[Paper Review] A refinement of multi-dimensional persistence

K. Knudson|ArXiv.org|Jun 18, 2007

Topological and Geometric Data Analysis5 references3 citations

TL;DR

This paper refines multi-dimensional persistent homology by introducing higher Tor-modules as discrete invariants, enabling a geometric and algebraic classification of persistence modules over polynomial rings. It constructs a projective variety structure on isomorphism classes of these modules, providing a finer invariant than previous methods and offering a geometric interpretation of hypertor groups in multi-filtered simplicial complexes.

ABSTRACT

We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes.

Motivation & Objective

• To develop a finer classification of multi-dimensional persistence modules beyond the barcode invariant.
• To address the limitation of existing invariants in multi-dimensional persistence, which lack completeness and fail to distinguish non-isomorphic modules with identical birth/death multisets.
• To introduce higher Tor-modules as discrete invariants that capture more refined algebraic structure in persistence modules.
• To endow the quotient space of isomorphism classes with a projective variety structure, enabling geometric and algebraic analysis.

Proposed method

• Define the multi-dimensional persistence module $ M = \bigoplus_{v \in \mathbb{N}^n} H_i(X_v; k) $ as a module over the polynomial ring $ A_n = k[x_1, \dots, x_n] $.
• Use the higher Tor-modules $ \operatorname{Tor}_i^{A_n}(M, k) $ as discrete invariants, indexed by $ i = 0, \dots, n $, to refine the classification of $ M $.
• Partition the space $ \mathcal{RF}(\xi_0, \xi_1) $ of modules with fixed birth and death multisets into subspaces $ \mathcal{RF}(\xi_2, \dots, \xi_n) $ based on higher Tor invariants.
• Construct a map $ \varphi: GL(F(\xi_0)) \backslash \mathcal{RF}(\xi_2, \dots, \xi_n) \to Y_{\xi_2, \dots, \xi_n} $, where $ Y_{\xi_2, \dots, \xi_n} $ is a projective variety, to endow the quotient with a geometric structure.
• Use spectral sequences to compute $ \mathbf{Tor}_\bullet^{A_n}(C_\bullet(X_\bullet), k) $, and show that the resulting complex $ T_\bullet $ quasi-isomorphically represents the chain complex $ C_\bullet(X; k) $.
• Demonstrate that the inclusion $ \varphi: C_\bullet(X; k) \to T_\bullet $ is a quasi-isomorphism, validating the algebraic construction as topologically meaningful.

Experimental results

Research questions

• RQ1 Can higher Tor-modules provide a finer invariant than the multiset of birth and death times in multi-dimensional persistence?
• RQ2 How can the space of isomorphism classes of persistence modules be endowed with a geometric structure when the classical barcode fails?
• RQ3 What is the geometric and topological meaning of higher Tor-modules in the context of multi-filtered simplicial complexes?
• RQ4 Under what conditions is the map $ \varphi $ from the quotient space to the projective variety $ Y_{\xi_2, \dots, \xi_n} $ injective, and when does it fail?
• RQ5 Can the spectral sequence of $ \mathbf{Tor}_\bullet^{A_n}(C_\bullet(X_\bullet), k) $ be used to reconstruct the homology of the filtered complex?

Key findings

• The higher Tor-modules $ \operatorname{Tor}_i^{A_n}(M, k) $, for $ i = 0, \dots, n $, form a finite family of discrete invariants that refine the classification of multi-dimensional persistence modules.
• The space $ \mathcal{RF}(\xi_0, \xi_1) $ of modules with fixed birth and death multisets $ \xi_0, \xi_1 $ is partitioned into subspaces $ \mathcal{RF}(\xi_2, \dots, \xi_n) $ based on higher Tor invariants.
• A map $ \varphi: GL(F(\xi_0)) \backslash \mathcal{RF}(\xi_2, \dots, \xi_n) \to Y_{\xi_2, \dots, \xi_n} $ is constructed, where $ Y_{\xi_2, \dots, \xi_n} $ is a projective variety, giving the quotient a geometric structure.
• In the example of the filtered circle, the complex $ T_\bullet $ has homology $ k $ in degrees 0 and 1, and 0 in degree 2, confirming that $ \varphi $ is a quasi-isomorphism.
• For the filtered sphere, the complex $ T_\bullet $ has trivial homology, and $ \varphi: C_\bullet(X; k) \to T_\bullet $ is a quasi-isomorphism, validating the construction.
• The map $ \varphi $ is not always injective, indicating that the variety structure may lose information about certain degeneracies in the original parameter space.