The Shape of Data: Intrinsic Distance for Data Distributions

TL;DR

This paper introduces an intrinsic, multi-scale method based on spectral Gromov-Wasserstein distance for comparing data manifolds, enabling better analysis of data structure and generative model quality.

Contribution

It presents a novel intrinsic and multi-scale approach for data manifold comparison, addressing limitations of existing extrinsic, uni-scale methods.

Findings

Effectively discerns data manifold structures on unaligned, differently dimensional data.

Demonstrates efficacy in evaluating generative model quality.

Outperforms existing methods in capturing data geometry.

Abstract

The ability to represent and compare machine learning models is crucial in order to quantify subtle model changes, evaluate generative models, and gather insights on neural network architectures. Existing techniques for comparing data distributions focus on global data properties such as mean and covariance; in that sense, they are extrinsic and uni-scale. We develop a first-of-its-kind intrinsic and multi-scale method for characterizing and comparing data manifolds, using a lower-bound of the spectral variant of the Gromov-Wasserstein inter-manifold distance, which compares all data moments. In a thorough experimental study, we demonstrate that our method effectively discerns the structure of data manifolds even on unaligned data of different dimensionalities; moreover, we showcase its efficacy in evaluating the quality of generative models.