[Paper Review] Similarity of Neural Network Models: A Survey of Functional and Representational Measures

Max Klabunde, Tobias Schumacher+2 more|arXiv (Cornell University)|May 10, 2023

Model Reduction and Neural Networks14 citations

TL;DR

A comprehensive survey of representational and functional similarity measures for neural networks, unifying terminology, properties, and open challenges.

ABSTRACT

Measuring similarity of neural networks to understand and improve their behavior has become an issue of great importance and research interest. In this survey, we provide a comprehensive overview of two complementary perspectives of measuring neural network similarity: (i) representational similarity, which considers how activations of intermediate layers differ, and (ii) functional similarity, which considers how models differ in their outputs. In addition to providing detailed descriptions of existing measures, we summarize and discuss results on the properties of and relationships between these measures, and point to open research problems. We hope our work lays a foundation for more systematic research on the properties and applicability of similarity measures for neural network models.

Motivation & Objective

• Define the problem of comparing neural networks along two complementary dimensions: representations in hidden layers and output behavior in tasks such as classification.
• Provide a systematic and unified overview of existing similarity measures for both representations and outputs.
• Clarify preprocessing, invariances, and practical properties of measures to guide application and interpretation.
• Analyze relationships between representational and functional similarity and discuss when they align or diverge.
• Highlight open research challenges and future directions in neural network similarity research.

Proposed method

• Formalize neural networks as layered compositions and define two similarity perspectives: representational (activations) and functional (outputs).
• Classify representational measures into categories (e.g., CCA-based, alignment-based, and others) and summarize their invariances and preprocessing needs.
• Describe alignment-based approaches (e.g., Orthogonal Procrustes, Generalized Shape Metrics) and their invariance properties.
• Discuss preprocessing steps (normalization, dimensionality adjustment, flattening) and how they affect measure invariance.
• Outline the interplay between representational and functional similarity, including when they may diverge or corroborate.

Experimental results

Research questions

• RQ1 What are the existing representational and functional similarity measures for neural networks?
• RQ2 How are these measures related, and in what scenarios do they agree or conflict in characterizing model similarity?
• RQ3 What are the practical properties (robustness, invariances, preprocessing needs) of these measures and how do they affect applicability?
• RQ4 What open problems and research gaps remain in the systematic study of neural network similarity?

Key findings

• The survey provides a formal, unified framework distinguishing representational and functional similarity and surveys a wide range of measures for each.
• Representational measures are categorized (e.g., CCA-based like SVCCA and PWCCA; alignment-based like Orthogonal Procrustes) with explicit invariances and preprocessing requirements.
• Functional similarity measures are discussed mainly in the context of multi-class classification and can often operate with black-box access, highlighting practical applicability.
• There is no guaranteed correlation between representational and functional similarity; high functional similarity does not imply similar representations, and vice versa, necessitating cautious interpretation.
• The paper identifies open research challenges and emphasizes the need for systematic analysis of measure properties, invariances, and applicability across tasks and architectures.