# Dictionary Learning with BLOTLESS Update

## TL;DR

This paper introduces BLOTLESS, a novel block total least squares algorithm for dictionary update in sparse representation, which improves recovery accuracy especially with limited training data.

## Contribution

The paper proposes BLOTLESS, a new dictionary update method that updates blocks of elements simultaneously, with theoretical recovery conditions and advantages over existing algorithms.

## Key findings

- BLOTLESS achieves accurate dictionary recovery with fewer training samples.
- Theoretical bounds closely match empirical data requirements.
- Numerical experiments show improved performance over state-of-the-art methods.

## Abstract

Algorithms for learning a dictionary to sparsely represent a given dataset typically alternate between sparse coding and dictionary update stages. Methods for dictionary update aim to minimise expansion error by updating dictionary vectors and expansion coefficients given patterns of non-zero coefficients obtained in the sparse coding stage. We propose a block total least squares (BLOTLESS) algorithm for dictionary update. BLOTLESS updates a block of dictionary elements and the corresponding sparse coefficients simultaneously. In the error free case, three necessary conditions for exact recovery are identified. Lower bounds on the number of training data are established so that the necessary conditions hold with high probability. Numerical simulations show that the bounds approximate well the number of training data needed for exact dictionary recovery. Numerical experiments further demonstrate several benefits of dictionary learning with BLOTLESS update compared with state-of-the-art algorithms especially when the amount of training data is small.