide variety of computational contexts, such as geometric inference [18, 6], signal processing [40, 36],…”
BackgroundContinuous Indexing of Fibrosis (CIF): Improving the Assessment and Classification of MPN PatientsPreprint
Ryou, H., Sirinukunwattana, K., Aberdeen, A., et al. (2022) · medRxiv · doi:10.1101/2022.06.06.22276014 · Cited by 7
Discrete Morse theory for filtrations
Nanda, V. (2012) · Rutgers University Community Repository (Rutgers University) · doi:10.7282/t3q52n6n · Cited by 8
“4 surveys the relatively new field of persistent homology [15, 7, 40, 10] which is defined for filtered cell complexes.”
BackgroundTipping point for pulsatile oscillations in dynamical networks
Qin, B.-W., & Lin, W. J. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.043209 · Cited by 6
Quotient Complex (QC)-Based Machine Learning for 2D Hybrid Perovskite Design
Hu, C.-S., Mayengbam, R., Xia, K., & Sum, T. C. (2025) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.4c02033 · Cited by 8
Category-Theoretical and Topos-Theoretical Frameworks in Machine Learning: A Survey
Jia, Y., Peng, G., Yang, Z., et al. (2025) · Axioms · doi:10.3390/axioms14030204 · Cited by 6
Clustering and Topological Data Analysis: Comparison and Application
Combs, K., & Bihl, T. (2023) · Hawaii International Conference on System Sciences (HICSS) · doi:10.24251/hicss.2023.102 · Cited by 6
“Before the application of persistent homology, data is converted into simplicial complexes commonly visualized through a Czech or (Vietoris-)Rips diagram (Ghrist, 2008).”
The hitchhiker's guide to the galaxy of mathematical tools for shape analysis
Biasotti, S., Falcidieno, B., Giorgi, D., & Spagnuolo, M. (2012) · ACM SIGGRAPH Courses · doi:10.1145/2343483.2343499 · Cited by 6
Topological Analysis of Rough Surfaces Using Persistent Homology
Yamamoto, K. (2015) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.84.113001 · Cited by 6
Congestion barcodes: Exploring the topology of urban congestion using persistent homology
Wu, Y., Shindnes, G., Karve, V., Yager, D., Work, D. B., Chakraborty, A., & Sowers, R. B. (2017) · IEEE International Conference on Intelligent Transportation Systems (ITSC) · doi:10.1109/itsc.2017.8317777 · Cited by 6
“invariants have been used to study feasibility sets in robotics; see [20].”
MethodCoexistence holes fill a gap in community assembly theory
Letten, A. D. (2021) · Nature Ecology & Evolution · doi:10.1038/s41559-021-01461-9 · Cited by 5
Shape-based Feature Engineering for Solar Flare Prediction
Berger, T. E., Meiss, J. D., Bradley, E., et al. (2020) · CU Scholar (University of Colorado Boulder) · Cited by 6
On Topological Data Analysis for SHM: An Introduction to Persistent Homology
Gowdridge, T., Dervilis, N., & Worden, K. (2021) · Conference proceedings of the Society for Experimental Mechanics · doi:10.1007/978-3-030-76004-5_20 · Cited by 6
Zhang, Z., Xu, N., & Liu, Y. (2022) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2022.115110 · Cited by 6
Beshkov, K., & Tiesinga, P. H. E. (2021) · Biological Cybernetics · doi:10.1007/s00422-021-00906-5 · Cited by 6
Chung, M. K., Azizi, T., Hanson, J. L., Alexander, A. L., Pollak, S. D., & Davidson, R. J. (2023) · Network Neuroscience · doi:10.1162/netn_a_00355 · Cited by 11
“Persistent homology offers one possible solution to the multiscale problem [12,21,24,35,36,57,58,89].”
BackgroundMethodInfluentialFormation of features based on computational topology methods
Chukanov, S. N. (2023) · Computer Optics · doi:10.18287/2412-6179-co-1190 · Cited by 6
Large deviation principle for geometric and topological functionals and associated point processes
Hirsch, C., & Owada, T. (2023) · The Annals of Applied Probability · doi:10.1214/22-aap1914 · Cited by 6
Optimally fast incremental Manhattan plane embedding and planar tight span construction
Eppstein, D. (2011) · Journal of computational geometry · doi:10.20382/jocg.v2i1a8 · Cited by 6
“ecent algorithmic works in computational topology have involved two simplicial complexes dened from a set of points and a distance threshold (radius), the Cech complex and the Vietoris-Rips complex [7,20,21,34]. The Cech complex has a simplex for every set of balls of the given radius that have a common intersection,…”
BackgroundOn certain optimal diffeomorphisms between closed curves
Cerri, A., & Fabio, B. D. (2011) · Forum Mathematicum · doi:10.1515/form.2011.172 · Cited by 6
Statistical analysis of protein ensembles
Máté, G., & Heermann, D. W. (2014) · Frontiers in Physics · doi:10.3389/fphy.2014.00020 · Cited by 6
Topology, Big Data and Optimization
Vejdemo‐Johansson, M., & Škraba, P. (2016) · Studies in big data · doi:10.1007/978-3-319-30265-2_7 · Cited by 6
Anomaly Detection Using Persistent Homology
Bruillard, P., Nowak, K., & Purvine, E. (2016) · doi:10.1109/cybersec.2016.009 · Cited by 6
Open Problems in Computational Topology
Gasarch, W. I., Fasy, B. T., & Wang, B. (2017) · ACM SIGACT News · doi:10.1145/3138860.3138867 · Cited by 6
“For the interested reader, we suggest [6, 18, 21] for a gentle introduction to computational topology, [15] for algorithmic foundations, and [19] for a (free online) textbook with many examples.”
BackgroundIterated Integrals and Population Time Series Analysis
Giusti, C., & Lee, D. (2020) · Abel symposia · doi:10.1007/978-3-030-43408-3_9 · Cited by 9
De-Anonymization of Dynamic Online Social Networks via Persistent Structures
Gao, T., & Li, F. (2019) · International Conference on Communications · doi:10.1109/icc.2019.8761563 · Cited by 6
“Persistent homology is presented in the form of barcodes [3].”
BackgroundVariability in higher order structure of noise added to weighted networks
Blevins, A. S., Kim, J. Z., & Bassett, D. S. (2021) · Communications Physics · doi:10.1038/s42005-021-00725-x · Cited by 6
Bayesian topological signal processing
Oballe, C., Cherne, A., Boothe, D., Kerick, S. E., Franaszczuk, P. J., & Maroulas, V. (2021) · Discrete and Continuous Dynamical Systems - S · doi:10.3934/dcdss.2021084 · Cited by 6
“A comprehensive survey of persistence barcodes may be found in [18].”
BackgroundDark soliton detection using persistent homology
Leykam, D., Rondón, I., & Angelakis, D. G. (2022) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0097053 · Cited by 6
Classification of stochastic processes with topological data analysis
Güzel, İ., & Kaygun, A. (2023) · Concurrency and Computation Practice and Experience · doi:10.1002/cpe.7732 · Cited by 6
“The information that persistent homology yields on the change of topological features of a given point cloud can be presented in various different ways such as barcodes [14], persistence diagrams [10], landscapes [4], images [1], terraces [26], entropy [25] and curves [9].”
BackgroundQuantifying Extrinsic Curvature in Neural Manifolds
Acosta, F., Sanborn, S., Duc, K. D., Madhav, M. S., & Miolane, N. (2023) · IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) · doi:10.1109/cvprw59228.2023.00068 · Cited by 6
“Learn the Topology: The topology of M can be determined using topological data analysis methods such as persistent homology [22].”
MethodSchwabe, O., Shehab, E., & Erkoyuncu, J. A. (2016) · Procedia CIRP · doi:10.1016/j.procir.2016.08.017 · Cited by 6
“cost variance due to quantity, schedule and engineering, changes) then topological data analysis can be considered applicable since the minimum number of dimensions for such has been arrived at (therefore three) otherwise the cost estimator must depend on using analogies and expert opinion for proceeding with the cost…”
MethodA Novel Method for Topological Embedding of Time-Series Data
Kennedy, S. M., Roth, J. D., & Scrofani, J. W. (2018) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco.2018.8553502 · Cited by 6
“However, since the point cloud is a discrete sampling of this path, the homology of the point cloud may differ from the true homology of the path based on the assumed resolution at which the sampling was performed [2].”
BackgroundMethodVandaele, R., Bie, T. D., & Saeys, Y. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-10928-8_2 · Cited by 6
Assessing the Resilience of the Texas Power Grid Network
Ofori-Boateng, D., Dey, A. K., Gel, Y. R., Li, B., Zhang, J., & Poor, H. V. (2019) · IEEE Data Science Workshop (DSW) · doi:10.1109/dsw.2019.8755787 · Cited by 8
“Finally, the persistent diagram gives the birth and death times of topological features in the (x, y)-coordinate system, where x-coordinate represents the birth time and y-coordinate represents the death time in filtration [22].”
BackgroundPersistent Homology on Streaming Data
Moitra, A., Malott, N. O., & Wilsey, P. A. (2020) · IEEE ... International Conference on Data Mining workshops · doi:10.1109/icdmw51313.2020.00090 · Cited by 6
Unsupervised trajectory compression
Corcoran, P., Mooney, P., & Huang, G. P. (2016) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra.2016.7487479 · Cited by 6
“Trajectories may be subject to the following two forms of noise which are topological in nature [11].”
BackgroundTwo Measures for the Homology Groups of Binary Volumes
Gonzalez-Lorenzo, A., Bac, A., Mari, J.-L., & Real, P. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-32360-2_12 · Cited by 7
“Persistent homology can also be visualized in terms of barcodes [Ghr08], where each point ( i, j ) is represented as an interval in the real line.”
Combinatorics of Place Cell Coactivity and Hippocampal MapsPreprint
Babichev, A., Ji, D., Mémoli, F., & Dabaghian, Y. A. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1509.01677 · Cited by 6
“Using the methods of persistent homology [18, 19] it was shown that the topological structure of T captures the topological properties of the environment, if the range of place cell spiking rates and place field sizes happen to parallel biological”
MethodPersistence Landscape based Topological Data Analysis for Personalized Arrhythmia ClassificationPreprint
Yan, Y., Ivanov, K., Cen, J., Liu, Q.-H., & Wang, L. (2019) · Preprints.org · doi:10.20944/preprints201908.0320.v1 · Cited by 6
“Each bar in the Barcode graph means a homology of the point cloud.”
BackgroundMethodInfluentialThe stability of persistent homology of hypergraphsPreprint
Ren, S., & Wu, J. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2002.02237 · Cited by 6
Capturing Dynamics of Time-Varying Data via TopologyPreprint
Xian, L., Adams, H., Topaz, C. M., & Ziegelmeier, L. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.05780 · Cited by 6
“[19], development economics [3], urban planning [26], political science [27], natural language processing [55], and much more. Classic works that build and review the fundamental ideas of TDA include [56,31,28,23,8,44]. While TDA was originally developed with the study of static data in mind, in recent years, it has f…”
BackgroundInfluentialThe topological properties of the protein universe
Madsen, C. D., Barbensi, A., Zhang, S. Y., Ham, L., David, A., Pires, D. E. V., & Stumpf, M. P. H. (2025) · Nature Communications · doi:10.1038/s41467-025-61108-2 · Cited by 7
“Persistent homology 13,20 is a method in computational topology for analysing the shape of data via topological features.”
InfluentialBayer, M., Zajakina, J., Schäfer, M., Salmiņa, K., Rūmnieks, F., Jansons, J., Bestvater, F., Kurg, R., Erenpreisa, J. A., & Hausmann, M. (2025) · Cancers · doi:10.3390/cancers17152480 · Cited by 5
“SMLM in combination with Ripley statistics [35], persistent homology [36,37], persistent imaging [38], and after principal component analysis (PCA) [39] has been successfully applied to discriminate cell types according to their chromatin organization [33,40] and measure radiation treatment effects…”
Topological classification of tumour-immune interactions and dynamics
Yang, J., Fang, H., Dhesi, J., Yoon, I. H. R., Bull, J. A., Byrne, H. M., Harrington, H. A., & Grindstaff, G. (2025) · Journal of Mathematical Biology · doi:10.1007/s00285-025-02253-6 · Cited by 11
“For a comprehensive introduction to the theory of persis - tent homology, see Carlsson (2009) or Ghrist (2008); we refer additionally to Adams et al. (2017) for background on the persistence image vectorisation.”
Risk reduced sparse index tracking portfolio: A topological data analysis approach
Goel, A., Pasricha, P., & Kanniainen, J. (2025) · Omega · doi:10.1016/j.omega.2025.103432 · Cited by 5
Ren, Y., & Wei, G. (2025) · Advanced Intelligent Discovery · doi:10.1002/aidi.202500207 · Cited by 5
“By constructing a multiscale filtration (growing sequence of simplicial complexes) from data, PH captures enduring homological invariants and summarizes them in persistence diagrams or barcodes [21].”
TDA-Clustering Strategies for the Characterization of Brain Organoids
Martin, C. B., Simon-Chane, C., Clouchoux, C., et al. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-23223-7_10 · Cited by 5
Zabaleta-Ortega, Á., Masoller, C., & Guzmán-Vargas, L. (2023) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0167523 · Cited by 5
Catanzaro, M. J., Rizzo, S., Kopchick, J. J., Chowdury, A., Rosenberg, D. R., Bubenik, P., & Diwadkar, V. A. (2023) · Neuroinformatics · doi:10.1007/s12021-023-09645-3 · Cited by 6
“…overview of some key concepts underlying persistent homology (the Appendix contains a more thorough presentation of these ideas, and readers are directed to several survey articles in the literature (Carlsson, 2009; Chazal & Michel, 2017; Edelsbrunner & Harer, 2008; Ghrist, 2008; Munch, 2017)).”
Persistent directed flag Laplacian
Jones, B. R., & Wei, G.-W. (2024) · Foundations of Data Science · doi:10.3934/fods.2024048 · Cited by 8
“They can be analyzed using persistent barcodes [13], persistent diagrams, persistent images [1], persistence landscapes [3], and persistent Betti numbers.”
Background
Metadata
- Authors
- Ghrist, R.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Ghrist, Robert
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2007-10-26
- Volume
- 45
- Issue
- 1
- Pages
- 61-75
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2071391326
- Semantic Scholar
- cad0bff789bb0e626245b21a47791ef76a9cd5c2
- Cited by
- 1,236
- References
- 32
- Citations by year
- 2026: 78 · 2025: 98 · 2024: 81 · 2023: 105 · 2022: 93 · 2021: 111 · 2020: 121 · 2019: 121 · 2018: 86 · 2017: 66 · 2016: 76 · 2015: 46 · 2014: 41 · 2013: 37 · 2012: 18
- FWCI
- 16.70
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- ScholarlyCommons (University of Pennsylvania) (submittedVersion)
- Scholarly Commons (University of Pennsylvania) (submittedVersion)
- Funders
- Defense Advanced Research Projects Agency (HR0011-07-1-0002)
- University of Illinois at Urbana-Champaign
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistent homology, barcodes, computational algebraic topology, feature detection, shape recognition, high-dimensional data, point clouds, natural image classification
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