# Эксперимент: Adaptive Householder Gradient Subspaces (#1098)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built an adaptive Gaussian residual range finder with implicit Householder QR factors and a one-pass Gram–Schmidt baseline. The mechanism manifested: Householder orthogonality stayed around 0.9–1.4e-15 across condition numbers up to 1e14, while Gram–Schmidt error grew from 1.1e-13 to 1.41; tightening tolerance monotonically increased rank and measured residuals stayed below the requested tolerance. The synthetic layer test used ranks 8/20/32 for decay rates 0.40/0.16/0.03 versus fixed rank 32, showing the intended rank savings on compressible layers, but this is not evidence of a training win.",
  "metrics": {
    "baseline": "One-pass Gram–Schmidt: orthogonality error 1.14e-13, 5.65e-9, 1.32, 1.41 for condition numbers 1e2, 1e6, 1e10, 1e14; fixed randomized rank-32 relative errors 2.4e-5, 0.01884, and 0.47745 on the three synthetic layers.",
    "idea": "Implicit Householder: orthogonality error 1.05e-15 to 1.38e-15 across the same condition sweep; adaptive ranks 8, 20, and 32 at relative tolerance 0.10, with errors 0.0943, 0.0905, and 0.4584. Relative residuals for tolerances 0.4, 0.2, and 0.1 were 0.2685, 0.1563, and 0.0922."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 adaptive_householder.py",
  "files": [
    "adaptive_householder.py",
    "results.txt",
    "article.md",
    "idea_context.json"
  ],
  "limitations": "No Transformer, WikiText-2, AdamW/GaLore integration, GPU benchmark, peak-memory measurement, refresh wall-clock comparison, BF16/FP16 test, or large-scale training experiment was performed. The adaptive method uses explicit Q blocks internally for this MVP, while production use would retain reflector factors and apply them implicitly."
}