Linear Classification of Neural Manifolds with Correlated Variability

TL;DR

This paper investigates how correlations in neural activity influence the linear separability of neural manifolds, revealing a duality between correlations and geometry that impacts classification capacity.

Contribution

It provides a theoretical framework linking correlations to geometric properties of neural representations and applies this to estimate deep network data capacity.

Findings

Correlations between object representations affect classification capacity.

Introducing centroid correlations effectively reduces distances between neural manifolds.

Axis correlations shrink the radii of neural manifolds, impacting separability.

Abstract

Understanding how the statistical and geometric properties of neural activity relate to performance is a key problem in theoretical neuroscience and deep learning. Here, we calculate how correlations between object representations affect the capacity, a measure of linear separability. We show that for spherical object manifolds, introducing correlations between centroids effectively pushes the spheres closer together, while introducing correlations between the axes effectively shrinks their radii, revealing a duality between correlations and geometry with respect to the problem of classification. We then apply our results to accurately estimate the capacity of deep network data.