Bridging Functional and Representational Similarity via Usable Information
This work addresses the challenge of comparing neural representations for researchers in machine learning and neuroscience, providing a foundational framework that clarifies the relationship between different similarity metrics.
The paper tackles the problem of quantifying similarity between neural representations by introducing a unified framework based on usable information, linking functional and representational similarity through theoretical proofs and empirical analysis, such as showing that stitching performance relates to conditional mutual information and that similarity depends on predictive capacity.
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.