When Are Two Networks the Same? Tensor Similarity for Mechanistic Interpretability
For researchers in mechanistic interpretability, this provides a principled, algebraic solution to measuring similarity and verifying faithfulness, replacing empirical approximations.
The authors introduce tensor similarity, a weight-based metric invariant to weight-space symmetries, for comparing neural network components in mechanistic interpretability. It captures global functional equivalence and tracks training dynamics like grokking and backdoor insertion with higher fidelity than existing metrics.
Mechanistic interpretability aims to break models into meaningful parts; verifying that two such parts implement the same computation is a prerequisite. Existing similarity measures evaluate either empirical behaviour, leaving them blind to out-of-distribution mechanisms, or basis-dependent parameters, meaning they disregard weight-space symmetries. To address these issues for the class of tensor-based models, we introduce a weight-based metric, tensor similarity, that is invariant to such symmetries. This metric captures global functional equivalence and accounts for cross-layer mechanisms using an efficient recursive algorithm. Empirically, tensor similarity tracks functional training dynamics, such as grokking and backdoor insertion, with higher fidelity than existing metrics. This reduces measuring similarity and verifying faithfulness into a solved algebraic problem rather than one of empirical approximation.