Differentiability and Optimization of Multiparameter Persistent Homology

TL;DR

A general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors is developed and it is shown that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape.

Abstract

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

Figures (8)

• Figure 1: An example, from Loiseaux2023, of a filtered simplicial complex, the Hilbert function of its $0$th persistent homology, and the corresponding Hilbert decomposition signed measure.
• Figure 2: An example of a simplicial complex $K$, a two-parameter filtration $f$ of $K$, the grid of $f$ according to \ref{['construction:grid-from-filtration']}, and the aligned grid inclusion associated with $f$ according to \ref{['construction:grid-from-filtration', 'definition:iota']}.
• Figure 3: Factoring a descriptor $\alpha$ through each cell allows us to deal with functions over linear, open subsets of a Euclidean space.
• Figure 4: Optimizing the holes of point clouds. The colors indicate the $\log$-values of the density estimator.
• Figure 5: Rows correspond to datasets (top has less background noise), and columns correspond to no topological regularization, one-parameter regularization, and multiparameter regularization. One-parameter regularization is very susceptible to noise points, while no topological regularization can fail to close the circles and preserve topology. Interestingly, no topological regularization behaves better with many noise points, possibly due to the metric loss having more distances to work with. Multiparameter topological regularization ensures the preservation of topology in both cases.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Example 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Definition 3.1
• Definition 3.3