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

```python
# 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