Sparsification of the Generalized Persistence Diagrams for Scalability through Gradient Descent

2024-12-08

arXiv.org

AI Summary

Generalized persistence diagrams (GPDs) in multiparameter persistent homology capture rich topological structures but suffer from exponential blowup in the number of intervals, rendering computation intractable for large-scale data. Method: This paper introduces, for the first time, gradient descent into GPD sparsification. We propose a differentiable loss function grounded in erosion stability theory, enabling joint optimization over the domain space to balance computational efficiency and topological discriminability. Our end-to-end framework learns sparse support points directly—without requiring pre-specified interval subsets—thereby drastically reducing GPD construction complexity. Contribution/Results: Evaluated on multiple benchmark datasets, our method achieves substantial speedups in computation time while maintaining classification accuracy comparable to that of the full GPD baseline. This confirms the topological discriminability preservation of the learned sparse representation. To our knowledge, this is the first differentiable and scalable sparsification framework for GPDs, paving the way for their practical deployment on large-scale problems.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological information than its single-parameter counterpart. However, computing the GPD is computationally prohibitive due to the sheer size of the interval set. Restricting the GPD to a subset of intervals provides a way to manage this complexity, compromising discriminating power to some extent. However, identifying and computing an effective restriction of the domain that minimizes the loss of discriminating power remains an open challenge. In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale.

Problem

• Optimizing GPD domain via gradient descent for scalability
• Balancing computational efficiency with discriminative accuracy
• Reducing GPD computation time while maintaining classification accuracy

Innovation

• Optimizes GPD domain via gradient descent
• Uses loss function for interval selection
• Maintains accuracy while reducing computation time