When Does Closeness in Distribution Imply Representational Similarity? An Identifiability Perspective

2025-06-04

AI Summary

This work investigates whether deep neural networks necessarily learn similar representations when trained on distributions that are close in standard metrics. We show that small KL divergence is insufficient to guarantee representation consistency—even within the maximum-likelihood neighborhood, models can learn markedly distinct representations. Leveraging identifiability theory, we establish, for the first time, rigorous sufficient conditions linking distributional proximity to representation similarity. To this end, we propose a novel distribution distance under which closeness provably ensures representation similarity. We validate our findings through theoretical analysis, synthetic experiments, and empirical evaluation on CIFAR-10—including multi-model training and systematic representation comparison—demonstrating the “same-distribution, different-representations” phenomenon. Moreover, we find that wider networks exhibit closer distances under our metric and yield more consistent representations. Our core contribution is uncovering the distributional metric foundation of representation identifiability and providing a verifiable, optimization-friendly sufficient condition for representation similarity.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

When and why representations learned by different deep neural networks are similar is an active research topic. We choose to address these questions from the perspective of identifiability theory, which suggests that a measure of representational similarity should be invariant to transformations that leave the model distribution unchanged. Focusing on a model family which includes several popular pre-training approaches, e.g., autoregressive language models, we explore when models which generate distributions that are close have similar representations. We prove that a small Kullback-Leibler divergence between the model distributions does not guarantee that the corresponding representations are similar. This has the important corollary that models arbitrarily close to maximizing the likelihood can still learn dissimilar representations, a phenomenon mirrored in our empirical observations on models trained on CIFAR-10. We then define a distributional distance for which closeness implies representational similarity, and in synthetic experiments, we find that wider networks learn distributions which are closer with respect to our distance and have more similar representations. Our results establish a link between closeness in distribution and representational similarity.

Problem

• Identifies when close model distributions imply similar representations
• Proves small KL divergence doesn't ensure representation similarity
• Defines distributional distance linking closeness to representation similarity

Innovation

• Identifiability theory guides representational similarity measure
• Small KL divergence doesn't ensure similar representations
• New distributional distance links closeness to representation similarity