SynthCore

Kac-rotated fast projection / REPORT.md

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Kac-rotated fast projection MVP

Implementation

run_experiment.py implements the exact streamed update z[a]=cos(theta)*u+sin(theta)*v, z[b]=-sin(theta)*u+cos(theta)*v, followed by scaled coordinate subsampling. It compares the Kac product with Gaussian and Haar projections for n=64, m=16, using fixed seeds.

Run:

/home/maxwelhelp/main/bin/python3 run_experiment.py

The complete machine-readable output is in results.json.

Mechanism checks

  1. Orthogonality: every finite product stayed orthogonal, with maximum entrywise defect in Q.T @ Q-I of 1.33e-15 across all tested T.
  2. Quadratic mean relaxation: for a fixed unit vector, the measured means at T=0, 16, 32, 64, 128, 256, 512, 1024 were respectively 0.8388, 0.8845, 0.9031, 0.9571, 0.9636, 1.0001, 0.9934, 0.9987. They matched the analytic prediction 1 + (initial_scaled_energy - 1)*((n-1)/n)^T within at most 0.0159 absolute error over the sweep.
  3. Degree-two pseudo-mixing: the standard deviation of the scaled coordinate energy divided by the Haar reference standard deviation was 0, 0.568, 0.740, 0.847, 0.958, 0.969, 1.002, 1.023 over the same T values. Thus the quadratic statistic reached Haar-like spread around 8n rotations, as predicted qualitatively by the mechanism.

Secondary comparison

At T=8n, pair-distance distortion on 64 random normalized vectors was:

  • Kac: mean 1.0478, MAE 0.2678, 95th-percentile absolute error 0.5414
  • Gaussian: mean 1.0469, MAE 0.2749, 95th-percentile absolute error 0.6406
  • Haar: mean 1.0421, MAE 0.2898, 95th-percentile absolute error 0.6618

The literal NumPy streamed implementation took 0.0173 s per batch versus 4.39e-5 s for dense Gaussian multiplication. This is not a speed win without fused/vectorized kernels. The explicit (i,j,theta) representation at T=8n uses 8192 bytes in this setup versus 32768 bytes for a full dense orthogonal matrix, a 4x reduction; the runtime arrays additionally store precomputed sine/cosine and use 12288 bytes.

Conclusion

The mathematical mechanism manifested clearly: exact orthogonality, the predicted first-moment relaxation, and convergence of the relevant quadratic statistic toward Haar behavior. The experiment does not establish an end-to-end speedup; Python-level sequential updates are much slower than optimized dense BLAS. No CIFAR/MLP accuracy experiment was run.