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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Maximally persistent cycles in random geometric complexes
Bobrowski, O., Kahle, M., & Škraba, P. (2017) · The Annals of Applied Probability · doi:10.1214/16-aap1232 · Cited by 65
“Commonly (see [16, 33]), the persistence of a cycle is measured by the length of the corresponding bar in the barcode, namely by the difference δ(γ ) := γdeath − γbirth.”
BackgroundTopological data analysis of task-based fMRI data from experiments on schizophrenia
Stolz, B. J., Emerson, T., Nahkuri, S., Porter, M. A., & Harrington, H. A. (2021) · Journal of Physics Complexity · doi:10.1088/2632-072x/abb4c6 · Cited by 70
Significance mode analysis (SigMA) for hierarchical structures
Ratzenböck, S., Großschedl, J. E., Möller, T. B., Alves, J., Bomze, I. M., & Meingast, S. (2023) · Astronomy and Astrophysics · doi:10.1051/0004-6361/202243690 · Cited by 79
Holzinger, A. (2014) · Cited by 65
Exact topological inference of the resting-state brain networks in twins
Chung, M. K., Lee, H., DiChristofano, A., Ombao, H., & Solo, V. (2019) · Network Neuroscience · doi:10.1162/netn_a_00091 · Cited by 64
“…the single linkage dendrogram (SLD) (Lee et al., 2012), the Dulmage-Mendelsohn decomposition (Chung, Adluru, Dalton, Alexander, & Davidson, 2011; Pothen & Fan, 1990), or the simplical complex method (Carlsson & Memoli, 2008; de Silva & Ghrist, 2007; Edelsbrunner, Letscher, & Zomorodian, 2002).”
BackgroundClassifying Clustering Schemes
Carlsson, G., & Mémoli, F. (2013) · Foundations of Computational Mathematics · doi:10.1007/s10208-012-9141-9 · Cited by 85
“Moreover, the functoriality of the construction of path-connected components is a key feature in all aspects of topology, and the functoriality of clustering constructions should play a similar role in the computational topology of finite datasets [3, 8, 9, 11, 12].”
BackgroundPersistent homology and microlocal sheaf theory
Kashiwara, M., & Schapira, P. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-018-0019-z · Cited by 64
“See in particular the survey papers (Edelsbrunner and Harer 2008; Ghrist 2008; Oudot 2015).”
Pike, J. A., Khan, A. O., Pallini, C., Thomas, S. G., Mund, M., Ries, J., Poulter, N. S., & Styles, I. B. (2019) · Bioinformatics · doi:10.1093/bioinformatics/btz788 · Cited by 61
Applications of Topological Data Analysis in Oncology
Bukkuri, A., Andor, N., & Darcy, I. K. (2021) · Frontiers in Artificial Intelligence · doi:10.3389/frai.2021.659037 · Cited by 74
“For more details regarding persistent homology and barcodes (please see Ghrist, 2008; Carlsson, 2009; Edelsbrunner and Harer, 2010; Otter et al., 2017).”
BackgroundGeometrical and topological approaches to Big Data
Snåšel, V., Nowaková, J., Xhafa, F., & Barolli, L. (2016) · Future Generation Computer Systems · doi:10.1016/j.future.2016.06.005 · Cited by 64
“Efficient computational algorithms such as the pairing algorithm [25, 35], Smith normal form [26, 119] and Morse reduction [27, 38, 39], have been proposed to track topological variations during the filtration pro-”
MethodDecoding of Neural Data Using Cohomological Feature Extraction
Rybakken, E., Baas, N. A., & Dunn, B. A. (2018) · Neural Computation · doi:10.1162/neco_a_01150 · Cited by 61
“ings by approximations called Rips complexes, see Appendix D. 10In topological data analysis, persistent homology is often used instead, but we will need cohomology for the decoding step. We refer to Ghrist, 2008; Edelsbrunner and Harer, 2010 for introductions to persistence, and Ghrist, 2014; Carlsson, 2009 for intro…”
MethodGeometry Score: A Method For Comparing Generative Adversarial NetworksPreprint
Khrulkov, V., & Oseledets, I. V. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.02664 · Cited by 76
“Information about how homology is changing with respect to ε can be conveniently encoded in the so-called persistence barcodes (Ghrist, 2008; Zomorodian & Carlsson, 2005).”
BackgroundMethodInfluentialTopological consistency via kernel estimation
Bobrowski, O., Mukherjee, S., & Taylor, J. (2016) · Bernoulli · doi:10.3150/15-bej744 · Cited by 58
“We refer the reader to [22, 23, 26, 47] for more details and formal definitions.”
BackgroundSFCM: A Fuzzy Clustering Algorithm of Extracting the Shape Information of Data
Bui, Q., Vo, B., Snåšel, V., Pedrycz, W., Hong, T., Nguyên, N. T., & Chen, M. (2020) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2020.3014662 · Cited by 60
“Besides the very powerful techniques built on persistent homology [12]–[14], the Mapper algorithm is one of the most effective TDA tools to identify topological shape characteristics of data.”
MethodVisual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology
Hajij, M., Wang, B., Scheidegger, C., & Rosen, P. (2018) · IEEE Pacific Visualization Symposium · doi:10.1109/pacificvis.2018.00024 · Cited by 57
“A persistence diagram can be visualized as persistence barcodes [32] (see Figures 2 and 5), where each bar starts at time u and ends at time v.”
BackgroundObject-oriented persistent homology
Wang, B., & Wei, G. (2015) · Journal of Computational Physics · doi:10.1016/j.jcp.2015.10.036 · Cited by 50
Hypergraph-based persistent cohomology (HPC) for molecular representations in drug design
Liu, X., Wang, X., Wu, J., & Xia, K. (2020) · Briefings in Bioinformatics · doi:10.1093/bib/bbaa411 · Cited by 60
Phase Coexistence in Insect Swarms
Sinhuber, M., & Ouellette, N. T. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.178003 · Cited by 56
A functorial Dowker theorem and persistent homology of asymmetric networks
Chowdhury, S., & Mémoli, F. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-018-0020-6 · Cited by 75
Arai, M., Brandt, V. L., & Dabaghian, Y. A. (2014) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1003651 · Cited by 40
“The pattern of cycles is referred to as a barcode [16] that can be easily read to give topological information about a given environment (Figure 1C) [6,7].”
BackgroundMethodTopological methods for genomics: Present and future directions
Cámara, P. G. (2016) · Current Opinion in Systems Biology · doi:10.1016/j.coisb.2016.12.007 · Cited by 55
“A convenient representation of persistent homology is provided by barcodes [27] (Figure 2d).”
BackgroundPhinyomark, A., Ibáñez‐Marcelo, E., & Petri, G. (2017) · IEEE Transactions on Big Data · doi:10.1109/tbdata.2017.2734883 · Cited by 55
“, Mapper [80], [81]), graph-theoretic and network tools amenable to statistical mechanical treatments [82], and finally full fledged topological data analysis tools, in particular persistent homology [83].”
BackgroundSimplicial Neural NetworksPreprint
Ebli, S., Defferrard, M., & Spreemann, G. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.03633 · Cited by 113
“The advance of topological methods in machine learning [11, 12, 13], and the earlier establishment of topological data analysis (TDA) [14, 15, 16, 17] as a field in its own right, have confirmed the usefulness of viewing data as topological spaces in general, or in particular as simplicial complexes.”
BackgroundStratifying Multiparameter Persistent Homology
Harrington, H. A., Otter, N., Schenck, H., & Tillmann, U. (2019) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/18m1224350 · Cited by 52
“ameter case. 1. Introduction Persistent homology was introduced about 15 years ago, and has become a standard tool in the analysis of datasets with complicated structure; for overviews see [5], [14], [20], [35], and [25] for an introduction to computations. Persistent homology (PH) has been successful in attacking man…”
BackgroundTopological clustering of multilayer networks
Yuvaraj, M., Dey, A. K., Lyubchich, V., Gel, Y. R., & Poor, H. V. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2019994118 · Cited by 54
Consistent manifold representation for topological data analysis
Berry, T., & SAUER, T. D. (2019) · Foundations of Data Science · doi:10.3934/fods.2019001 · Cited by 53
“A second approach, known as persistent homology [4, 12, 14], produces a series of unweighted graphs that reconstructs topology one scale at a time, tracking homology generators as a scale parameter is varied.”
BackgroundOne-dimensional reduction of multidimensional persistent homology
Cagliari, F., Fabio, B. D., & Ferri, M. (2010) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-10-10312-8 · Cited by 52
A physics perspective on collective animal behavior
Ouellette, N. T. (2022) · Physical Biology · doi:10.1088/1478-3975/ac4bef · Cited by 51
“They further used techniques from topological data analysis [154, 155] to split the swarm into a dense inner core and a more dilute surrounding shell, which they argued were analogous to a condensed phase and a vapor phase.”
A Structural Average of Labeled Merge Trees for Uncertainty Visualization
Yan, L., Wang, Y., Munch, E., Gasparovic, E., & Wang, B. (2019) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2019.2934242 · Cited by 50
“For data modeled as scalar fields, the most popular descriptors include vector-based such as persistence diagrams [25] and barcodes [13, 32], graph-based such as merge trees [7], contour trees [15], and Reeb graphs [64], as well as complex-based such as Morse and Morse-Smale complexes [23, 24, 31].”
MethodTopological Complexity of Smooth Random Functions
Adler, R. J., & Taylor, J. (2011) · Lecture notes in mathematics · doi:10.1007/978-3-642-19580-8 · Cited by 51
“A very lively discussion of this trend can be found in Rob Ghrist’s review [40], book in progress [41] and website on a project on sensor topology for minimal planning [42].”
BackgroundPersistence-based clustering in riemannian manifolds
Chazal, F., Guibas, L. J., Oudot, S., & Škraba, P. (2011) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1998196.1998212 · Cited by 48
Measuring and computing natural generators for homology groups
Chen, C., & Freedman, D. (2009) · Computational Geometry · doi:10.1016/j.comgeo.2009.06.004 · Cited by 47
Topological Analysis of Variance and the Maxillary Complex
Heo, G., Gamble, J., & Kim, P. T. (2012) · Journal of the American Statistical Association · doi:10.1080/01621459.2011.641430 · Cited by 49
“Survey articles on this topic can be found in Edelsbrunner and Harer (2008) and Carlsson (2009), while a motivational overview of the topic was provided by Ghrist (2008), see also Adler et al. (2010).”
BackgroundTopological portraits of multiscale coordination dynamics
Zhang, M., Kalies, W. D., Kelso, J. A. S., & Tognoli, E. (2020) · Journal of Neuroscience Methods · doi:10.1016/j.jneumeth.2020.108672 · Cited by 49
“In practice, persistent homology (Zomorodian and Carlsson, 2005; Carlsson, 2009; Ghrist, 2007) is a natural choice for computing such a multiscale topological portrait.”
InfluentialExploring the topology of dynamical reconstructions
Garland, J., Bradley, E., & Meiss, J. D. (2016) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2016.03.006 · Cited by 52
“A simpler implementation of the fuzzy witness complex gives a “clique” or “flag” complex, analogous to the Rips complex [17], that consists of simplices whose pairs of vertices have a common witness (this is called a “lazy” complex in [3]).”
BackgroundIntegrated multimodal network approach to PET and MRI based on multidimensional persistent homology
Lee, H., Kang, H., Chung, M. K., Lim, S., Kim, B., & Lee, D. S. (2016) · Human Brain Mapping · doi:10.1002/hbm.23461 · Cited by 47
“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].”
MethodBlockchain networks: Data structures of Bitcoin, Monero, Zcash, Ethereum, Ripple, and Iota
Akçora, C. G., Gel, Y. R., & Kantarcıoğlu, M. (2021) · Wiley Interdisciplinary Reviews Data Mining and Knowledge Discovery · doi:10.1002/widm.1436 · Cited by 48
“Anotheremergingpromisingapproach for ransomware payment detection is topological data analysis (TDA) [30, 31].”
BackgroundContinuous Indexing of Fibrosis (CIF): improving the assessment and classification of MPN patients
Ryou, H., Sirinukunwattana, K., Aberdeen, A., Grindstaff, G., Stolz, B. J., Byrne, H. M., Harrington, H. A., Sousos, N., Godfrey, A. L., Harrison, C. N., et al. (2022) · Leukemia · doi:10.1038/s41375-022-01773-0 · Cited by 52
Topological data analysis in investment decisions
Goel, A., Pasricha, P., & Mehra, A. (2020) · Expert Systems with Applications · doi:10.1016/j.eswa.2020.113222 · Cited by 48
Topology-Based Machine Learning Strategy for Cluster Structure Prediction
Chen, X., Chen, D., Weng, M., Jiang, Y., Wei, G., & Pan, F. M. (2020) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.0c00974 · Cited by 46
Crackle: The Homology of Noise
Adler, R. J., Bobrowski, O., & Weinberger, S. (2014) · Discrete & Computational Geometry · doi:10.1007/s00454-014-9621-6 · Cited by 42
“Some recent references are [2,3,9,15,19] with two reviews, from different aspects, in [1] and [12].”
BackgroundParametric Inference using Persistence Diagrams: A Case Study in Population GeneticsPreprint
Emmett, K., Rosenbloom, D. I. S., Cámara, P. G., & Rabadán, R. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1406.4582 · Cited by 47
“For detailed background, see the reviews (Carlsson, 2009; Ghrist, 2008) and the books (Edelsbrunner & Harer, 2010; Zomorodian, 2005).”
BackgroundZomorodian, A. J. (2009) · Chapman & Hall/CRC applied algorithms and data structures series · doi:10.1201/9781584888215-c3 · Cited by 46
Levy, E. R. J., Segura-Carrillo, S., Park, E., Redman, W. T., Hurtado, J., Chung, S., & Fenton, A. A. (2023) · Cell Reports · doi:10.1016/j.celrep.2023.113142 · Cited by 51
Sebastian, E. R., Quintanilla, J. P., Sánchez-Aguilera, A., Esparza, J., Cid, E., & de la Prida, L. M. (2023) · Nature Neuroscience · doi:10.1038/s41593-023-01471-9 · Cited by 43
“First, we sought to examine the topology of the SWR cloud in the 127D space by estimating the presence of discontinuous components, holes and cavities with persistent homology 21 .”
MethodMultiresolution persistent homology for excessively large biomolecular datasets
Xia, K., Zhao, Z., & Wei, G. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4931733 · Cited by 38
Persistent topology for cryo‐EM data analysis
Xia, K., & Wei, G. (2015) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.2719 · Cited by 36
“Barcodes provide a systematic representation of topological persistence.(32) Figure 2 shows the intrinsic topological patterns of fullerene C20.”
BackgroundBounds on spectral norms and barcodes
Kislev, A., & Shelukhin, E. (2021) · Geometry & Topology · doi:10.2140/gt.2021.25.3257 · Cited by 42
“ersistence modules and boundary depth. We recall brie y the category pmod of persistence modules that we work with, together with their relevant properties. For detailed treatment of these topics see [12, 22, 23, 34, 49, 121]. Let K be a eld. A persistence module over K is a pair (V;ˇ) where, fVtg t2R is a family of n…”
BackgroundAnalyzing Prospects for Quantum Advantage in Topological Data Analysis
Berry, D. W., Su, Y., Gyurik, C., King, R., Basso, J., Barba, A. D. T., Rajput, A., Wiebe, N., Dunjko, V., & Babbush, R. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.010319 · Cited by 74
“[49] Robert Ghrist, “Barcodes: the persistent topology of data,” Bulletin of the American Mathematical Society 45, 61–75 (2008).”
BackgroundMultivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures
Rieck, B., Mara, H., & Leitte, H. (2012) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2012.248 · Cited by 42
“Currently, there are two common methods for visualizing these intervals: Drawing them as intervals in the plane, which is done by persistence barcodes [14], or drawing them as points in the plane, which is done by persistence diagrams [6].”
MethodMittal, K., & Gupta, S. (2017) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4983840 · Cited by 44
Demonstration of topological data analysis on a quantum processor
Huang, H.-L., Wang, X., Rohde, P. P., Luo, Y.-H., Zhao, Y., Liu, C., Li, L., Liu, N.-L., Lu, C., & Pan, J.-W. (2018) · Optica · doi:10.1364/optica.5.000193 · Cited by 47
F-Mapper: A Fuzzy Mapper clustering algorithm
Bui, Q., Vo, B., Nguyen, H.-A. T., Nguyen, Q. V. H., & Snåšel, V. (2019) · Knowledge-Based Systems · doi:10.1016/j.knosys.2019.105107 · Cited by 44
Topological structures in the equities market network
Leibon, G., Pauls, S. D., Rockmore, D. N., et al. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0802806106 · Cited by 44
“Thedetection of a well-articulated topology in this network is in a similar spirit to that of [20, 21] where the computation of the homology of large datasets is used to topologically classify the data configuration.”
MethodBrain activity: Conditional dissimilarity and persistent homology
Cassidy, B., Rae, C. D., & Solo, V. (2015) · IEEE International Symposium on Biomedical Imaging (ISBI) · doi:10.1109/isbi.2015.7164127 · Cited by 41
“Here we demonstrate the formation of a ‘barcode’ from DISCOH, which describes the rate of formation of connected graph components as the dissimilarity threshold is adjusted [12].”
BackgroundClustering Comparison of Point Processes, with Applications to Random Geometric Models
Błaszczyszyn, B., & Yogeshwaran, D. (2014) · Lecture notes in mathematics · doi:10.1007/978-3-319-10064-7_2 · Cited by 43
Topological Data Analysis for Image Tampering Detection
Asaad, A., & Jassim, S. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-64185-0_11 · Cited by 43
“The features that are not persistent by gradually increasing the threshold considered to be noise [7, 19].”
BackgroundConnectivity in fMRI: Blind Spots and Breakthroughs
Solo, V., Poline, J., Lindquist, M. A., Simpson, S. L., Bowman, F. D., Chung, M. K., & Cassidy, B. (2018) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2018.2831261 · Cited by 41
“Aside from the persistence diagram, various persistent homological features such as barcodes [89] and persistent landscapes [90] are available.”
BackgroundDistance functions, critical points, and the topology of random Čech complexes
Bobrowski, O., & Adler, R. J. (2014) · Homology Homotopy and Applications · doi:10.4310/hha.2014.v16.n2.a18 · Cited by 43
“ological, homological, point of view of random structures such as graphs and simplicial complexes. Some recent references are [3, 5, 7, 10, 18, 24] with two reviews, from dierent aspects, in [1] and [12]. Many of these papers nd their raison d’^etre in essentially statistical problems, in which data generates these st…”
BackgroundPersistent Homology Machine Learning for Fingerprint Classification
Giansiracusa, N., Giansiracusa, R., & Moon, C. (2019) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2019.00201 · Cited by 42
Topology‐Aware Surface Reconstruction for Point Clouds
Gabrielsson, R. B., Ganapathi‐Subramanian, V., Škraba, P., & Guibas, L. J. (2020) · Computer Graphics Forum · doi:10.1111/cgf.14079 · Cited by 41
DLPFC transcriptome defines two molecular subtypes of schizophrenia
Bowen, E. F. W., Burgess, J. L., Granger, R., et al. (2019) · Translational Psychiatry · doi:10.1038/s41398-019-0472-z · Cited by 42
Multiscale Persistent Functions for Biomolecular Structure Characterization
Xia, K., Li, Z., & Mu, L. (2017) · Bulletin of Mathematical Biology · doi:10.1007/s11538-017-0362-6 · Cited by 40
Homological percolation and the Euler characteristic
Bobrowski, O., & Škraba, P. (2020) · Physical review. E · doi:10.1103/physreve.101.032304 · Cited by 39
“To describe these two topics and the connection between them, we will use the language of persistent homology [13, 15, 16, 41], which is main workhorse in the field of Topological Data Analysis (TDA) [8, 17].”
MethodTowards quantum advantage via topological data analysis
Gyurik, C., Cade, C., & Dunjko, V. (2022) · Quantum · doi:10.22331/q-2022-11-10-855 · Cited by 70
“Figure 1: Example of a clique complex with three 1-dimensional holes (adapted from [14]).”
BackgroundMethodHofmann, A., Krufczik, M., Heermann, D. W., & Hausmann, M. (2018) · International Journal of Molecular Sciences · doi:10.3390/ijms19082263 · Cited by 43
“In order to overcome these restrictions, a novel mathematical approach is presented here, which analyzes SMLM data with methods of persistent homology [44].”
BackgroundMethodInfluentialAesthetic Highlight Detection in Movies Based on Synchronization of Spectators’ Reactions
Muszyński, M., Κωστούλας, Θ., Lombardo, P., Pun, T., & Chanel, G. (2018) · ACM Transactions on Multimedia Computing Communications and Applications · doi:10.1145/3175497 · Cited by 40
“Finally, we introduce the basics of persistent homology: filtrations and persistence diagrams [32, 63, 64].”
MethodTopological Data Analysis of Functional MRI Connectivity in Time and Space Domains
Anderson, K. L., Anderson, J. S., Palande, S., & Wang, B. (2018) · Lecture notes in computer science · doi:10.1007/978-3-030-00755-3_8 · Cited by 40
“TDA [3,13] of networks goes beyond graph-theoretic analysis by utilizing tools from computational topology to describe the architecture of networks or data structures in more flexible ways.”
InfluentialThe topological Dirac equation of networks and simplicial complexes
Bianconi, G. (2021) · Journal of Physics Complexity · doi:10.1088/2632-072x/ac19be · Cited by 48
“H̃[3] = ⎜⎜⎜⎝ m0IN[0] b[1]B[1] 0N[0]×N[2] 0N[0]×N[3] b [1]B † [1] 0N[1]×N[1] b[2]B[2] 0N[1]×N[3] 0N[2]×N[0] b [2]B † [2] 0N[2]×N[2] b[3]B[3] 0N[3]×N[0] 0N[3]×N[1] b [3]B † [3] −m0IN[3] ⎟⎟⎟⎠ .”
Multiscale topology characterizes dynamic tumor vascular networks
Stolz, B. J., Kaeppler, J., Markelc, B., Braun, F., Lipsmeier, F., Muschel, R. J., Byrne, H. M., & Harrington, H. A. (2022) · Science Advances · doi:10.1126/sciadv.abm2456 · Cited by 51
“These multiscale and multidimensional topological features are summarized in a barcode ( 49 ) (see Fig.”
Structure-Function Network Mapping and Its Assessment via Persistent Homology
Liang, H., & Wang, H. (2017) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1005325 · Cited by 38
“We further introduce a novel measure of network similarity based on persistent homology [25–27] for assessing the goodness of fit for the mapping; such a measure enables the complete comparison of network topological changes across all possible thresholds, thereby effectively circumvents the…”
BackgroundMethodRen, S., Wu, C., & Wu, J. (2018) · Rocky Mountain Journal of Mathematics · doi:10.1216/rmj-2018-48-8-2661 · Cited by 38
“First, we give a review of persistence [14, 25], with generalizations to the weighted case.”
Simplicial Models and Topological Inference in Biological Systems
Nanda, V., & Sazdanović, R. (2013) · Natural computing series · doi:10.1007/978-3-642-40193-0_6 · Cited by 42
“Numerous applications of persistent homology to various problems in the experimental sciences have been thoroughly documented elsewhere [4, 13, 18].”
BackgroundTransfer learning for autonomous chatter detection in machining
Yesilli, M. C., Khasawneh, F. A., & Mann, B. P. (2022) · Journal of Manufacturing Processes · doi:10.1016/j.jmapro.2022.05.037 · Cited by 40
“[61, 62, 63, 64, 65, 66] for further details.”
Persistent homology of toroidal sliding window embeddings
Perea, J. A. (2016) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2016.7472916 · Cited by 39
“While defining persistent homology (for the Rips filtration and coefficients on a field F) is out of the scope of this paper, we refer the reader to [10] for a terse introduction.”
BackgroundTopological image modification for object detection and topological image processing of skin lesions
Vandaele, R., Nervo, G. A., & Gevaert, O. (2020) · Scientific Reports · doi:10.1038/s41598-020-77933-y · Cited by 39
Persistent Homology of Geospatial Data: A Case Study with Voting
Feng, M., & Porter, M. A. (2021) · SIAM Review · doi:10.1137/19m1241519 · Cited by 39
Path Topology in Molecular and Materials Sciences
Chen, D., Liu, J., Wu, J., Wei, G., Pan, F. M., & Yau, S. (2023) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.2c03706 · Cited by 40
On the topology of random complexes built over stationary point processes
Yogeshwaran, D., & Adler, R. J. (2015) · The Annals of Applied Probability · doi:10.1214/14-aap1075 · Cited by 60
“s of the Vietoris–Rips complex. 1.3. Some implications for topological data analysis. Perhaps the core tool of TDA is persistent homology, as visualized through barcodes and persistence diagrams; cf. [11, 15, 19, 42]. See also the very accessible recent survey [12]. While here is not the place to go into the details o…”
BackgroundInfluentialMorphological brain network assessed using graph theory and network filtration in deaf adults
Kim, E., Kang, H., Lee, H., Lee, H., Suh, M., Song, J., Oh, S. H., & Lee, D. S. (2014) · Hearing Research · doi:10.1016/j.heares.2014.06.007 · Cited by 40
Persistent Homology Guided Force-Directed Graph Layouts
Suh, A., Hajij, M., Wang, B., Scheidegger, C., & Rosen, P. (2019) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2019.2934802 · Cited by 35
“Finally, the set of all PH features produces a compressed description of the graph that can be represented using a persistence barcode [35], which our approach uses as a graphical user interface to manipulate the graph via PH features , instead of direct node manipulation.”
BackgroundMethodLi, Y., Islambekov, U., Akçora, C. G., Smirnova, E., Gel, Y. R., & Kantarcıoğlu, M. (2020) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611976236.59 · Cited by 40
“The most widely used topological summaries of persistent features are the Betti numbers, barcode plots, persistent diagrams, and persistent landscapes [19].”
BackgroundCoreduction homology algorithm for inclusions and persistent homology
Mrożek, M., & Wanner, T. (2010) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2010.09.036 · Cited by 38
“See for example [10] for an introduction.”
Scalable brain network construction on white matter fibers
Chung, M. K., Adluru, N., Dalton, K. M., Alexander, A. L., & Davidson, R. J. (2011) · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · doi:10.1117/12.874245 · Cited by 35
Reconstructing relational information
Horecka, K. M., Dulas, M. R., Schwarb, H. D., Lucas, H. D., Duff, M. C., & Cohen, N. J. (2017) · Hippocampus · doi:10.1002/hipo.22819 · Cited by 36
“Topological data analysis (TDA) techniques may prove to be a better analysis framework for determining the degree to which high-level relational information is preserved in a reconstruction (Carlsson, 2009; Ghrist, 2007).”
The fiber of the persistence map for functions on the interval
Curry, J. M. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-019-00024-z · Cited by 35
A topological approach to selecting models of biological experiments
Ulmer, M. P., Ziegelmeier, L., & Topaz, C. M. (2019) · PLoS ONE · doi:10.1371/journal.pone.0213679 · Cited by 36
Anand, D. V., Meng, Z., Xia, K., & Mu, Y. (2020) · Scientific Reports · doi:10.1038/s41598-020-66710-6 · Cited by 41
Quantum utility – definition and assessment of a practical quantum advantage
Herrmann, N., Arya, D., Doherty, M. W., Mingare, A., Pillay, J. C., Preis, F., & Prestel, S. (2023) · IEEE International Conference on Quantum Software (QSW) · doi:10.1109/qsw59989.2023.00028 · Cited by 53
“In essence, it involves the estimation of the persistent homology [108] of a given data graph, which contains the changing pattern of the graph’s Betti numbers.”
BackgroundStatistical Analysis of Persistence Intensity FunctionsPreprint
Chen, Y., Wang, D., Rinaldo, A., & Wasserman, L. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1510.02502 · Cited by 37
“of TDA is to extract useful topological information from a given data and make inferences using these information. TDA has been applied various fields such as pattern recognition [8], computer vision [15], and biology [23, 22]. One of the most popular and powerful approaches to TDA is persistent homology. This is a mul…”
BackgroundActive phase discovery in heterogeneous catalysis via topology-guided sampling and machine learning
Zheng, S., Zhang, X.-M., Liu, H.-S., Liang, G.-H., Zhang, S.-W., Zhang, W., Wang, B., Yang, J., Jin, X., Pan, F. M., et al. (2025) · Nature Communications · doi:10.1038/s41467-025-57824-4 · Cited by 37
“Topological persistence is usually represented by persistent barcodes, where different horizontal line segments or bars denote homology generators 41 .”
Serrano, D. H., & Gómez, D. S. (2020) · Applied Mathematics and Computation · doi:10.1016/j.amc.2020.125331 · Cited by 36
“The applications of TDA have been fruitful and varied during the last 15 years and have mainly (but not only) focused on the study of network topology and its evolution using homology and persistent homology of simplicial complexes; see for instance [5, 7, 6, 11, 30, 27, 28, 34] for applications in social systems, [13…”
BackgroundQuantum algorithm for persistent Betti numbers and topological data analysis
Hayakawa, R. (2022) · Quantum · doi:10.22331/q-2022-12-07-873 · Cited by 40
“Estimating the persistent Betti numbers allows one to recover the topological summary of data such as the persistent Barcodes.”
BackgroundGasparovic, E., Gommel, M., Purvine, E., Sazdanović, R., Wang, B., Wang, Y., & Ziegelmeier, L. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_3 · Cited by 35
Kuang, L., Han, X., Chen, K., Caselli, R. J., Reiman, E. M., Wang, Y., & Initiative, A. D. N. (2018) · Human Brain Mapping · doi:10.1002/hbm.24383 · Cited by 33
“In a graph filtration (defined in Appendix A), the zeroth Betti numbers plot (BNP) (defined in Appendix B) shows how the number of connected components (i.e., the zeroth Betti number) varies over different filtration values (Ghrist, 2008).”
BackgroundMethodInfluentialTopological Data Analysis for Multivariate Time Series Data
El-Yaagoubi, A. B., Chung, M. K., & Ombao, H. (2023) · Entropy · doi:10.3390/e25111509 · Cited by 45
“The reader is refered to [18] for an introduction to the notion of persistence, [17] and [20] for a survey on persistent homology and barcodes, and [13] for a review of TDA in general.”
BackgroundTopological Schemas of Cognitive Maps and Spatial Learning
Babichev, A., Cheng, S., & Dabaghian, Y. A. (2016) · Frontiers in Computational Neuroscience · doi:10.3389/fncom.2016.00018 · Cited by 28
“With methods drawn from persistent homology theory (Zomorodian and Carlsson, 2005; Ghrist, 2008) one can determine the minimal period Tmin required for removing the spurious loops, which provides a theoretical estimate of the time required to learn the environment (Dabaghian et al., 2012; Arai,…”
MethodFeng, M., & Porter, M. A. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033426 · Cited by 34
Simplicial degree in complex networks. Applications of topological data analysis to network science
Serrano, D. H., Serrano, J. H., & Gómez, D. S. (2020) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2020.109839 · Cited by 34
“The application of TDA on the study of higher-order structures in complex systems has produced significant advances in a wide variety of domains; see for instance [3, 7, 9, 31, 27, 28, 35] for social systems, [15, 16, 25, 39, 40] for biomolecular systems, [12, 18, 19, 26, 37, 38] for brain networks, [32] for network c…”
BackgroundMoraleda, R. R., Xiong, W., Halama, N., Breitkopf‐Heinlein, K., Dooley, S., Salinas, L., Heermann, D. W., & Valous, N. A. (2017) · Medical Image Analysis · doi:10.1016/j.media.2017.02.009 · Cited by 33
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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