Decomposition and Stability of Multiparameter Persistence Modules

Cheng Xin, Purdue University

Abstract

In the contemporary computing landscape, the effective utilization of data has emerged as a primary driving force. Nonetheless, handling data with intricate structures, non-Euclidean properties, has introduced novel challenges for data scientists and researchers working in machine learning and artificial intelligence domains. Topological data analysis (TDA) has demonstrated its proficiency in mitigating these issues by offering advanced techniques capable of unveiling hidden patterns and high-level connectivity within complex data. 1- parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To further enrich the representations of topological features, multiparameter persistence modules are studied. This dissertation delves into the decomposition, stability and summary problems of multiparameter persistence modules.First, we study the decomposition structure of multiparameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module M defined over the multiparameter Z d . The algorithm first assumes that the module is presented with a set of N generators and relations that are distinctly graded. Based on a generalized matrix reduction technique it runs in O(N2ω+1) time where ω < 2.373 is the exponent of matrix multiplication. This is much better than the well-known algorithm called Meataxe which runs in O˜(N6(d+1)) time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of n simplices, our algorithm runs in O(n (d−1)(2ω+1)) time for d ≥ 2. For the special case of 0-th homology, it runs in time O(n 2ω+1).Next, we consider a reasonably complicated class called d-parameter interval decomposable modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called dimension distancethat bounds it from below. We further study the stability-property of rectangle decomposable modules and derive a stability-like property for 2-parameter persistence modules. For this, first we consider interval decomposable modules and their optimal approximations with rectangle decomposable modules with respect to the bottleneck distance. We provide a polynomial time algorithm to exactly compute this optimal approximation which, together with the polynomial-time computable bottleneck distance among interval decomposable modules, provides a lower bound on the interleaving distance. Next, we leverage this result to derive a polynomial-time computable distance for general multi-parameter persistence modules which enjoy similar stability-like property. This distance can be viewed as a generalization of the matching distance defined in the literature.Last but not the least, we study the vector representations of multiparameter persistence modules for machine learning tasks. All the previous vectorization methods of multiparameter persistence modules have discriminating power limited by a well-known incomplete discrete invariant called rank invariant. We propose a novel vector representation called Generalized Rank Invariant Landscape (Gril) for 2-parameter persistence modules that in theory has more discriminating power than all previous methods.

Degree

Ph.D.

Advisors

Dey, Purdue University.

Subject Area

Artificial intelligence

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