# Эксперимент: Covariance-Adaptive Hermite Latent Bottleneck (#1385)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built Hermite multi-index evaluation, Gaussian density-ratio projection, adaptive degree selection, and exact diagonal-Gaussian verification. The heat-semigroup identity matched product Gauss-Hermite quadrature with maximum relative error 8.0e-8, and tails decayed geometrically with covariance defect q. Adaptive selection used 5 versus 70 coefficients at q=0.04 and 35 versus 70 at q=0.20 while meeting the 0.08 target; it lost badly at q=0.55, so the benefit is real but limited to near-Gaussian covariances. This is a promising math/toy result, not evidence of a trained VAE win.",
  "metrics": {
    "baseline": "Fixed total degree 4: 70 coefficients; relative tails 4.86e-5, 7.46e-3, and 0.1223 for q=0.04, 0.20, and 0.55.",
    "idea": "Adaptive degrees 1, 3, and 8: 5, 35, and 495 coefficients; relative tails 0.0367, 0.0390, and 0.0324 respectively. At the 0.08 target, coefficient ratios versus baseline were 0.071x, 0.5x, and 7.07x."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 mini_experiment.py",
  "files": [
    "hermite_bottleneck.py",
    "verify.py",
    "mini_experiment.py",
    "verify_result.json",
    "mini_result.json"
  ],
  "limitations": "No MNIST/CIFAR VAE, differentiable training loop, decoder FLOP benchmark, EMA covariance estimation, non-diagonal covariance, mean-shift effects, or per-sample transmission protocol was tested. The exact tail test uses small diagonal Gaussian systems and distribution-level coefficients."
}