Separated Digital-Net Codebook Initialization

Integrate this at initialization for the codebook matrix C in a VQ-VAE or online k-means layer. First run the encoder on a small calibration set, estimate robust per-coordinate bounds [a_j,b_j] from the 1st and 99th percentiles, and optionally PCA-project the latent vectors to d dimensions when the original latent dimension is large. Generate K binary digital-net points u_k in [0,1]^d with matrix or linear scrambling, then initialize c_{k,j}=a_j+(b_j-a_j)u_{k,j}; decode through the inverse PCA transform if projection was used. Keep the standard nearest-code assignment and commitment loss, with no repulsion term in the first experiment so the initialization effect is isolated. Pseudocode: warm up encoder; estimate PCA,a,b; generate scrambled net u; set C=decode(a+(b-a)u); train the VQ model. At initialization and every 1,000 steps, compute R_infty(C), estimate h_infty(C) using latent samples or a probe grid, calculate rho_infty(C), count assignments per codeword, and record codebook perplexity. The paper supplies the expected geometric ordering, while latent-distribution coverage and code utilization are empirical. Test a CIFAR-10 VQ-VAE with K=512 and d=8 against Gaussian, random-calibration-example, k-means++, and Owen-scrambled Sobol initialization at equal warm-up cost. Success means fewer dead codes during the first 20,000 steps, lower reconstruction loss at equal updates, higher assignment perplexity, and no worse final FID or reconstruction error.