Score-based pullback Riemannian geometry

Willem Diepeveen, Georgios Batzolis, Zakhar Shumaylov, Carola-Bibiane Schönlieb

Keywords: Data-driven Riemannian geometry, interpretable representation learning, pullback Riemannian geometry, generative models, closed-form geodesics, manifold dimension estimation

TL;DR: This paper introduces a scalable framework for data-driven Riemannian geometry that integrates pullback Riemannian geometry and generative models to reliably estimate intrinsic data manifold dimensions and construct high-quality geodesics.

Abstract:

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on various datasets, we demonstrate that our framework not only produces high-quality geodesics through the data support, but also reliably estimates the intrinsic dimension of the data manifold and provides a global chart of the manifold, even in high-dimensional ambient spaces.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 8253