Controllability-Rank Regularizer

Attach the regularizer to a structured orthogonal mixer with trainable skew-symmetric matrices \(A\) and \(B\). During each training step, compute \(C_0=B\) and recursively form \(C_1,\ldots,C_K\) using matrix commutators. Normalize each \(C_k\) by its Frobenius norm, construct the small \((K+1)\times(K+1)\) Gram matrix \(G\), and add \(\lambda\mathcal{L}_{\mathrm{rank}}\) to the task loss. Use \(K=2\) or \(3\) initially so the extra cost is modest; evaluate the log determinant with a Cholesky factorization of \(G+\delta I\), using \(\delta=10^{-4}\). Stop gradients through normalization only if exploding gradients occur. The exact mathematical computation is the iterated adjoint family and its Gram determinant; \(\lambda\), \(K\), normalization, and whether to apply the term only during warm-up are empirical choices. Run a cheap ablation on a CIFAR-10 MLP, a small transformer feed-forward mixer, or sequential MNIST: compare no regularizer, orthogonality-only, and the Lie-rank penalty at matched parameter count. Log the singular values of the flattened family \([\operatorname{vec}(C_0),\ldots,\operatorname{vec}(C_K)]\), task accuracy, gradient variance, and loss across random seeds. The expected signal is a larger minimum singular value and less generator collapse, followed by improved validation accuracy or robustness under hidden-width reduction. Reject the method if the determinant increases without task benefit or if its overhead exceeds the structured layer's savings.