🤖 AI Summary
This work addresses the lack of a unified theoretical foundation for classical and modern signal transforms, which has hindered systematic understanding and automated selection. By leveraging group representation theory, the authors unify a broad class of transforms—including the DFT, DCT, Walsh–Hadamard, Haar wavelets, KLT, spherical harmonics, and fractional Fourier transform—as eigenbases of covariance matrices that are covariant under specific group actions. Central to this framework is the identification of a “matching group” that leaves the signal covariance invariant, combined with the Peter–Weyl theorem and the Algebraic Diversity (AD) formalism. The study further introduces a novel, data-driven polynomial-time algorithm to automatically discover the optimal matching group without expert intervention, enabling automatic transform selection. This approach naturally extends to cutting-edge domains such as massive MIMO systems, graph neural networks, and Transformer attention mechanisms.
📝 Abstract
We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Loève transform, and several others along with their continuous counterparts (Fourier transform, Fourier series, spherical harmonics, fractional Fourier transform) under one representation-theoretic principle: each is the eigenbasis of every covariance invariant under a specific finite or compact group, with columns constructed from the irreducible matrix elements of the group via the Peter-Weyl theorem. The unification rests on the Algebraic Diversity (AD) framework, which identifies the matched group of a covariance as the foundational object of second-order signal processing. The data-dependent KLT emerges as the trivial-matched-group limit; classical transforms emerge as the cyclic, dihedral, elementary abelian, iterated wreath, and hybrid wreath cases. Composition rules cover direct, wreath, and semidirect products. The Reed-Muller and arithmetic transforms appear as related change-of-basis transforms on the matched group of Walsh-Hadamard. A polynomial-time algorithm for matched-group discovery, the DAD-CAD relaxation cast as a generalized eigenvalue problem in double-commutator form, closes the operational loop: the matched group of any empirical covariance is discovered without expert judgment, with noise-aware variants via the commutativity residual $δ$ and algebraic coloring index $α$ for finite-SNR settings. The fractional Fourier transform is treated as the metaplectic $SO(2)$ case with Hermite-Gauss matched basis, and a structural principle relates matched group size inversely to transform resolution. Modern applications (massive-MIMO, graph neural networks, transformer attention, point cloud and 3D vision, brain connectivity, single-cell genomics, quantum informatics) are sketched with their matched groups.