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Poster E in Poster Session E: Thursday, August 6, 10:30 am – 12:15 pm, Kimmel Center, Shorin & Rosenthal Rooms

Common Representation Similarity Metrics Have Dimension-Free Error Bounds

Shujun Xiong¹, Sarah E Harvey², Jacob A Zavatone-Veth³, Alex H Williams¹; ¹New York University, ²Flatiron Institute, ³Harvard University

Presenter: Shujun Xiong

Methods like centered kernel alignment (CKA), representational similarity analysis (RSA), and Procrustes shape distance have become popular tools for quantifying similarity in the geometry of neural population codes, with well-characterized asymptotic behavior. However, neural recording techniques with single-cell resolution can only access a small fraction of cells within a brain region of interest. Moreover, recordings are inherently limited in duration, and one can therefore only measure responses to a small subset of possible experimental conditions (e.g. natural images). Here, we present novel, non-asymptotic error bounds to quantify how estimates of representational similarity simultaneously depend on the number of sampled neurons (N) and experimental conditions (M). Under mild assumptions, we show that estimation error decays in proportion to 1/M + 1/N for several measures related to CKA and RSA. Strikingly, this bound does not depend on the dimensionality of the space of experimental conditions. Thus, similarity estimation does not suffer from the curse of dimensionality, which plagues similar problems in high-dimensional statistics. We verify our results empirically in random shallow neural networks.

Topic Area: Methods, Tools, Theory & Neural Coding