Marchenko-Pastur law for tensor powers of exchangeable unconditional vectors
arXiv:2607.21759
2026
Theory
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper extends the Marchenko–Pastur spectral law from independent-coordinate vectors to isotropic, exchangeable, unconditional vectors and tensor powers such as X^{\otimes d}. The transferable asset is a predictable covariance spectrum despite substantial coordinate dependence and tensor-induced correlations. This supports principled initialization, width selection, and conditioning diagnostics for tensorized random features and polynomial feature maps. The most practical experiments use the predicted spectral edges to rescale tensor features or regularize covariance outliers during training.
Ideas from this paper
Unverified
2026
Use Marchenko–Pastur spectral edges to calibrate tensorized random features even when the base vector has exchangeable, sign-symmetric dependent coordinates. Rescale the tensor features and select their retained dimension so the predicted covariance bulk remains well-conditioned instead of assuming independent Gaussian coordinates.
Useful6/10
Difficulty5/10
Novelty7/10
Unverified
2026
Add a spectral regularizer that prevents tensorized feature batches from developing covariance outliers or a collapsed lower edge. The target is the Marchenko–Pastur bulk predicted for the current feature-to-sample ratio, rather than an arbitrary identity-covariance penalty that may suppress useful anisotropy.
Useful5/10
Difficulty6/10
Novelty6/10