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Fast Structured Orthogonal Dictionary Learning using Householder Reflections

Anirudh Dash, Aditya Siripuram

TL;DR

This work tackles fast, structured orthogonal dictionary learning by exploiting Householder reflections. For a single Householder, the authors prove $\ell_\infty$-consistent recovery of the generating vector with $p = \Omega(\log n)$ columns and $O(np)$ time, under a Bernoulli-sparse coefficient model with nonzeros in $[1,2]$ and mean $\mu$. They extend to a product of $m$ Householder matrices via a non-iterative sequential update with $O(n p m)$ complexity, enabling scalable recovery while maintaining accuracy in the sample-limited regime. Theoretical guarantees are complemented by simulations showing robust performance to sparsity $\theta$ and noise, and favorable comparisons against Procrustes-based and prior fast-structure methods when $p$ is small. Overall, the approach delivers faster dictionary learning with provable guarantees under structural constraints, with practical impact for large-scale signal processing and graph-based applications.

Abstract

In this paper, we propose and investigate algorithms for the structured orthogonal dictionary learning problem. First, we investigate the case when the dictionary is a Householder matrix. We give sample complexity results and show theoretically guaranteed approximate recovery (in the $l_{\infty}$ sense) with optimal computational complexity. We then attempt to generalize these techniques when the dictionary is a product of a few Householder matrices. We numerically validate these techniques in the sample-limited setting to show performance similar to or better than existing techniques while having much improved computational complexity.

Fast Structured Orthogonal Dictionary Learning using Householder Reflections

TL;DR

This work tackles fast, structured orthogonal dictionary learning by exploiting Householder reflections. For a single Householder, the authors prove -consistent recovery of the generating vector with columns and time, under a Bernoulli-sparse coefficient model with nonzeros in and mean . They extend to a product of Householder matrices via a non-iterative sequential update with complexity, enabling scalable recovery while maintaining accuracy in the sample-limited regime. Theoretical guarantees are complemented by simulations showing robust performance to sparsity and noise, and favorable comparisons against Procrustes-based and prior fast-structure methods when is small. Overall, the approach delivers faster dictionary learning with provable guarantees under structural constraints, with practical impact for large-scale signal processing and graph-based applications.

Abstract

In this paper, we propose and investigate algorithms for the structured orthogonal dictionary learning problem. First, we investigate the case when the dictionary is a Householder matrix. We give sample complexity results and show theoretically guaranteed approximate recovery (in the sense) with optimal computational complexity. We then attempt to generalize these techniques when the dictionary is a product of a few Householder matrices. We numerically validate these techniques in the sample-limited setting to show performance similar to or better than existing techniques while having much improved computational complexity.
Paper Structure (13 sections, 2 theorems, 14 equations, 5 figures, 3 algorithms)

This paper contains 13 sections, 2 theorems, 14 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and result summary
  3. Prior work
  4. Recovery for the structured orthogonal (Householder) dictionary
  5. Recovery for the general orthogonal dictionary
  6. Simulations
  7. Data Generation
  8. Results for the case m=1
  9. Results for the general case
  10. Proofs and other details
  11. Proof of Theorem \ref{['thm:1']}
  12. Proof of Lemma \ref{['lem1']}
  13. Details on Algorithm \ref{['find_H_X2']}

Key Result

Theorem 1

(Householder Recovery) Consider $\mathbf{Y=HX}$ and the model described in eq:1* for $\mathbf{X}$. Suppose then $\mathbf{u}$ can be recovered (up to sign) with the following recovery guarantee: We get $\mathbb{P} \left ( \lVert \mathbf{u} - \hat{\mathbf{u}} \rVert_\infty > t\right ) \rightarrow 0.$ The estimate $\hat{\mathbf{u}}$ is computed via Algorithm find_H_X, and the computational complexi

Figures (5)

  • Figure 1: Frobenius norm error in the estimated orthogonal dictionary for a varying number of Householder matrices (n=1000; p=20)
  • Figure 2: Frobenius norm error in the estimated orthogonal dictionary for a varying number of columns $(n=1000; m=10)$
  • Figure 3: $l_{\infty}$ norm error in $\mathbf{u}$ for a varying number of columns and different sparsity levels ($\theta$) for $\mathbf{Y=HX}$$(n=1000)$
  • Figure 4: Frobenius norm error per entry in $\mathbf{X}$ for a varying number of columns and varying sparsity levels ($\theta$) for $\mathbf{Y=HX}$$(n=1000)$
  • Figure 5: $l_{\infty}$ norm error in $\mathbf{u}$ for a varying number of columns and different sparsity levels ($\theta$) for $\mathbf{Y=HX}$ under noisy conditions (SNR is in dB) $(n=1000)$

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1