Spectral quadrature features

Integrate this at the input embedding or kernel layer of a kernelized model, a random-feature MLP, or a neural tangent-kernel approximation. Given inputs $X\in\mathbb R^{M\times d}$, first select a fixed quadrature rule $(w_i,a_i)_{i=1}^N$ for the feature domain: use Gauss-Hermite nodes for Gaussian Fourier features in low-dimensional Euclidean inputs, or spherical-design/quasi-uniform sphere nodes for features whose parameter domain is a sphere. Normalize weights to be nonnegative and sum to one. Construct the feature matrix with the following pseudocode: nodes, weights = quadrature_rule(N); weights = weights / weights.sum(); for minibatch Xb: F[:,i] = sqrt(weights[i]) * G(nodes[i], Xb); H = F @ theta, where $G$ is the original Fourier, sigmoid, ReLU, or other feature function and $\theta$ is the trainable linear readout. For complex Fourier features, either retain complex tensors or concatenate real and imaginary parts. The formula for $Z$ and $\widehat K=ZZ^*$ is computed exactly; quadrature nodes and weights are fixed before training. On a held-out subset, compute the exact kernel Gram matrix using the original integral formula or a very large reference feature set, then estimate $\lambda_j(\mathbf K)$ and $\lambda_j(\widehat{\mathbf K})$ with Lanczos. Also measure the operator-norm proxy, leading-eigenvalue relative error, effective rank, kernel ridge validation error, and condition number. The first cheap experiment should use MNIST or CIFAR-10 reduced to 20--50 PCA dimensions, compare $N\in\{64,128,256,512\}$ deterministic Gauss or spherical quadrature features against equal-width iid random Fourier features, and train the same linear ridge classifier. Success means lower top-20 eigenvalue error and equal or better test accuracy at the same $N$; a stronger result is achieving the random-feature accuracy with at least 2x fewer features, reducing both feature memory and matrix-multiplication cost.