Isotropic Decompositions via Inverse Eigenvectors

What the math gives to ML

The paper constructs a nonlinear analogue of an eigenbasis from vectors satisfying the coordinatewise inverse-eigenvector equation Mα = α⁻¹, and uses these vectors to decompose the identity into positive rank-one terms. For real correlation matrices, the resulting vectors and weights form a centered discrete isotropic probability measure, equivalently a weighted tight frame with Σᵢ cᵢαᵢαᵢᵀ = I and Σᵢcᵢ = 1. This provides a principled way to generate finite prototype or feature dictionaries whose covariance is exactly controlled without applying an explicit whitening transform. The most plausible ML transfer is a structured initialization or trainable codebook for embeddings, attention directions, or MoE routers, with approximate inverse-eigenvector solving performed offline and the tight-frame residual monitored during training.