Canonicalizing Multimodal Contrastive Representation Learning

TL;DR

This paper demonstrates that independently trained multimodal contrastive models like CLIP and FLAVA are related by an orthogonal transformation, enabling alignment of their embedding spaces and facilitating model compatibility and privacy considerations.

Contribution

The work reveals a universal orthogonal relationship between different multimodal contrastive models' embedding spaces, supported by theoretical proof and empirical validation.

Findings

Embedding spaces are related by an orthogonal map Q.

The same Q aligns both image and text encoders.

The relationship holds across different model architectures and training distributions.

Abstract

As models and data scale, independently trained networks often induce analogous notions of similarity. But, matching similarities is weaker than establishing an explicit correspondence between the representation spaces, especially for multimodal models, where consistency must hold not only within each modality, but also for the learned image-text coupling. We therefore ask: given two independently trained multimodal contrastive models (with encoders $(f, g)$ and $(\widetilde{f},\widetilde{g})$) -- trained on different distributions and with different architectures -- does a systematic geometric relationship exist between their embedding spaces? If so, what form does it take, and does it hold uniformly across modalities? In this work, we show that across model families such as CLIP, SigLIP, and FLAVA, this geometric relationship is well approximated by an orthogonal map (up to a global mean shift), i.e., there exists an orthogonal map $Q$ where $Q^\top Q = I$ such that $\widetilde{f}(x)\approx Q f(x)$ for paired images $x$. Strikingly, the same $Q$ simultaneously aligns the text encoders i.e., $\widetilde{g}(y)\approx Q g(y)$ for texts $y$. Theoretically, we prove that if the multimodal kernel agrees across models on a small anchor set i.e. $\langle f(x), g(y)\rangle \approx \langle \widetilde{f}(x), \widetilde{g}(y)\rangle$, then the two models must be related by a single orthogonal map $Q$ and the same $Q$ maps images and text across models. More broadly, this finding enables backward-compatible model upgrades, avoiding costly re-embedding, and has implications for the privacy of learned representations. Our project page: https://canonical-multimodal.github.io/