Topology and data

In the paper

“Although clustering is clearly a very important part of data analysis, the ways in which it is formulated and implemented are fraught with ambiguities.In particular, the arbitrariness of various threshhold choices and lack of robustness are difficulties one confronts.”

What the sources say

• Partly supported

  The Introduction describes clustering as widely used and says its consistency guarantees are unsatisfactory. The discussion following Proposition 24 shows that unnormalized spectral clustering can fail and that its reliability can be difficult to assess. These passages support the general concern about reliability, but they do not identify arbitrary threshold choices as a difficulty or establish a lack of robustness for clustering generally; no page numbers are available in our copy. (LLM, conf=0.82)

  cites [43] von Luxburg, Ulrike, 2008, Consistency of spectral clustering, Ann. Statist. sourceExperimental — accuracy not yet measured
• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [39] kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.

In the paper

“In particular, in [41], indications were found that the data was concentrated around an annulus.”

What the sources say

• Supported

  Section 5.1 describes the ideal-edge manifold using a circular orientation parameter and a bounded displacement parameter, giving it an annulus-like parametrization. Section 5.2 finds evidence that optical data points concentrate near that manifold. The paper does not use the word “annulus,” but the citing sentence is a reasonable, hedged paraphrase. (LLM, conf=0.82)

  cites [41] lee A.B. Lee, K.S. Pedersen, and D. Mumford, The nonlinear statistics of high-contrast patches in natural images, International Journal of Computer Vision (54), No. 1-3, August 2003, pp. 83-103. source

In the paper

“By using persistence, one is able to develop a systematic way of addressing such questions.That methodology has been described in [65].”

What the sources say

• Supported

  Section 1.2 describes a method for localizing topological attributes relative to a cover. Section 4.4 gives the persistence algorithm that computes representatives of homology classes, supplying descriptions of those attributes. Together, these passages support the paraphrase about systematically addressing where features come from. (LLM, conf=0.90)

  cites [65] Zomorodian, Afra, 2008, Localized homology, Comput. Geom. source

In the paper

“The algorithm Ξ is the algorithm which associates to a finite metric space (X, d X ) the persistent set {π 0 (V R(X, ))} ≥0 .”

What the sources say

• Supported

  Example 2.1 (Section 2, p. 3) associates a finite metric space with a persistent set whose blocks at scale r contain exactly the points connected by chains of distances at most r. Those blocks are the connected components of the Vietoris–Rips complex at that scale, as the claim describes. (LLM, conf=0.90)

  cites [9] memolicarlsson G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg , Preprint, March 2008. source

In the paper

“Also, results about barcodes under the general heading have now begun to appear (see [15] and [16]).”

What the sources say

• Supported

  The Introduction (p. 2) says the paper makes concrete statements about the stability of barcodes. The Applications section (p. 16) states a Barcode Theorem, making this paper an example of the barcode results the citing sentence says have begun to appear. (LLM, conf=0.90)

  cites [15] Cohen-Steiner, David, 2007, Stability of persistence diagrams, Discrete Comput. Geom. source
• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [16] stabilitytwo D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz functions have 𝐿_{𝑝}-stable persistence, Found. Comput. Math., to appear.

In the paper

“It is readily checked that θ is a functor from N pers (Ab) to the category of graded Z[t]-modules, and is in fact an equivalence of categories, since an inverse functor can be given by M * → {M n }, where the morphisms ψ mn are given by multiplication by t n-m .”

In the paper

“A first observation is that the points are scattered throughout the 7-sphere, in the sense that no point on S 7 is very far from the set, but that the density appears to vary a great deal.”

In the paper

“We note that there are a number of short lines, and one long one.According to the philosophy mentioned above, this suggests that the first Betti number should be estimated to be one.”

In the paper

“We note that in this case, there are many short segments and 5 longer segments.It is again the case that this barcode is stable over the choices of landmark points and samples from M. This suggests that the first Betti number of the putative underlying space should be five.”

In the paper

“Note that while the two smaller holes have now been “filled in”, a new hole has been introduced in the lower right-hand portion of the figure.Consequently, if we were to compute the homology of this complex, we would obtain a first Betti number of two.”

In the paper

“We note that the vertical circles intersect the horizontal primary circle in two points, and do not intersect each other, so the picture in question produces a natural embedding of the three circle model in the Klein bottle.”

In the paper

“We note that the β 0 barcode (the upper picture) clearly shows the single component, the β 1 shows two lines from threshhold parameter values .15 to .35, and finally the β 2 barcode shows a single line on roughly the same interval.This gives β 0 = 1, β i = 2, and β 2 = 1.These are the mod 2 Betti numbers for the Klein bottle.”

In the paper

“At both scales, there is a central dense core, and two”

In the paper

“These two are distinguished by mod 3 homology, and we have performed the computation to show that the mod 3 homology is consistent with the Klein bottle and not with the torus.”

In the paper

“We note that given only the Mapper output, one might suspect that the small feature (the array of orange nodes) could simply be an artifact, but examination of the data shows that they correspond to essentially distinct contact maps.”

In the paper

“Some experimentation with the three circle model results in the following picture in which the horizontal segments form the primary circle and the vertical segments form the secondary circles, as in the diagram.”

In the paper

“We have informally confirmed that the indicated patches are the ones which occur in the high density portions of the data set.”

In the paper

“By far the most frequently occurring signatures were (1, 1, 0) and (1, 0, 1), corresponding to a circle and a sphere, respectively.”

In the paper

“Note that the two secondary circles each intersect the primary circle in two points, and that they do not intersect each other.”

In the paper

“Note also that the data are obtained by simply examining the states occurring in the simulation, and that it does not include any dynamic information which would show how the states are traversed in the folding process.”

In the paper

“The set S exhibits the three circle model clearly, but enlarging it still does not exhibit the non-trivial β 2 which would be characteristic of the Klein bottle.”

In the paper

“It is easy to check (a) that the factorization is a homeomorphism and (b) that the orbit space is homeomorphic to a Klein bottle.”

In the paper

“A M iso -functorial clustering algorithm determines an choice p ι ∈ P G ι ι for every ι, and conversely an arbitrary choice of such p ι 's determines an M iso -functorial clustering algorithm.”

In the paper

“Each of these notions creates the structure of a category whose objects are the finite metric spaces, since one can readily observe that each of the classes of morphisms is closed under composition and contains the identity.”

In the paper

“Some work in clustering theory has been done in trying to determine the optimal choice of , but it is now well understood that it is much more informative to maintain the entire dendrogram of the set, which provides a summary of the behavior of clustering under all possible values of the parameter at once.”

In the paper

“In the biological context, notions of distance are constructed using some intuitively attractive measures of similarity (such as BLAST scores or their relatives), but it is far from clear how much significance to attach to the actual distances, particularly at large scales.”

In the paper

“Topology studies only properties of geometric objects which do not depend on the chosen coordinates, but rather on intrinsic geometric properties of the objects.As such, it is coordinate-free.”

In the paper

“Moreover, it is understood that most of the information about topological spaces can be obtained through diagrams of discrete sets, via a process of simplicial approximation.”

In the paper

“We then demonstrate that persistence can be generalized in several different directions, providing more structure and information about the data sets in question.”

In the paper

“If the clusterings are consistent, i.e. if the clusters in C(X 1 ) and C(X 2 ) in C(X 1 ∪ X 2 ) correspond well under these maps, then one can regard that as evidence that the subsample clusterings actually correspond to clusterings on the full data set X.”

In the paper

“For any topological space X with a finite number of path components, β 0 (X) is the number of path components.”

In the paper

“We begin with a discussion of persistent homology, which is a mathematical formalism which permits us to infer topological information from a sample of a geometric object, and we show how it can be applied to particular data sets arising from natural image statistics and neuroscience.”

In the paper

“Functoriality is central in algebraic topology in that the functoriality of homological invariants is what permits one to compute them from local information, and that functoriality is at the heart of most of the interesting applications within mathematics.”

In the paper

“We then show that the philosophy of functoriality can be used to reason about the nature of clustering methods, and we conclude by speculating about theorems one might hope to prove and discussing how the subject might develop more generally.”

In the paper

“One important goal of data analysis is to allow the user to obtain knowledge about the data, i.e. to understand how it is organized on a large scale.”

In the paper

“Although we often receive data in the form of vectors of real numbers, it is frequently the case that the coordinates, like the metrics mentioned above, are not natural in any sense, and that therefore we should not restrict ourselves to studying properties of the data which depend on any particular choice of coordinates.”

In the paper

“The second method is more efficient, since it is precisely adapted to the complex at hand.Furthermore, it contains complete information about the original R-vector space.”

In the paper

“In interpreting the output, one now finds that long intervals in the output barcode indicate the presence of a homology class which “persists” over a long range of parameter values, while short intervals indicate cycles which are “born” at a given parameter value and then “die” at a nearby parameter value.”

In the paper

“This is due to the fact that for finite metric spaces, it is generically the case that each value of the distance is taken only for one pair of points, so one does not have any points which are equidistant between a pair of landmarks.”

In the paper

“Intuitively, we regard these two holes as coming from faulty sampling or other errors in the recovery of the data.”

In the paper

“The pictures and discussion above allow us to formulate the intuition that long intervals correspond to large scale geometric features in the space and that short intervals correspond to noise or inadequate sampling.”

In the paper

“Both the the Čech and Vietoris-Rips complexes typically produce simplices in dimensions much higher than the dimension of the space.”

In the paper

“It turns out that H simp k (X, Z) is always canonically isomorphic to the singular homology of the space |X|.”

In the paper

“The conclusion is that for simplicial complexes, homology is algorithmically computable, although standard methods for computing Smith normal form can often be painfully slow.”

In the paper

“If further is sufficiently small, then all the balls will be geodesically convex, and the complex will compute the homology of M correctly.”

In the paper

“Note that if the centers actually lie on a submanifold M ⊆ R n , and the set X is sufficiently dense in M , then this complex is the Čech complex attached to a covering of M .”

In the paper

“It is clearly the case that if we have 0 ≤ ≤ , then we have an inclusion W s (X, L, ) ↩→ W s (X, L, ) and similarly for W s V R , W w , and W w V R .”

In the paper

“However, for finite metric spaces, it generically produces degenerate (i.e.discrete) complexes, with no one dimensional simplices.”

In the paper

“In rough terms, δ k for large values of k computes density using points in large neighborhoods of x, and for small values uses small neighborhoods.So, δ k for large k corresponds to a smoothed out notion of density, and for small k corresponds to a version which carries more of the detailed structure of the data set.”

In the paper

“It was observed in [38] that in an informal sense, the images in the different conditions appeared to be quite similar, and a statistical analysis strongly suggested that the behavior of V1 in the spontaneous condition was consistent with a behavior which consisted in moving through a family of evoked images corresponding to responses to angular boundaries, without any particular order.”

What the sources say

• ?Not checked — no full text of the sourcecites [38] kenet T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, Spontaneously emerging cortical representations of visual attributes, Nature 425 (2003), 954-956. source

In the paper

“This result is consistent with what was obtained by using ISOMAP in [61].”

What the sources say

• ?Not checked — no full text of the sourcecites [61] isomapJ.B. Tenenbaum, V. de Silva and J.C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290 (2000) (December), pp. 2319-2323. source

In the paper

“The arrays of neurons studied in [59] are from the primary visual cortex or V1 in Macaque monkeys.”

What the sources say

• ?Not checked — no full text of the sourcecites [59] visionG. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach, Topological Structure of Population Activity in Primary Visual Cortex, Journal of Vision, Volume 8, Number 8, Article 11, pp. 1-18, 2008. source

In the paper

“since the intuition is that features which appear at several levels in such a multiresolution diagram would be more intrinsic to the data set than those which appear at a single level.”

In the paper

“The core consists of normal or near-normal patients, and the two flares consist of patients with the two different forms of diabetes.”

In the paper

“Z[m, n]-persistence vector spaces are examples of quiver representations, a highly developed area of algebra.See [29] for a complete description.”

What the sources say

• ?Not checked — no full text of the sourcecites [29] Gabriel, P., 1997 source

In the paper

“These functions are clearly invariants of the isomorphism class of the module M , and we refer to them as the dimension and rank invariants, respectively.”

In the paper

“If the question is instead whether or not an element arises from a union of k elements of S, then we do not obtain complete information this way, but we do obtain partial information in that we can preclude the possibility that a class arises from such a union.”

In the paper

“We note that the correspondence X → π 0 (X) can actually be viewed as a functor (see [44]) from the category of topological spaces to the category of sets, in the sense that a continuous map f : X → Y induces a map of sets π 0 (f ) : π 0 (X) → π 0 (Y )”

What the sources say

• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [44] Mac Lane, Saunders, 1998

In the paper

“As we have seen above, algebraic topology is capable of producing signatures which indicate the presence of topological features within a space.As it stands, however, it is not capable of describing the source of the feature, i.e.where in the space the hole or other feature is located.”

In the paper

“Theorem 4.5.Let M denote any Z[m, n]-persistent F -vector space, so that every vector space V i is finite dimensional.Then there is an isomorphism”

What the sources say

• ?Not checked — no full text of the sourcecites [29] Gabriel, P., 1997 source

In the paper

“As Kleinberg points out, it is possible to do so in a number of different ways by specifying cost functions on particular clusterings, and prove a uniqueness result for optimal choices of clusterings with respect to this cost function”

What the sources say

• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [39] kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.

In the paper

“This machinery is not available in the multivariable case, but it turns out that it can be replaced by the Gröbner basis methodology (see [18] or [47]).”

What the sources say

• ?Not checked — no full text of the sourcecites [18] Cox, David, 1998 source
• ?Not checked — no full text of the sourcecites [47] Miller, Ezra, 2005 source

In the paper

“Any finite Z-persistence F -vector space M can be decomposed as”

In the paper

“The properties of trees guarantee that ≤ T is a partial ordering.”

In the paper

“In the paper [59], this was shown by simulation, which is of course always an option.”

What the sources say

• ?Not checked — no full text of the sourcecites [59] visionG. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach, Topological Structure of Population Activity in Primary Visual Cortex, Journal of Vision, Volume 8, Number 8, Article 11, pp. 1-18, 2008. source

In the paper

“Not only is the correspondence π 0 (-) a functor from spaces to sets, it is equipped with a natural surjective map of sets η X : X → π 0 (X).”

• Also stated in: § 5. Reasoning about clustering · plain

  “Further, these surjective maps have the property that for every continuous map f : X → Y , the diagram of sets (below) commutes.”

In the paper

“While the observation is of course suggestive, to prove that it indicates structure in the data set one must show that segments of the indicated length cannot occur in a random model associated to a null hypothesis.”

In the paper

“This means that Kleinberg's conditions also make sense in this context.”

In the paper

“We point out that in the context of ordinary homology (i.e.not persistent homology), M. Penrose and others have developed a theory of “geometric random graphs” and have proved various results concerning the Betti numbers of complexes attached to such graphs (see [54]).”

What the sources say

• ?Not checked — no full text of the sourcecites [54] Penrose, Mathew, 2003 source

In the paper

“It is clear that a C-functorial clustering algorithm is a clustering algorithm in the sense of Kleinberg, since the surjective map of sets from X to χ(X, d X ) yields a partition of X, namely the partition whose blocks are the sets η -1 X (z), as z ranges over all elements of χ(X, d X ).”

In the paper

“Each such interval can be considered as a point in the set D = {(x, y) ∈ R 2 |x ≤ y}, and the output is therefore a finite subset of the set D. As such, we now have a finite spatial point process (see [20], [3]).”

What the sources say

• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [20] Daley, D. J., 1988
• ?Not checked — no full text of the sourcecites [3] Baddeley, Adrian, 2007, Spatial point processes and their applications source