Dictionary Learning with BLOTLESS Update
This work addresses the challenge of improving dictionary learning efficiency for sparse coding applications, offering incremental advancements in algorithm performance.
The paper tackles the problem of dictionary learning for sparse representation by proposing a block total least squares (BLOTLESS) algorithm for dictionary update, which simultaneously updates dictionary elements and sparse coefficients. Numerical experiments show that BLOTLESS outperforms state-of-the-art algorithms, particularly with small training datasets, and establishes theoretical bounds for exact dictionary recovery.
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.