Manifold Approximation leads to Robust Kernel Alignment

TL;DR

This paper introduces Manifold approximated Kernel Alignment (MKA), a novel method that incorporates manifold geometry into kernel alignment, leading to more robust and consistent measurement of representations across different data scales.

Contribution

The paper proposes MKA, a new manifold-aware kernel alignment method, along with a theoretical framework and empirical validation demonstrating its robustness over existing approaches.

Findings

MKA outperforms traditional CKA in robustness across data scales.

Manifold-aware alignment provides more consistent similarity measures.

Empirical results on synthetic and real datasets validate MKA's effectiveness.

Abstract

Centered kernel alignment (CKA) is a popular metric for comparing representations, determining equivalence of networks, and neuroscience research. However, CKA does not account for the underlying manifold and relies on numerous heuristics that cause it to behave differently at different scales of data. In this work, we propose Manifold approximated Kernel Alignment (MKA), which incorporates manifold geometry into the alignment task. We derive a theoretical framework for MKA. We perform empirical evaluations on synthetic datasets and real-world examples to characterize and compare MKA to its contemporaries. Our findings suggest that manifold-aware kernel alignment provides a more robust foundation for measuring representations, with potential applications in representation learning.