Matched Generators for the Karhunen--Loève Transform: A Double-Commutator Eigenvalue Theory

The Karhunen--Lo\`eve transform (KLT) diagonalizes the covariance of a second-order process and is optimal for mean-square truncation. Which classical transform it reduces to is governed by the symmetry commutant of the covariance: when the kernel commutes with a group action, the KLT eigenfunctions are the irreducible representation functions of that group, recovering the Fourier, cosine, Mellin, and spherical-harmonic systems. We study the inverse question. Given a covariance $R$ and a finite-dimensional space of candidate generators, the generator nearest to commuting with $R$, the minimizer of $\delta(A,R)=\|[R,A]\|_F/(\|R\|_F\|A\|_F)$, is the smallest-eigenvalue solution of a double-commutator eigenvalue problem $\mathrm{ad}_R^2(A^\ast)=\lambda A^\ast$, a Hermitian generalized eigenvalue problem of size the number of generators, independent of dimension. The framework recovers hidden transforms as well as classical ones: a variational characterization turns the existence of a commuting generator into a spectral condition, and a tridiagonal commutant-uniqueness result yields the prolate spheroidal, cosine, and discrete orthogonal-polynomial bases as exact recoveries, with matrix-valued extensions, and produces a continuum of transforms interpolating between and beyond the classical families. When symmetry is approximate, the coding penalty of the symmetry-adapted blockwise transform equals the multi-information among the sectors, an exact threshold between the fixed and data-driven transforms. We further give a graph-automorphism characterization of permutation structure, a sequential deflation for non-Abelian symmetry, and stability bounds under estimation error. As an application, the KLT of a two-paradigm covariance is synthesized from its two known generators, without forming the mixed covariance, reaching the full-data transform's compaction from few observations.