Learning Dictionaries with Bounded Self-Coherence

TL;DR

This paper introduces a dictionary learning method that effectively controls self-coherence, balancing sparsity and approximation quality for signal processing tasks.

Contribution

It proposes a novel approach to regulate self-coherence in learned dictionaries, enabling better trade-offs between sparsity and signal approximation.

Findings

The method achieves low self-coherence in learned dictionaries.

It improves the balance between sparsity and approximation accuracy.

The approach approximates an equiangular tight frame.

Abstract

Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training data matrix into a dictionary and a sparse coding matrix. The learned dictionary is characterized by two properties: the coherence of the dictionary to observations of the signal class, and the self-coherence of the dictionary atoms. A high coherence to the signal class enables the sparse coding of signal observations with a small approximation error, while a low self-coherence of the atoms guarantees atom recovery and a more rapid residual error decay rate for the sparse coding algorithm. The two goals of high signal coherence and low self-coherence are typically in conflict, therefore one seeks a trade-off between them, depending on the application. We present a dictionary learning method with an effective control over the self-coherence of the trained dictionary, enabling a trade-off between maximizing the sparsity of codings and approximating an equiangular tight frame.