The DOI matches its Crossref record; 32 of the 32 references it cites were found in a registry or an index; 8 of the 28 sources its claims cite were checked for support.
Barcodes: The persistent topology of data
Open accessAbstract
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— persistent homology —and a novel representation of this algebraic characterization— barcodes . We sketch an application of these techniques to the classification of natural images.
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Claims the paper makes
17 Own interpretation · 13 Sources not found · 12 No basis given · 5 Supported · 3 Partly supported · 2 Not checked yet · 2 Own result · 1 No full text of the sources · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“R ↩→ C √ 2 ↩→ R √ 2 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.1. Persistence.
What the sources say
- Supportedcites [10] DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358. source
Theorem 2.5 (§2.3; page not provided) gives a chain of inclusions from a Rips complex through a Čech complex to a Rips complex at the larger parameter. Its required scale ratio is √(2d/(d+1)), which is at most √2, so the claimed √2-scaled chain follows for a positive parameter. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“Indeed, as shown in [7], barcodes are stable in the presence of noise added to a [Morse] filtration.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
What the sources say
- Supportedcites [7] CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271. source
The Main Theorem (Section 3.1, p. 7) says persistence diagrams are stable under small-amplitude perturbations of a function. The Introduction (p. 2) says its assumptions are satisfied by Morse functions, and Applications (p. 16) identifies barcodes as persistence diagrams represented by intervals. (LLM, conf=0.92)
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- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“H * (C; F ) ∼ = i x t i • F [x] ⊕ ⎛ ⎝ j x r j • (F [x]/(x s j • F [x])) ⎞ ⎠ .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
What the sources say
- Supportedcites [22] Zomorodian, Afra, 2005, Computing persistent homology, Discrete Comput. Geom. source
Section 3.2 (p. 12) says that finite-type persistence modules over a field correspond to finitely generated graded modules over F[t] and describes their decomposition into free modules and quotients by powers of t. Section 2.1 (p. 6) identifies these as the free and torsional portions of the decomposition. (LLM, conf=0.91)
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“However, the Čech complex and various topologically equivalent subcomplexes (e.g., the alpha complex of [13]) are delicate objects to compute, relying on the precise distances between the nodes in E n .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
What the sources say
- Supportedcites [13] EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72. source
The citation identifies the alpha complex as an example of the subcomplexes being discussed. Section 3.3 defines alpha-complex membership using circumsphere radii, and Section 6 (Implementation) describes the need for exact arithmetic and symbolic perturbation to ensure correct computation. The paper need not independently establish the sentence’s broader claim about Čech complexes for this example citation to be supported. (LLM, conf=0.88)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The full data set consists of roughly 8,000,000 points in E 9 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Partly supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. sourceExperimental — accuracy not yet measured
Section 2 (page unavailable) reports about 4.2 million optical patches and 0.79 million range patches—roughly 5 million total, not 8 million. Section 3 (page unavailable) represents each original 3 × 3 patch as a vector in nine-dimensional space, although the preprocessed data lie on a seven-dimensional sphere. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The witness complex of Carlsson and de Silva [8,9,14] is one solution to this problem.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
What the sources say
- Partly supportedcites [8] D V. de Silva, “A weak definition of Delaunay triangulation”, preprint (2003). sourceExperimental — accuracy not yet measured
Section 6 (p. 8) describes using a subsample as the vertices of a topology-approximating simplicial complex and using weak witnesses to construct it. That supports the witness-complex alternative, but our copy does not discuss Rips complexes or say that their unmanageable size is the problem this construction solves. (LLM, conf=0.82)
- Not checked — no full text of the sourcecites [9] DC V. de Silva and G. Carlsson, “Topological estimation using witness complexes”, in SPBG’04 Symposium on Point-Based Graphics (2004), 157-166. source
- Not matchedcites [14] GO L. Guibas and S. Oudot, “Reconstruction using witness complexes”, in Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms (2007).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“By normalizing with respect to mean intensity and restricting attention to high-contrast images (those away from the origin), the data set is projected to a set of points M on a topological seven-sphere S 7 ⊂ E 8 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.1
What the sources say
- Supportedcites [18] MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103. source
Section 3 (Preprocessing; page unavailable) says the authors subtract each patch’s mean, normalize its contrast, and transform the resulting points onto a seven-dimensional sphere in R⁸. The Introduction (page unavailable) says the study focuses on high-contrast data, including regions far from the origin. More precisely, preprocessing uses the logarithms of intensity values. (LLM, conf=0.92)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The Čech theorem (or, equivalently, the “nerve theorem”) states that C has the homotopy type of the union of closed radius /2 balls about the point set {x α }.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As with β k , the barcode for H k does not give any information about the finer structure of the homology, but merely a continuously parameterized rank.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The torsional elements correspond to those homology generators which appear at parameter r j and disappear at parameter r j + s j .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Probably not a substantial claim. 10
- It may state two separate claims as one. 16
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“A collection of 4,167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3. Example: natural images
What the sources say
- Partly supportedcites [20] HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366. sourceExperimental — accuracy not yet measured
The Images section describes a collection photographed with a digital camera, but gives its size as 4,212 images, not 4,167. It says the images came from woods, open landscapes and urban areas, often in consecutive series, rather than describing them as random outdoor scenes. The paper was published in 1998, but the passage does not specify when the photographs were taken. (LLM, conf=0.90)
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Partly supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, in Figure 4, one sees (from a very coarse sampling) that the point cloud likely represents a connected object with one or two significant 'holes' as measured by H 1 and no significant higher homology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Theorem 2.4 ([22]).
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From the figure, it certainly appears as though an ideal choice of , if it exists, is rare: by the time is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
hedgeddirectSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“So coarse a reduction of natural images (into three-by-three squares of greyscale intensities) still leads to a point cloud of too high a dimension to visualize.Worse still, what structure there is is blurred and foggy: points appear at first to be distributed over the entire S 7 .”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.2. "Hover through the fog".
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The resulting homology H * (C) retains the structure of an R[x]-module, but, unlike the chain module, is not necessarily free.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the combinatorics of the 1-skeleton completely determines the complex, and the Rips complex can be stored as a graph and reconstituted instead of storing the entire boundary operator needed for a Čech complex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The reason is that the Rips complex is a flag complex: it is maximal among all simplicial complexes with the given 1-skeleton.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For sufficiently small, the complex is a discrete set; for sufficiently large, the complex is a single highdimensional simplex.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Nor is it easily classified: the Artin-Rees theory from commutative algebra implies that the problem of classifying (finite-type) persistence modules such as H * (C) is equivalent to classifying finitely generated nonnegatively graded R[x]-modules.This is known to be very difficult in, say, Z[x].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.2. Persistent homology.
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In general, it is neither a subcomplex of E n nor does it necessarily behave like an n-dimensional space at all (Figure 2).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“From a computational point of view, the Rips complex is less expensive that the corresponding Čech complex, even though the Rips complex has more simplices (in general).”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > Definition 1.2.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Despite being both computable and insightful, the homology of a complex associated to a point cloud at a particular is insufficient: it is a mistake to ask which value of is optimal.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2. Algebraic topology for data
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of the data point corresponding to the primary circle reveals a pattern of 3-by-3 patches with one light region and one dark region separated by a linear transition.This nodal curve between light and dark is linear and appears in a circular family parameterized by the angle of the nodal line, as shown in Figure 5.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This necessitates efficient sampling or reduction of the complex with accurate topology.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These persistent generators, upon close examination, do correspond to meaningful structures in the data, inspiring a sensible parametrization of the global structure of the data set.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A close examination of these three circles reveals that each intersects the highk primary twice, yet the two secondary circles are disjoint.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“[3] reports that a repetition of the experiment with a camera held at a constant angle π/4 yields a data set whose secondary persistent H 1 generators exhibit a bias towards true vertical and true horizontal: the axis of pixellation appears less relevant than the axis of gravity in natural image data.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is worth noting that for chain filtrations arising from realistic data sets, the Rips complexes are of an unmanageable size.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.4. Computation.
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The computation of persistent relative homology is more subtle, since the ensuing parameterized chain complex C is no longer free as an F [x]-module.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Following this analogy, the H 1 primary generator fills the role of a next term in the expansion of the homotopy type of the data set, collating the nodal curve between two contrasting patches.The secondary circles, interpolating between single and dual nodal curves, act as higher-order terms in the expansion, in which horizontal and vertical biases arise.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.4. "Come like shadows, so depart!"
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“However, there are fundamental algebraic difficulties in constructing an analogous theory of persistence modules in this setting [24].”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 2.5. Other directions.
What the sources say
- Not checked — no full text of the sourcecites [24] ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193. source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As noted in [3], each secondary circle regulates images with three contrasting regions and interpolates between these states and the primary circle.The difference between the two secondary circles lies in their bias for horizontal and vertical stratification respectively.”
§ BARCODES: THE PERSISTENT TOPOLOGY OF DATA > 3.3. "When shall we three meet again?"
What the sources say
- Not matchedcites [3] CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
The authors state this as an assumption.
Each chain complex in the persistence module is finitely generated as a module over the polynomial ring, and the sequence stabilizes in the index.
“One assumes a finite-type condition that each C i * is finitely generated as an R[x]-module and that the sequence stabilizes in i (in the case of an infinite sequence of chain complexes).”
Citations
Cites
Density Estimation for Statistics and Data Analysis
Silverman, B. W. (2018) · Routledge eBooks · doi:10.1201/9781315140919 · Cited by 15,938
Density Estimation for Statistics and Data Analysis
(1986) · Springer eBooks · doi:10.1007/978-1-4899-3324-9 · Cited by 8,569
Density Estimation for Statistics and Data Analysis
Dehnad, K., & Silverman, B. W. (1987) · Technometrics · doi:10.2307/1269475 · Cited by 5,479
(2001) · Cited by 5,374
Three-dimensional alpha shapes
Edelsbrunner, H., & Mücke, E. P. (1994) · ACM Transactions on Graphics · doi:10.1145/174462.156635 · Cited by 2,488
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Stability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280
Independent component filters of natural images compared with simple cells in primary visual cortex
van Hateren, J. H., & van der Schaaf, A. (1998) · Proceedings of the Royal Society of London. Series B: Biological Sciences · doi:10.1098/rspb.1998.0303 · Cited by 1,106 · cited 1×
“A collection of 4167 digital photographs of random outdoor scenes was assembled in the late 1990s by van Hateren and van der Schaaf [20].”
BackgroundTopological persistence and simplification
Edelsbrunner, H., Letscher, D., & Zomorodian, A. J. (2002) · Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.2000.892133 · Cited by 716 · cited 2×
“Persistence, as introduced by Edelsbrunner, Letscher, and Zomo-rodian [12] and refined by Carlsson and Zomorodian [22], is a rigorous response to this problem.”
Method
Cited by
Sorted by citations
Items 1,001–1,100 of 1,274
Sap, D., & Szabo, D. P. (2025) · Journal of Computer Languages · doi:10.1016/j.cola.2025.101365 · Cited by 1
The self-similar evolution of stationary point processes via persistent homologyPreprint
Spitz, D., & Wienhard, A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.05751 · Cited by 5
The growing topology of the C. elegans connectomePreprint
Helm, A., Blevins, A. S., & Bassett, D. S. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2101.00065 · Cited by 10
Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise.Preprint
Masoomy, H., Askari, B., Najafi, M. N., & Movahed, M. S. (2021) · arXiv (Cornell University) · Cited by 1
Interleaving by parts for persistence in a posetPreprint
Kim, W., Mémoli, F., & Stefanou, A. (2019) · arXiv (Cornell University) · Cited by 1
Zhao, Y., & Zhang, H. (2021) · arXiv (Cornell University) · Cited by 1
Persistence Homology of TEDtalk: Do Sentence Embeddings Have a Topological Shape?Preprint
Das, S., Haque, S. A., & Tanveer, M. I. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.14131 · Cited by 5
“Some notable examples of these methods include persistence barcodes (Ghrist, 2008), persistence landscape (Bubenik, 2015) etc.”
BackgroundParallel Decomposition of Persistence Modules Through Interval BasesPreprint
Gregorio, A. D., Guerra, M., Scaramuccia, S., & Vaccarino, F. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4690895 · Cited by 1
Convergence of persistence diagram in the sparse regimePreprint
Owada, T. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.12943 · Cited by 1
“A more rigorous description of persistent homology can be found in [16], [49], and [15], while [1] and [18] provide good introductory reading.”
BackgroundHodge-Laplacian of Brain Networks and Its Application to Modeling Cycles.Preprint
Anand, D. V., & Chung, M. K. (2021) · arXiv (Cornell University) · Cited by 1
Topological conditions for propagation of spatially-distributed neural activityPreprint
Tudoras, À., & Reyes, A. D. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.11.30.470616 · Cited by 1
“Typically, the Čech or Vietoris-Rips complexes are calculated by increasing the diameter of ‘balls’ and then constructing simplices and documenting the persistence of Betti numbers [19, 20], with the longer lasting Betti numbers being more indicative of the true topology.”
BackgroundTopological Data Analysis for Directed Dependence Networks of Multivariate Time Series DataPreprint
El-Yaagoubi, A. B., & Ombao, H. (2023) · arXiv (Cornell University) · doi:10.48550/arxiv.2306.07912 · Cited by 1
“Over the past twenty years, topological data analysis (TDA) has witnessed significant advances that aim to study and understand various patterns in the data, thanks to the pioneering work of [11], [13], [14] and [19].”
BackgroundNeural computation in the brainstem for visceral sensation-driven haemodynamicsPreprint
Lee, J., Mun, J., & Park, S. (2023) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2023.08.27.555024 · Cited by 1
Bujía, G., Ruarte, G., Sclar, M., Solovey, G., & Kamienkowski, J. E. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.01.14.632909 · Cited by 1
Interpreting topology in the context of social science
Feng, M. (2025) · Proceedings of symposia in applied mathematics · doi:10.1090/psapm/080/00731 · Cited by 1
Hansen, E. C. A., Sanderson, N., Nourin, S., Candat, V., Curto, C., & Sumbre, G. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.10.26.684564 · Cited by 1
“…can be more robust to noise, (iii) topology is a tool for converting local features into global properties, and (iv) topological summaries of data can span a range of values for a parameter, such as a threshold, yielding results that are less sensitive to arbitrary parameter choices [32, 33, 34].”
Robinson, M. (2026) · Mathematical engineering · doi:10.1007/978-3-032-08634-1_4 · Cited by 1
Optimisation de la couverture de communication et de mesure dans les réseaux de capteurs
Zhang, M. (2015) · theses.fr (ABES) · doi:10.70675/12fc1af4zc66ez43b3z8b0dzb81ae83b232f · Cited by 1
Pre-string Organizational Channels from Spectral LocalityPreprint
Nesen, O. (2026) · doi:10.24108/preprints-3116428 · Cited by 1
An Analytic investigation into self organizing maps and their network topologies
Baltimore, R. (2010) · RIT Scholar Works (Rochester Institute of Technology) · Cited by 1
Lau, A., & Yin, C. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1012.0520
New Pseudodistances Associated with Reparametrization Invariant SeminormsPreprint
Ferrero, M. M. A. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0903.3401
“nish with an example where the use of this lower bound is illustrated. 1 Introduction Topological Persistence is attracting increasing attention from the mathematical community (cf. [BFlFa], [EH] and [G]). It studies the properties of a scalar function ϕ(defined on a topological space X) which are invariant with respec…”
BackgroundEstimating persistent Betti numbers for discrete shape analysis
Cavazza, N. (2011) · AMS Dottorato Institutional Doctoral Theses Repository (University of Bologna) · doi:10.6092/unibo/amsdottorato/3468
Computing interesting topological features
Erickson, J., & Chambers, E. W. (2008)
Shape from Functions:Enhancing Geometrical-Topological Descriptors
Fabio, D. C. (2009) · AMS Dottorato Institutional Doctoral Theses Repository (University of Bologna) · doi:10.6092/unibo/amsdottorato/1961
“Therefore, the persistent homology of a filtered topological space can be portrayed as a collection of open intervals, called persistence intervals or barcode [11, 39] (see Figure 1.”
BackgroundYan, Z., Yi, D., Zhao, C., Jing, C., & Kang, S. G. (2013) · International Conference on Computational and Information Sciences · doi:10.1109/iccis.2013.437
Geometric Search and Geometric Processing
Chen, L. M. (2014) · Springer eBooks · doi:10.1007/978-3-319-12099-7_12
Discrete Time Evolution of Proteomic Biomarkers
Gnabasik, D., & Alaghband, G. (2014) · International Conference on Computational Science and Computational Intelligence (CSCI) · doi:10.1109/csci.2014.87
Foreword to the Special Focus on Mathematics, Data and Knowledge
Chen, X., Wang, D., & Zhang, X. (2013) · Mathematics in Computer Science · doi:10.1007/s11786-013-0169-2
“To answer the questions of how to infer high-dimensional structure from low-dimensional representations and how to assemble discrete points into global structure, one may employ topological methods to replace sets of data points by families of simplicial complexes, indexed with proximity parameters, to analyze the com…”
MethodCosta, J. P., Vejdemo‐Johansson, M., & Škraba, P. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1409.8613
“For approachable introductions to the field and its applications, we recommend [8], [17] and [22], and for an accessible introduction to algebraic topology [24].”
BackgroundZhang, J., & Du, Q. (2012) · Communications in Computational Physics · doi:10.4208/cicp.261211.290612a
“For in-depth discussions on the latter, we refer to [10, 23], though the diffuse interface approach differs from those through direct discrete topological constructions.”
BackgroundAn Information Theoretic Persistent Barcode
Chintakunta, H., Gentimis, T., González-Dı́az, R., Jiménez, M.-J., & Krim, H. (2014)
Algorithms and hardness results in computational homology
Freedman, D. Z., & Chen, C. (2009)
Geometric theory of inversion and seismic imagingPreprint
Lau, A., & Yin, C. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1506.00463
An Algebraic Topological Approach to Privacy: Numerical and Categorical DataPreprint
Alberto Speranzon, B., & Bopardikar, S. D. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1602.06643
“[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]”
BackgroundMethodTopological characterization of S[B] systems: From data to models of complexityPreprint
Merelli, E., Rucco, M., Sloot, P. M. A., & Tesei, L. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1505.05768
“For a more formal definition we refer to [20].”
BackgroundOn the Strengthening of Topological Signals in Persistent Homology through Vector Bundle Based Maps
Hanson, E., Motta, F. C., Peterson, C., & Ziegelmeier, L. (2012)
Transient cell assembly networks encode persistent spatial memoriesPreprint
Babichev, A., & Dabaghian, Y. A. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.02765
Ludu, A. (2016) · Springer series in synergetics · doi:10.1007/978-3-662-49078-5_8
Masood, A., Pan, W., & Doshi‐Velez, F. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.06250
“However, these are impractical in large scenarios or when the modes of the posterior distribution are unknown.”
BackgroundMultiradial (multi)filtrations and persistent homology
Martin, J. M. (2016) · NC Digital Online Collection of Knowledge and Scholarship (The University of North Carolina at Greensboro)
Pietrosanu, M. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1611.09435
An Algorithmic Way to Generate Simplexes for Topological Data Analysis
Rykaczewski, K., Wiśniewski, P., & Stencel, K. (2016) · CS&P
Hamiltonian dynamics and persistent homology
Zhang, J. (2016)
Congestion Barcodes: Exploring the Topology of Urban Congestion Using Persistent HomologyPreprint
Wu, Y., Shindnes, G., Karve, V., et al. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.08557
“invariants have been used to study feasibility sets in robotics; see [20].”
MethodCategorification of Gromov-Hausdorff Distance and Interleaving of FunctorsPreprint
Bubenik, P., de Silva, V., & Scott, J. (2017) · arXiv (Cornell University)
Levanger, R. (2017) · PhDT · doi:10.7282/t3057jsd
“We direct the reader to [2, 3, 4, 5] for an introduction to this rich topic.”
Topological data analysis and the MAA national study of college calculus
Stevens, I. E. (2016)
Reconstruction of compacta by finite approximations and inverse persistence
Ruiz, D. M., & Morón, M. A. (2020) · Revista Matemática Complutense · doi:10.1007/s13163-020-00356-w · Cited by 3
“In the last years, there has been a renewed interest in the approximation and reconstruction of topological properties of compact metric spaces, in part because of the development and importance of Computational Topology ([19,48]) and, more concretely, Topological Data Analysis (see the surveys by Carlsson [13] and Gh…”
BackgroundA topological characterization of S[B] systemsPreprint
Merelli, E., Rucco, M., Sloot, P. M. A., & Tesei, L. (2015) · arXiv (Cornell University)
Piecewise linear sheavesPreprint
Kashiwara, M., & Schapira, P. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.00349
Persistent Homology on Time series
Zhou, Y. (2016) · ERA: Education and Research Archive (University of Alberta) · doi:10.7939/r3k931f13
Equivariant Simplicial Reconstruction
Carbone, L., Nanda, V., & Naqvi, Y. (2020) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/20m1327483
“Examples of this phenomenon include, but are by no means restricted to, the recent employment of Laplacian eigenfunctions for spectral embedding [3], of rough paths in machine learning [9], of sheaf theory for linear programming [23], of Morse theory for image processing [13] and of (co)homology for data analysis [17].”
Vejdemo-Johansson, M. (2018) · Georgian Mathematical Journal · doi:10.1515/gmj-2018-0057
Path Space Cochains and Population Time Series AnalysisPreprint
Giusti, C., & Lee, D. (2018) · arXiv (Cornell University)
Complex and High‐Dimensional Inference in Astrostatistics
Cisewski-Kehe, J. (2017) · Wiley StatsRef: Statistics Reference Online · doi:10.1002/9781118445112.stat07934
Stable Persistent Homology Features of Dynamic Metric Spaces.Preprint
Kim, W., & Mémoli, F. (2018) · arXiv (Cornell University)
A flat persistence diagram for improved visualization of persistent homologyPreprint
Wadhwa, R., Dhawan, A., Williamson, D. F. K., & Scott, J. G. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1812.04567
“The most successful of these are the persistence barcode [Carlsson et al., 2005, Ghrist, 2008] and the persistence diagram [Edelsbrunner et al., 2002].”
BackgroundDistributions of Matching Distances in Topological Data AnalysisPreprint
Han, S. M., Okonek, T., Yadav, N., & Zheng, X. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1812.11258
“We give here a brief introduction to persistent homology in both the one- and two-parameter settings; more detailed surveys of the subject are found in [6, 7].”
BackgroundTopological data analysis: applications in machine learning
Calcina, S. G. S. (2019) · doi:10.11606/t.55.2019.tde-03012019-094950
A Topological Approach to Understanding Location-Based Data
McAbee, C., Wakefield, M. D., Roth, J. D., et al. (2018) · 2018 52nd Asilomar Conference on Signals, Systems, and Computers · doi:10.1109/acssc.2018.8645120
“The union of all these horizontal lines is called a barcode [15].”
BackgroundModeling a recurrent, hidden dynamical system using energy minimization and kernel density estimates
Karn, T. K., Petrone, S., & Griffin, C. H. (2019) · Physical review. E · doi:10.1103/physreve.100.042137
PHom-WAE: Persitent Homology for Wasserstein Auto-Encoders.Preprint
Charlier, J., Petit, F., Ormazábal, G., State, R., & Hilger, J. (2019) · arXiv (Cornell University)
Topologia computacional para análise de série temporal
Miranda, V. L. D. (2019) · doi:10.11606/d.59.2019.tde-13052019-175021
Tuning the Performance of a Computational Persistent Homology PackagePreprint
Hylton, A., Henselman‐Petrusek, G., Sang, J., & Short, R. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1809.04424
“The topological data produced by using persistent homology can be visualized through a barcode (25).”
BackgroundPersistent topology for cryo-EM data analysisPreprint
Xia, K., & Wei, G. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1412.2370
“Barcodes provide a systematic representation of topological persistence.(32) Figure 2 shows the intrinsic topological patterns of fullerene C20.”
BackgroundEmT: Locating empty territories of homology group generators in a dataset
Xu, X., & Cisewski-Kehe, J. (2019) · Foundations of Data Science · doi:10.3934/fods.2019010
“construction [34], which requires pairwise intersections to form a simplex [18, 11], as was used in the previous example.”
MethodQuantifying Homology ClassesPreprint
Chen, C., & Freedman, D. (2008) · arXiv (Cornell University) · doi:10.4230/lipics.socg.2020.32
Jaccard Filtration and Stable Paths in the Mapper.Preprint
Arendt, D., Broussard, M., Krishnamoorthy, B., & Saul, N. (2019) · arXiv (Cornell University)
Filtrations induced by continuous functions
Fabio, B. D., & Frosini, P. (2012) · AMS Acta (University of Bologna) · doi:10.48550/arxiv.1304.1268
“…introducing a nested collection of sets (usually carriers of simplicial complexes), the latter of giving a continuous function from a topological space to R or Rn (called a filtering function), whose sub-level sets represent the elements of the considered filtration (cf., e.g., [11, 15]).”
BackgroundIntegrated multimodal network approach to PET and MRI based on multidimensional persistent homologyPreprint
Lee, H., Kang, H., Chung, M. K., Lim, S., Kim, B.-N., & Lee, D. S. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1410.4620
“In Algebraic Topology, the Betti number is used to determine the shape of topological spaces including networks and to distinguish topological spaces [Adler et al., 2010; Carlsson et al., 2005; Edelsbrunner and Harer, 2008; Ghrist, 2008].”
MethodMultidimensional persistence in biomolecular dataPreprint
Xia, K., & Wei, G. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1412.7679
Multiresolution topological simplificationPreprint
Xia, K., Zhao, Z., & Wei, G. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1504.00033
“, Betti numbers, of the underlying copies can be tracked by using either persistent diagrams or barcode representation.(13) Appropriate mathematical apparatus has been deviced to organize simplicial complexes generated via the filtration process into homology groups.”
BackgroundStable comparison of multidimensional persistent homology groups with torsionPreprint
Frosini, P. (2010) · AMS Acta (University of Bologna) · doi:10.48550/arxiv.1012.4169 · Cited by 1
“Recent surveys on the topic include [18, 19, 23, 26].”
Spectral Sequences, Exact Couples and Persistent Homology of filtrationsPreprint
Basu, S., & Parida, L. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1308.0801
“We refer the reader to survey articles [6, 11, 16] for details regarding these applications.”
BackgroundCang, Z., & Wei, G. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1703.10966
“As a result, the birth, death, and persistence of topological invariants over the filtration give rise to the barcode representation of a given data set.(29) When persistent homology is used to analyze three dimensional (3D) protein structures, one-dimensional (1D) persistent homology barcodes are obtained as topologi…”
BackgroundA topological approach for protein classificationPreprint
Cang, Z., Mu, L., Wu, K., et al. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1510.00953
“The importance of retaining geometric information in topological analysis has been recognized [10], and topology has been advocated as a new approach for tackling big data sets [9, 11, 14, 54, 56].”
BackgroundObjective-oriented Persistent HomologyPreprint
Wang, B., & Wei, G. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1412.2368
Kundu, B., & Choudhury, S. K. (2017) · International conference natural language processing
Improving the Projection of Global Structures in Data through Spanning Trees
Alcaide, D., & Aerts, J. A. (2019) · Lirias (KU Leuven) · doi:10.48550/arxiv.1907.05783
“The stability of topological features is visualized in a so-called barcode [24] and analyzed to identify suitable parameterization [25, 26].”
BackgroundVisualization of AE's Training on Credit Card Transactions with Persistent HomologyPreprint
Charlier, J., Petit, F., Ormazábal, G., State, R., & Hilger, J. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.13020
“This family records how the topology of the space is changing when going through the filtration [20].”
BackgroundSensitivity Analysis with ManifoldsPreprint
Hernández, A., Solís, M., & Zúñiga, R. (2018) · arXiv (Cornell University)
From topological analyses to functional modeling: the case of hippocampusPreprint
Dabaghian, Y. A. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1909.07536
“…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).”
BackgroundTopological Stability of the Hippocampal Spatial Map and Synaptic Transience
Dabaghian, Y. A. (2021) · Springer proceedings in mathematics & statistics · doi:10.1007/978-981-16-0174-3_20
“), which captures the topological identity of X [48].”
BackgroundChiew, K. J. (2019) · DR-NTU (Nanyang Technological University) · doi:10.32657/10356/83241
“In addition, the difference between the death and the birth value, labelled as dy - dx is known as the persistence of the individual invariant.(25,27,31) This entire sequence of generating simplicial complex over a spatial scale is unique to persistent homology and referred to as filtration.”
BackgroundLeader Clusters and Shape Classes
Peters, J. F. (2019) · Intelligent systems reference library · doi:10.1007/978-3-030-22192-8_6
From Persistent Homology to Reinforcement Learning with Applications for Retail BankingPreprint
Charlier, J. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1911.11573
“This family records how the topology of the space is changing when going through the filtration [43].”
BackgroundRucco, M. (2019) · arXiv (Cornell University)
MVF Designer: Design and Visualization of Morse Vector FieldsPreprint
Zhou, Y., Lazovskis, J., Catanzaro, M. J., et al. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.09580
“in a simple and efficient way. In addition, inspired by topological data analysis, MVF Designer will encode and embrace the notion of persistence [ELZ02]; it will: 1.Compute and visualize the barcode [Ghr08] of a designer vector field to offer a global summary of its features; 2.Support the adjustment of function values…”
MethodPersistent Homotopy.Preprint
Batan, M. A., Pamuk, M., & Varlı, H. (2019) · arXiv (Cornell University)
Persistent Homology Techniques for Big Data and Machine Intelligence: A Survey
Joshi, M. A., Joshi, D., & Sharma, V. S. (2020) · Advances in intelligent systems and computing · doi:10.1007/978-981-15-1366-4_8
Hristopulos, D. T. (2020) · Advances in geographic information science · doi:10.1007/978-94-024-1918-4_15
Angarita, O., & Alejandra, M. (2019)
On algebraic and combinatorial properties of weighted simplicial complexesPreprint
Kara, S. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.11590
“For example, in TDA, one can encode the persistent homology of the data set in a “barcode” using Betti numbers (see [7, 9]).”
MethodMental Health Sensing Using Machine Learning
Taye, Y. M., Pingal, N. F., Seifu, Y. G., et al. (2020)
Grain Analysis of Atomic Force Microscopy Images via Topological Data Analysis
Duman, A. N. (2020) · Microscopy and Microanalysis · doi:10.1017/s1431927620020474
Persistent Homology in $\ell_{\infty}$ MetricPreprint
Beltramo, G., & Škraba, P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.02071
“In particular, the corresponding filtrations in Euclidean space may be related via a sandwiching argument [25].”
BackgroundHernández, A., Solís, M., & Zúñiga-Rojas, R. A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.05351
“The basic workflow described by [11] for TDA has three steps: 1.”
BackgroundMethodShape-based Feature Engineering for Solar Flare Prediction
Deshmukh, V., Berger, T. E., Meiss, J. D., et al. (2021) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v35i17.17795
“TDA handles this by filling in the gaps between the data points with different types of simplices.”
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
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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