Multi-Way Representation Alignment

TL;DR

This paper introduces GCPA, a novel method for aligning multiple neural network representations into a shared space, improving retrieval tasks while maintaining a consistent global reference.

Contribution

It adapts GPA for multi-model alignment, identifies limitations of isometric methods, and proposes GCPA with a correction step for better retrieval performance.

Findings

GCPA improves any-to-any retrieval accuracy.

GCPA maintains a practical shared reference space.

GCPA outperforms pairwise alignment methods.

Abstract

The Platonic Representation Hypothesis suggests that independently trained neural networks converge to increasingly similar latent spaces. However, current strategies for mapping these representations are inherently pairwise, scaling quadratically with the number of models and failing to yield a consistent global reference. In this paper, we study the alignment of $M \ge 3$ models. We first adapt Generalized Procrustes Analysis (GPA) to construct a shared orthogonal universe that preserves the internal geometry essential for tasks like model stitching. We then show that strict isometric alignment is suboptimal for retrieval, where agreement-maximizing methods like Canonical Correlation Analysis (CCA) typically prevail. To bridge this gap, we finally propose Geometry-Corrected Procrustes Alignment (GCPA), which establishes a robust GPA-based universe followed by a post-hoc correction for directional mismatch. Extensive experiments demonstrate that GCPA consistently improves any-to-any retrieval while retaining a practical shared reference space.