Local Identification of Overcomplete Dictionaries

TL;DR

This paper introduces a theoretical framework for the local identification of overcomplete, coherent dictionaries from training signals, along with a practical iterative algorithm, ITKM, demonstrating effective recovery under certain conditions.

Contribution

The paper provides the first theoretical results on stable local identification of overcomplete dictionaries and proposes a simple, efficient algorithm for practical recovery.

Findings

Stable identification possible for sparsity up to O(μ^{-2})

Asymptotic exact recovery for sparsity up to O(μ^{-1})

ITKM algorithm has complexity O(dKN) and demonstrates local efficiency

Abstract

This paper presents the first theoretical results showing that stable identification of overcomplete $\mu$-coherent dictionaries $\Phi \in \mathbb{R}^{d\times K}$ is locally possible from training signals with sparsity levels $S$ up to the order $O(\mu^{-2})$ and signal to noise ratios up to $O(\sqrt{d})$. In particular the dictionary is recoverable as the local maximum of a new maximisation criterion that generalises the K-means criterion. For this maximisation criterion results for asymptotic exact recovery for sparsity levels up to $O(\mu^{-1})$ and stable recovery for sparsity levels up to $O(\mu^{-2})$ as well as signal to noise ratios up to $O(\sqrt{d})$ are provided. These asymptotic results translate to finite sample size recovery results with high probability as long as the sample size $N$ scales as $O(K^3dS \tilde \varepsilon^{-2})$, where the recovery precision $\tilde \varepsilon$ can go down to the asymptotically achievable precision. Further, to actually find the local maxima of the new criterion, a very simple Iterative Thresholding and K (signed) Means algorithm (ITKM), which has complexity $O(dKN)$ in each iteration, is presented and its local efficiency is demonstrated in several experiments.