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Block-Decomposition for 3-Parameter Persistence Modules

Siheng Yi

TL;DR

This work extends block-decomposition theory to $3$-parameter persistence modules by introducing $3$-parameter strong exactness and proving a corresponding decomposition theorem. Building on Cochoy–Oudot’s 2D framework, the authors define cuboid blocks, develop a $3$-D analogue of the strong exactness condition, and show that any pointwise finite-dimensional module with this property decomposes uniquely into a direct sum of block modules, with multiplicities given by a counting functor $ ext{$ ext{CF}$}$. The paper also develops the required machinery for Im/Ker analysis on cuboids, constructs block submodules via a suitable complement in $V_B^{+}$, and uses disjointness/covering of sections along with duality to complete the decomposition, clarifying why block decompositions do not hold for arbitrary persistence modules. These results generalize the $2$-parameter theory and connect to recent multi-parameter exactness results, offering a constructive pathway to block-level descriptors in 3-parameter persistence. The findings provide a principled explanation for the structure of $3$-parameter modules and lay groundwork for higher-parameter extensions, with implications for robust invariants in multi-parameter topological data analysis.

Abstract

In 2020, Cochoy and Oudot got the necessary and sufficient condition of the block-decomposition of 2-parameter persistence modules $\mathbb{R}^2 \to \textbf{Vec}_{\Bbbk}$. And in 2024, Lebovici, Lerch and Oudot resolve the problem of block-decomposability for multi-parameter persistence modules. Following the approach of Cochoy and Oudot's proof of block-decomposability for 2-parameter persistence modules, we rediscuss the necessary and sufficient conditions for the block decomposition of the 3-parameter persistence modules $\mathbb{R}^3 \to \textbf{Vec}_{\Bbbk}$. Our most important contribution is to generalize the strong exactness of 2-parameter persistence modules to the case of 3-parameter persistence modules. What's more, the generalized method allows us to understand why there is no block decomposition in general persistence modules to some extent.

Block-Decomposition for 3-Parameter Persistence Modules

TL;DR

This work extends block-decomposition theory to -parameter persistence modules by introducing -parameter strong exactness and proving a corresponding decomposition theorem. Building on Cochoy–Oudot’s 2D framework, the authors define cuboid blocks, develop a -D analogue of the strong exactness condition, and show that any pointwise finite-dimensional module with this property decomposes uniquely into a direct sum of block modules, with multiplicities given by a counting functor ext{CF}. The paper also develops the required machinery for Im/Ker analysis on cuboids, constructs block submodules via a suitable complement in , and uses disjointness/covering of sections along with duality to complete the decomposition, clarifying why block decompositions do not hold for arbitrary persistence modules. These results generalize the -parameter theory and connect to recent multi-parameter exactness results, offering a constructive pathway to block-level descriptors in 3-parameter persistence. The findings provide a principled explanation for the structure of -parameter modules and lay groundwork for higher-parameter extensions, with implications for robust invariants in multi-parameter topological data analysis.

Abstract

In 2020, Cochoy and Oudot got the necessary and sufficient condition of the block-decomposition of 2-parameter persistence modules . And in 2024, Lebovici, Lerch and Oudot resolve the problem of block-decomposability for multi-parameter persistence modules. Following the approach of Cochoy and Oudot's proof of block-decomposability for 2-parameter persistence modules, we rediscuss the necessary and sufficient conditions for the block decomposition of the 3-parameter persistence modules . Our most important contribution is to generalize the strong exactness of 2-parameter persistence modules to the case of 3-parameter persistence modules. What's more, the generalized method allows us to understand why there is no block decomposition in general persistence modules to some extent.
Paper Structure (9 sections, 30 theorems, 56 equations, 2 figures)

This paper contains 9 sections, 30 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Definition of Persistence Modules
  4. Cuts, Cuboids, and Block
  5. The Results of Cochoy and Oudot
  6. Main Results and Their Proof
  7. Some Basic Definitions and Results
  8. Find Block Submodules in $\mathbb{M}$
  9. The Direct Sum Decomposition

Key Result

Theorem 1.1

Let $\mathbb{M}$ be a pointwise finite-dimensional 3-parameter persistence module satisfying the 3-parameter strong exactness. Then $\mathbb{M}$ may decompose uniquely (up to isomorphism and reordering of the terms) as a direct sum of block modules: in which $M_B\cong \bigoplus_{i=1}^{n_B}\Bbbk_{B}$ in which $n_B$ are determined by the counting functor $\mathcal{CF}$.

Figures (2)

  • Figure 2.1: From left to right: three classes of layer blocks, birth blocks, death blocks
  • Figure 2.2: From left to right: birth blocks, death blocks, horizontal blocks, vertical blocks

Theorems & Definitions (52)

  • Theorem 1.1
  • Example 1
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Example 2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.2
  • Lemma 3.3
  • ...and 42 more