Learning interpretable Schur forms of recurrent weight matrices

Recurrent neural networks are widely used in neuroscience and machine learning, but their weight matrices often appear dense and unstructured, defying clean interpretation. The Schur decomposition expresses any recurrent weight matrix in quasi-triangular form, allowing network dynamics to be interpreted as a latent circuit of "functionally feedforward" interactions between orthogonal modes. However, its application for interpreting recurrent weight matrices has been limited due to its well-known combinatorial non-uniqueness. We introduce SchurMO (Schur Manifold Optimization), a method for discovering structured Schur forms via Riemannian gradient descent on the manifold of orthogonal similarity transformations, enabling recovery of target motifs. Compared to a greedy baseline, SchurMO more reliably and more efficiently recovers ground-truth latent motifs (chain-like, banded, and modular structure) from noisy weight matrices. Applying SchurMO to nonlinear recurrent networks enables recovery of latent chain-like structure, providing a link between connectivity and the underlying dynamics of sequence production. Our results highlight SchurMO as a scalable, computationally efficient approach to discovering latent structure in the Schur forms of recurrent weights.