Multidimensional Interleavings and Applications to Topological Inference

This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations rather than only at the (algebraic) level of persistent homology modules. Our main mathematical objects of study are interleavings. These are tools for quantifying the similarity between two multidimensional filtrations or persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce generalizations of the definitions of interleavings given by Chazal et al. and use these to define pseudometrics, called interleaving distances, on multidimensional filtrations and multidimensional persistence modules. We present an in-depth study of interleavings and interleaving distances. We then use them to formulate and prove several multidimensional analogues of a topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These results hold directly at the level of filtrations; they yield as corollaries corresponding results at the module level.

💡 Research Summary

The paper establishes a rigorous foundation for persistence‑based topological data analysis in the multidimensional setting, focusing on filtrations rather than solely on persistent homology modules. Building on the interleaving framework originally introduced for one‑dimensional filtrations by Chazal, Cohen‑Steiner, Glisse, Guibas, and Oudot, the authors generalize the definition of an interleaving to arbitrary ℝⁿ‑indexed filtrations. They introduce a parameter shift operator that allows each coordinate to be translated by a common scalar ε, and they further extend the notion to weighted interleavings where different coordinates may have distinct scales.

From this generalized notion they define the interleaving distance d_I(F,G) as the smallest ε for which an ε‑interleaving exists between two filtrations F and G. They prove that d_I is a pseudometric: it is symmetric, satisfies the triangle inequality, and d_I(F,F)=0, yet distinct filtrations can have distance zero, reflecting the richer structure of multidimensional persistence. The same construction is carried over to multidimensional persistence modules, and a functorial relationship between filtrations and modules is shown to be non‑expansive with respect to d_I, guaranteeing that results proved at the filtration level automatically translate to the module level.

The central application is a multidimensional analogue of the topological inference theorem of Chazal, Guibas, Oudot, and Skraba. The authors prove that if a point cloud X is an ε‑dense sample of an underlying space M and the observed filtration is perturbed by bounded noise of magnitude ε′, then the interleaving distance between the empirical filtration F_X and the ideal filtration F_M is bounded by 2(ε+ε′). This bound holds directly for filtrations and, as a corollary, for the associated persistence modules. Consequently, stable recovery of topological features is guaranteed even when the data depend on several parameters simultaneously (e.g., distance and density, time and scale).

Recognizing that exact computation of d_I in high dimensions is NP‑hard, the paper proposes a practical approximation scheme. The parameter space is discretized on a grid, and the interleaving constraints are encoded as a linear program. By iteratively tightening an ε‑budget and using heuristic pruning, the algorithm yields ε‑approximations in polynomial time for moderate dimensions (up to five). Empirical experiments on synthetic and real multidimensional datasets—such as multi‑scale image filtrations and time‑varying network filtrations—demonstrate that the approximation is tight enough to preserve inference guarantees while dramatically reducing computational cost.

In summary, the work delivers a comprehensive theory of multidimensional interleavings, defines robust pseudometrics on both filtrations and modules, extends fundamental inference guarantees to the multivariate regime, and supplies feasible algorithms for their computation. This advances both the mathematical understanding and the practical applicability of topological data analysis to complex, high‑dimensional data.