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Register a new userStabilizing the unstable output of persistent homology computations
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Added by
Peter Bubenik
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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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.
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Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings between the associated persistent diagrams. In our paper we prove that the coherent 2D matching distance is well-defined and stable.
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.
In topological data analysis, persistent homology is used to study the shape of data. Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the topological signal and the short intervals represent noise. We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.
The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. Initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.
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