Spectral Analysis of Representational Similarity with Limited Neurons

TL;DR

This paper develops a spectral analysis framework using Random Matrix Theory to accurately measure neural representational similarity from limited neuron recordings, addressing sampling biases and proposing denoising methods.

Contribution

It introduces an analytical approach linking similarity measures to spectral properties, revealing underestimation issues and providing practical denoising techniques for small samples.

Findings

Neural similarities are systematically underestimated due to eigenvector delocalization.

The number of localized eigenvectors scales as the square root of recorded neurons.

A denoising method enables accurate population-level similarity inference from limited data.

Abstract

Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. Moreover, for power-law population spectra, we show that the number of localized eigenvectors scales as the square root of the number of recorded neurons, providing a simple rule of thumb for practitioners. To overcome sampling bias, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints.