Truncation-Corrected Local Pseudospectral Regularizer

Integrate this during training into the transition operator of an RNN, linear state-space model, or structured sequence layer, rather than into the input and output projections. Assume the layer applies h_{t+1} = phi(W h_t + U x_t), and use the same recurrent matrix W that is repeatedly applied across time. First choose a patch size n, such as 16 or 32, and construct K overlapping coordinate windows over the hidden dimensions. For convolutional, banded, or block-circulant W, use translated group patches; for a dense W, use contiguous principal submatrices or fixed random coordinate patches. At each regularization step, sample a small set Z of complex shifts, for example z = r exp(i theta), with r in {0.9, 1.0, 1.1} and eight equally spaced angles. For every z and patch k, form B_{n,k} = z I_n - W_{n,k}. Compute its smallest singular value using an exact differentiable SVD for the first MVP, or approximate it with two to five inverse-power iterations on B_{n,k}^* B_{n,k}; add lambda times the local regularizer to the task loss. Estimate c_hat empirically by measuring the statistic at patch sizes 8, 16, and 32, fitting an upper envelope of the observed discrepancy to c/n, and multiplying the fitted constant by 1.5 for safety. The paper supplies the 1/n scaling; the implementation estimates the constant and selects the patch family. Test first on permuted sequential MNIST with a one-layer tanh RNN of hidden size 128. Compare an unconstrained baseline and spectral-radius clipping at equal FLOPs. Record long-sequence gradient norms, loss after 50 and 200 updates, maximum hidden-state Jacobian amplification, and validation accuracy. Success means fewer exploding or vanishing-gradient runs and better long-horizon accuracy at comparable parameter count. Also verify that increasing n produces a tighter, more stable resolvent envelope.