On the Pseudo-Mixing of Kac's Walk

What the math gives to ML

The paper provides a constructive way to generate approximately Haar-random orthogonal transformations using only a sequence of two-coordinate rotations. The transferable asset is computational pseudo-mixing: low-degree observables of the resulting matrix can become Haar-like after roughly n polylog(n) rotations, even though the full matrix has not mixed in total variation. This suggests replacing dense Gaussian or Haar projections in neural networks with a streamed Kac rotation sequence, retaining exact norm preservation while reducing matrix storage and potentially projection cost. The cleanest first target is a random projection or bottleneck layer whose downstream behavior depends primarily on degree-two norm and inner-product statistics.