AI 中文总结
研究卡尔胡宁 - 洛伊夫变换逆问题，通过双换位子特征值问题找到与协方差最接近可交换的生成器，能恢复多种变换，给出对称近似时的相关特性及应用，如从已知生成器合成两范式协方差的KLT。
AI 中文摘要
卡尔胡宁 - 洛伊夫变换（KLT）可使二阶过程的协方差对角化，对均方截断是最优的。它归结为何种经典变换由协方差的对称换位子决定。本文研究逆问题。给定协方差\(R\)和候选生成器的有限维空间，与\(R\)最接近可交换的生成器，即\(\delta(A,R)=\|[R,A]\|_F/(\|R\|_F\|A\|_F)\)的极小值点，是双换位子特征值问题\(\mathrm{ad}_R^2(A^\ast)=\lambda A^\ast\)的最小特征值解。该框架能恢复隐藏变换和经典变换，给出了变分特征、三对角换位子唯一性结果等。当对称性近似时，给出了对称自适应分块变换的编码代价等。还给出了置换结构的图自同构特征等。作为应用，从两个已知生成器合成了两范式协方差的KLT。
英文摘要
The Karhunen--Loève 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 $δ(A,R)=\|[R,A]\|_F/(\|R\|_F\|A\|_F)$, is the smallest-eigenvalue solution of a double-commutator eigenvalue problem $\mathrm{ad}_R^2(A^\ast)=λA^\ast$, 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.
Commentsv1: 59 pages, 10 figures, 2 tables
