[Paper Review] A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions

David Loiseaux, Mathieu Carrière+1 more|arXiv (Cornell University)|Jun 19, 2023

Topological and Geometric Data Analysis4 citations

TL;DR

This paper introduces a novel framework for fast and stable representations of multiparameter persistent homology using interval module decompositions, enabling efficient computation of vector-space embeddings for machine learning. It provides theoretical stability guarantees, closed-form algorithms for key invariants (interleaving distance, volume, and slicings), and demonstrates superior performance on real data with fast running times and statistical convergence.

ABSTRACT

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Motivation & Objective

• To address the lack of stable, informative, and computationally efficient representations for multiparameter persistent homology in machine learning pipelines.
• To overcome limitations of existing methods that either reduce to single-parameter settings or lack theoretical stability under noise.
• To develop a general framework that leverages recent advances in interval module decompositions for richer, more informative descriptors.
• To provide theoretical stability guarantees for the proposed representations under perturbations of the input data.
• To design efficient algorithms for computing key invariants such as interleaving distance, volume, and slicings with closed-form formulas.

Proposed method

• The framework represents multiparameter persistent homology via interval modules, decomposing the multiparameter persistence module into simple geometric pieces with birth and death corners in R^n.
• It introduces a stable weight function based on the interleaving distance d_I(M, 0), computed via a closed-form formula over corners of interval modules.
• The volume of the largest rectangle fitting within the support of an interval module is computed using a closed-form expression involving differences between birth and death corner coordinates.
• Slicings of interval modules along diagonal lines are computed by projecting corners onto the line, yielding persistence barcodes for 1D slices.
• Algorithms 1–4 implement the restriction of interval modules to rectangles, lines, and the computation of key invariants using corner-based representations.
• The framework ensures theoretical stability by proving that the proposed representations are Lipschitz continuous with respect to the interleaving distance.

Figure 1 : Common pipelines for the use of multiparameter persistent homology in data science—our work provides new contributions to the arrow highlighted in red.

Experimental results

Research questions

• RQ1 Can we construct a general, stable, and computationally efficient representation framework for multiparameter persistent homology that preserves rich topological information?
• RQ2 How can we compute key invariants such as interleaving distance, volume, and slicings in closed form using interval module corners?
• RQ3 Does the proposed framework maintain theoretical stability under data perturbations, particularly in noisy settings?
• RQ4 Can the framework achieve fast computation and statistical convergence in practical machine learning applications?
• RQ5 How does the framework compare to existing heuristic or reduced representations in terms of predictive accuracy and running time?

Key findings

• The proposed framework achieves theoretical stability by ensuring that the representations are Lipschitz continuous with respect to the interleaving distance, guaranteeing robustness to noise.
• The closed-form algorithms for interleaving distance, volume, and slicings enable fast computation, with empirical results showing fast running times on real datasets.
• Numerical experiments demonstrate statistical convergence of the representations, indicating reliable learning from data.
• The framework outperforms existing methods in prediction accuracy across multiple UCR time series datasets, with classifiers like LightGBM achieving up to 99.0% accuracy on the Plane dataset.
• The S-CDR weight function and volume representation are shown to be computable in closed form using only the birth and death corner coordinates of interval modules.
• The framework successfully generalizes previous approaches by preserving more topological information than rank invariants, while remaining computationally tractable.

Figure 2 : (left) Bi-filtration of a noisy point cloud induced by both feature scale (using unions of balls with increasing radii) and (co)density. The cycle highlighted in the green zone can be detected as a large subset in the corresponding candidate decomposition computed by the MMA method [ 26 ]