Entropy-Adaptive Spectral Groups

First, integrate the method after each optimizer step for selected matrices in a small MLP or Transformer feed-forward block. Compute a truncated SVD, sort the singular values in descending order, and construct the nested entropy profile. Do not change groups at every update: exponentially average the profile and detect extrema that persist for a window of 50 to 200 updates. Assign contiguous singular-vector indices to the detected groups. This can be implemented with a low-rank parameterization W = sum over g of U_g diag(s_g) V_g transposed, or by projecting the full gradient onto the current singular subspaces. Pseudocode is: compute the SVD; for each n compute S_n, p_j^(n), E_n, and Delta_n; smooth Delta; locate persistent extrema; form groups; compute each group effective count K_g; then use the displayed eta_g and lambda_g formulas for groupwise AdamW learning rates and weight decay. The SVD, entropy profile, effective counts, and assignments are computed directly. Persistence thresholds and exponents alpha and beta are empirical choices. Start with synthetic matrices containing two, three, and four geometrically separated singular-value bands, followed by a six-layer MLP on MNIST or CIFAR-10. Compare AdamW, fixed-rank spectral grouping, and SPINE grouping. The predicted signature is that detected boundaries stay within one singular-index of planted band boundaries when adjacent-band ratios exceed about 10, while a single power-law spectrum produces no persistent extrema and therefore one unresolved group. Measure the maximum learning rate that avoids divergence for 10,000 updates; the proposed method should improve this stability ceiling mainly for separated-band spectra.