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Register a new userDimension Detection with Local Homology
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Added by
Tamal Dey
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithms proposed in various fields, a few can provide theoretical guarantee on the correctness of the estimated dimension. However, the correctness usually requires certain regularity of the input: the input points are either uniformly randomly sampled in a statistical setting, or they form the so-called $(varepsilon,delta)$-sample which can be neither too dense nor too sparse. Here, we propose a purely topological technique to detect dimensions. Our algorithm is provably correct and works under a more relaxed sampling condition: we do not require uniformity, and we also allow Hausdorff noise. Our approach detects dimension by determining local homology. The computation of this topological structure is much less sensitive to the local distribution of points, which leads to the relaxation of the sampling conditions. Furthermore, by leveraging various developments in computational topology, we show that this local homology at a point $z$ can be computed emph{exactly} for manifolds using Vietoris-Rips complexes whose vertices are confined within a local neighborhood of $z$. We implement our algorithm and demonstrate the accuracy and robustness of our method using both synthetic and real data sets.
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We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contractions. We prove the following: (i) For homology groups, edges satisfying the p- and (p-1)-link conditions can be contracted without disturbing the p-dimensional homology group. (ii) For relative homology groups, the (p-1)-, and the (p-2)-link conditions suffice to guarantee that the contraction does not introduce any new class in any of the resulting relative homology groups, though some of the existing classes can be destroyed. Unfortunately, the surjection in relative homolgy groups does not guarantee that no new relative torsion is created. (iii) For torsions, edges satisfying the p-link condition alone can be contracted without creating any new relative torsion and the p-link condition cannot be avoided. The results on relative homology and relative torsion are motivated by recent results on computing optimal homologous chains, which state that such problems can be solved by linear programming if the complex has no relative torsion. Edge contractions that do not introduce new relative torsions, can safely be availed in these contexts.
We develop a method for analyzing spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), a tool from TDA that allows one to algorithmically detect geometric voids in a data set and quantify the persistence of these voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (an approach for visualizing PH), to track how the locations of the anomalies change over time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code. Second, we study a year-long data set of COVID-19 case rates in neighborhoods in the city of Los Angeles.
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We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the diagram as a signature of the input, and the use of features derived from it in statistics and machine learning. However, these computations also produce other information of great interest to practitioners that is unfortunately unstable. For example, each point in the diagram corresponds to a simplex whose addition in the filtration results in the birth of the corresponding persistent homology class, but this correspondence is unstable. In addition, the persistence diagram is not stable with respect to other procedures that are employed in practice, such as thresholding a point cloud by density. We recast these problems as real-valued functions which are discontinuous but measurable, and then observe that convolving such a function with a suitable function produces a Lipschitz function. The resulting stable function can be estimated by perturbing the input and averaging the output. We illustrate this approach with a number of examples, including a stable localization of a persistent homology generator from brain imaging data.
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)$ induced by automorphisms of $K$ is abelian.
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