Bridging Functional and Representational Similarity via Usable Information

TL;DR

This paper introduces a unified framework linking functional and representational similarity using usable information, revealing their relationships, limitations, and the importance of bidirectional analysis for robust comparison.

Contribution

It provides a theoretical and empirical synthesis connecting functional and representational similarity through usable information, highlighting asymmetry and the role of predictive capacity.

Findings

Stitching performance relates to conditional mutual information.

Reconstruction-based metrics estimate usable information.

Representational similarity is sufficient but not necessary for functional similarity.

Abstract

We present a unified framework for quantifying the similarity between representations through the lens of \textit{usable information}, offering a rigorous theoretical and empirical synthesis across three key dimensions. First, addressing functional similarity, we establish a formal link between stitching performance and conditional mutual information. We further reveal that stitching is inherently asymmetric, demonstrating that robust functional comparison necessitates a bidirectional analysis rather than a unidirectional mapping. Second, concerning representational similarity, we prove that reconstruction-based metrics and standard tools (e.g., CKA, RSA) act as estimators of usable information under specific constraints. Crucially, we show that similarity is relative to the capacity of the predictive family: representations that appear distinct to a rigid observer may be identical to a more expressive one. Third, we demonstrate that representational similarity is sufficient but not necessary for functional similarity. We unify these concepts through a task-granularity hierarchy: similarity on a complex task guarantees similarity on any coarser derivative, establishing representational similarity as the limit of maximum granularity: input reconstruction.