Covariance-Adaptive Hermite Latent Bottleneck

Integrate the bottleneck after a VAE encoder or after an uncertainty-producing MLP, before the latent is passed to the decoder. The input is a minibatch z\in\mathbb{R}^{B imes d}; maintain exponential-moving-average estimates of the mean m and covariance \widehat\Sigma, whiten each sample to v=(z-m)(\widehat\Sigma+\epsilon I)^{-1/2}, and represent the normalized latent density or feature response with a truncated Hermite expansion. Use tensor-product or total-degree multivariate Hermites only for small d, such as d=8 or 16; for larger d use separable first- and second-order terms or random projections. A concrete loop is: (1) compute m and \widehat\Sigma; (2) compute eigenvalues of \widehat\Sigma and q=max_i|\lambda_i-1|; (3) choose N=min\{N_max:C_hat q^{(N+1)/2}\le au\}; (4) evaluate H_\alpha(v) recursively using H_0(x)=1 and H_{k+1}(x)=xH_k(x)-kH_{k-1}(x); (5) retain coefficients with |\alpha|\le N and reconstruct the decoder input from those coefficients. Train with reconstruction and KL losses plus a small covariance penalty \|\widehat\Sigma-I\|_F^2 so that the theorem's q<1 regime is encouraged. The mathematical quantities are the Hermite recursion, covariance-defect estimate, and degree rule; C_hat should be estimated on held-out batches as the maximum observed tail divided by q^{(N+1)/2}. First test on MNIST or CIFAR-10 with a 16-dimensional VAE, comparing a dense latent bottleneck and PCA truncation at equal transmitted coefficients. Measure reconstruction error versus retained coefficients, decoder FLOPs, and whether the predicted tail bound tracks actual error. Success means at least 2x fewer latent coefficients at matched reconstruction error, or improved reconstruction at fixed bottleneck size, without unstable training when covariance is near but not exactly identity.