Diffeomorphic interpolation for efficient persistence-based topological optimization

TL;DR

This paper introduces a diffeomorphic interpolation method to improve the efficiency of topological optimization in point clouds, enabling scalable and interpretable topological regularization for autoencoders.

Contribution

It proposes a novel diffeomorphic interpolation approach that transforms sparse gradients into smooth vector fields, facilitating scalable topological optimization and regularization in high-dimensional data.

Findings

Enables topological optimization on large point clouds efficiently.

Allows re-application of learned flows for new data in linear time.

Improves interpretability of topologically-regularized autoencoders.

Abstract

Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimizationvia gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice.Focusing on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show thatlearning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, yielding better interpretability of the model.