Estimating Shape Distances on Neural Representations with Limited Samples

TL;DR

This paper analyzes the statistical efficiency of shape distance estimators in high-dimensional neural data, introduces a new method-of-moments estimator with improved bias-variance tradeoff, and provides theoretical bounds for data-limited regimes.

Contribution

It derives bounds on estimator convergence, introduces a novel estimator with lower bias, and advances the statistical theory of high-dimensional shape analysis.

Findings

New bounds on estimator convergence in high dimensions

Proposed a method-of-moments estimator with lower bias

Demonstrated improved performance on neural data

Abstract

Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence of standard estimators of shape distance$\unicode{x2014}$a measure of representational dissimilarity proposed by Williams et al. (2021).These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a new method-of-moments estimator with a tunable bias-variance tradeoff. We show that this estimator achieves substantially lower bias than standard estimators in simulation and on neural data, particularly in high-dimensional settings. Thus, we lay the foundation for a rigorous statistical theory for high-dimensional shape analysis, and we contribute a new estimation method that is well-suited to practical scientific settings.