Estimating Shape Distances on Neural Representations with Limited Samples
This work addresses a foundational problem in neuroscience and deep learning for researchers analyzing neural representations, though it is incremental as it builds on existing shape distance measures.
The paper tackled the problem of measuring geometric similarity between high-dimensional neural representations with limited data, deriving bounds that reveal challenges and introducing a new estimator that achieves substantially lower bias in simulations and neural data.
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.