signal-transform-unification

name

signal-transform-unification

description

Unify signal transforms (DFT, DCT, wavelet, KLT, etc.) under representation-theoretic principles via the Algebraic Diversity framework. Covers matched group discovery, Peter-Weyl theorem applications, covariance-invariant transforms, and applications to MIMO, GNNs, transformers, and quantum informatics. Activation: signal transform theory, matched group discovery, Algebraic Diversity, DFT unification, representation theory signal processing, Peter-Weyl transform, covariance eigenbasis.

Unified Signal Transform Theory

Unify all classical and modern signal transforms under one representation-theoretic principle. Based on arXiv:2605.11589v1.

Core Principle

Every signal transform is the eigenbasis of every covariance invariant under a specific group.

Columns are constructed from irreducible matrix elements of the group via the Peter-Weyl theorem.

Algebraic Diversity (AD) Framework

Step 1: Identify the Matched Group

For any covariance matrix, find the group that leaves it invariant:

• DFT: Cyclic group C_n
• DCT: Dihedral group D_n
• Walsh-Hadamard: Elementary abelian 2-group
• Haar wavelet: Iterated wreath product
• KLT: Trivial matched group (data-dependent limit)

Step 2: Composition Rules

Transforms compose via:

• Direct products: Independent signal dimensions
• Wreath products: Hierarchical/multiscale structure
• Semidirect products: Mixed symmetry groups

Step 3: Matched Group Discovery Algorithm

# DAD-CAD relaxation cast as generalized eigenvalue problem
# Input: Empirical covariance matrix
# Output: Matched group identification

# Key metrics:
# - Commutativity residual δ (noise-aware)
# - Algebraic coloring index α (finite-SNR)

Application Mapping

Domain	Matched Group	Transform
Massive MIMO	Unitary group	DFT variants
Graph Neural Networks	Graph automorphism	Graph Fourier
Transformer attention	Permutation group	Learned basis
Brain connectivity	Structural symmetry	Connectivity spectrum
Quantum informatics	Symplectic group	Fractional Fourier

Key Insights

1. Data-dependent vs. fixed: KLT is the trivial-group limit; classical transforms are symmetry-matched
2. Resolution tradeoff: Matched group size inversely relates to transform resolution
3. Polynomial-time discovery: DAD-CAD algorithm discovers matched group without expert judgment
4. Reed-Muller/Arithmetic transforms: Change-of-basis on Walsh-Hadamard matched group

Practical Usage

When designing a signal processing pipeline:

1. Compute empirical covariance of input data
2. Run DAD-CAD to discover matched group
3. Select transform corresponding to discovered group
4. For noisy data, use δ and α metrics for robustness

Activation Keywords

• signal transform theory
• matched group discovery
• Algebraic Diversity
• Peter-Weyl theorem
• covariance eigenbasis
• DAD-CAD algorithm
• representation theory signal processing

References

• arXiv: 2605.11589v1 — "Unification of Signal Transform Theory"
• Author: Mitchell A. Thornton
• Published: 2026-05-12