Worst-Subset Conditioned Linear Layer

1. Integration point: use this regularizer on a trainable linear measurement or embedding matrix W\in\mathbb R^{N imes m}, such as the first layer of a phase-retrieval model, sensor encoder, channel-expansion layer, or token/channel projection that must tolerate row erasures. The forward path remains y=Wx; the extra term is evaluated during training. 2. Pseudocode: initialize W with Xavier or normalized Gaussian initialization; for each minibatch draw K subsets T_k of size m; form W_Tk; compute the smallest singular value using a differentiable eigendecomposition of W_Tk^ op W_Tk, with sigma_min=sqrt(max(lambda_min,epsilon)); compute the soft minimum -tau*logsumexp(-sigma_min/tau); add the isometry penalty and task loss; backpropagate and update W. Every R steps, replace half the random subsets by candidates found through greedy row deletion, repeatedly deleting the row whose removal most decreases the current smallest singular value. 3. The paper supplies the extreme-subset definition and Gaussian baseline exponent. The implementation estimates the minimum from K subsets and should measure the exact minimum by enumerating all subsets for small N. Constrain row norms or W's Frobenius norm so the objective cannot improve only by scaling W. 4. First experiment: train a two-layer MLP encoder on MNIST or CIFAR-10 with an N=1.5m linear measurement layer, then randomly erase N-m rows at test time. Compare standard Xavier initialization, row-normalized W, and this regularizer at equal task loss and FLOPs. Report the minimum singular value over all subsets for m\leq12, reconstruction or classification accuracy under erasures, and gradient stability. Success is a substantially larger first-percentile subset condition number and higher erased-input accuracy without increasing un-erased error.