Pick-Spectral Boundedness Loss

Use this at the output of a small complex-valued MLP, complex kernel regressor, or neural operator that maps normalized complex coordinates $z$ to complex predictions $g_\theta(z)$. Ensure $|z|\le1-\eta$ by mapping raw inputs through $z=(1-\eta)\tanh(r)e^{i\phi}$, or use one-dimensional real data embedded on the interval $[-1+\eta,1-\eta]$. In each minibatch of size $n$, compute the prediction vector $y_i=g_\theta(z_i)$ and form the dense Hermitian matrix $K_{ij}=(M^2-y_i\overline y_j)/(1-z_i\overline z_j)$. Symmetrize numerically with $K\leftarrow(K+K^*)/2$ and add $\delta I$. Compute its eigenvalues with a differentiable Hermitian eigensolver and backpropagate $\mathcal L_{\mathrm{Pick}}=n^{-1}\sum_r\operatorname{ReLU}(-\lambda_r)^2$. The mathematical component is the exact Pick matrix and positive-semidefinite certificate; $M$, batch size, jitter $\delta$, and penalty weight are hyperparameters. For larger batches, estimate the minimum eigenvalue with 5--10 Lanczos iterations and penalize only negative Ritz values. First test on analytic-function regression: sample $z$ uniformly in the disk, target $g(z)=0.6z+0.2z^3$, and compare an unconstrained complex MLP, output clipping, and the Pick loss at matched parameter count. Then evaluate extrapolation near $|z|=1$ and adversarial batches. Success is fewer boundedness violations on held-out Gram matrices, lower maximum $|g(z)|$, and equal or better interpolation error; a useful signal is that the constrained model maintains $\lambda_{\min}(K)\ge-10^{-4}$ on unseen batches without degrading in-domain MSE by more than 5%.