Topological data analysis (TDA) is a rapidly growing area of data science, whose most common descriptor is persistent homology, which tracks the topological changes in growing families of subsets of the data set itself, called filtrations, and encodes them in an algebraic object, called a persistence module. The algorithmic and theoretical properties of persistence modules are now well understood in the single-parameter case, that is, when there is only one filtration (e.g., feature scale) to study. In contrast, much less is known in the multi-parameter case, where several filtrations (e.g., scale and density) are used simultaneously. Since multi-parameter persistence modules usually encode information that is invisible to their single-parameter counterparts, it is critical to build tractable proxies for them, ideally with some theoretical robustness guarantees. In this article, we introduce a new parameterized family of topological descriptors, taking the form of candidate decompositions, for multi-parameter persistence modules, and we a identify a subfamily of these descriptors, that we call approximate decompositions, that are controllable approximations, in the sense that they preserve diagonal barcodes. Then, we introduce <Emphasis FontCategory="NonProportional">MMA</Emphasis> (Multipersistence Module Approximation): an algorithm based on matching functions for computing instances of candidate decompositions with some precision parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\delta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. By design, <Emphasis FontCategory="NonProportional">MMA</Emphasis> can handle an arbitrary number of filtrations, and has bounded complexity and running time. Moreover, we prove the robustess of <Emphasis FontCategory="NonProportional">MMA</Emphasis>: when computed with so-called compatible matching functions, we show that <Emphasis FontCategory="NonProportional">MMA</Emphasis> produces approximate decompositions (and we prove that such matching functions exist for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> filtrations). Next, we restrict the focus on modules that can be decomposed into interval summands. In that case, compatible matching functions always exist, and we show that, for small enough <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>, the approximate decompositions obtained with such compatible matching functions by <Emphasis FontCategory="NonProportional">MMA</Emphasis> have an approximation error (in terms of the standard interleaving and bottleneck distances) that is bounded by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>, and that reaches zero for an even smaller, positive precision <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta _{\textrm{exact}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mtext>exact</mtext> </msub> </math></EquationSource> </InlineEquation>. Finally, we present empirical evidence validating that <Emphasis FontCategory="NonProportional">MMA</Emphasis> has state-of-the-art performance and running time on several data sets.

Multi-parameter Module Approximation: an efficient and interpretable invariant for multi-parameter persistence modules with guarantees

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发表日期：

2025年10月25日

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David Loiseaux,
Mathieu Carrière,
Andrew J. Blumberg

摘要

Topological data analysis (TDA) is a rapidly growing area of data science, whose most common descriptor is persistent homology, which tracks the topological changes in growing families of subsets of the data set itself, called filtrations, and encodes them in an algebraic object, called a persistence module. The algorithmic and theoretical properties of persistence modules are now well understood in the single-parameter case, that is, when there is only one filtration (e.g., feature scale) to study. In contrast, much less is known in the multi-parameter case, where several filtrations (e.g., scale and density) are used simultaneously. Since multi-parameter persistence modules usually encode information that is invisible to their single-parameter counterparts, it is critical to build tractable proxies for them, ideally with some theoretical robustness guarantees. In this article, we introduce a new parameterized family of topological descriptors, taking the form of candidate decompositions, for multi-parameter persistence modules, and we a identify a subfamily of these descriptors, that we call approximate decompositions, that are controllable approximations, in the sense that they preserve diagonal barcodes. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm based on matching functions for computing instances of candidate decompositions with some precision parameter $\delta > 0$ $δ > 0$ . By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Moreover, we prove the robustess of MMA: when computed with so-called compatible matching functions, we show that MMA produces approximate decompositions (and we prove that such matching functions exist for $n=2$ $n = 2$ filtrations). Next, we restrict the focus on modules that can be decomposed into interval summands. In that case, compatible matching functions always exist, and we show that, for small enough $\delta $ $δ$ , the approximate decompositions obtained with such compatible matching functions by MMA have an approximation error (in terms of the standard interleaving and bottleneck distances) that is bounded by $\delta $ $δ$ , and that reaches zero for an even smaller, positive precision $\delta _{\textrm{exact}}$ $δ_{exact}$ . Finally, we present empirical evidence validating that MMA has state-of-the-art performance and running time on several data sets.