Complete Dictionary Learning via $\ell_p$-norm Maximization
Yifei Shen, Ye Xue, Jun Zhang, Khaled Letaief, Vincent Lau
Abstract
Dictionary learning is a classic representation learning method that has been widely applied in signal processing and data analytics. In this paper, we investigate a family of $\ell_p$-norm ($p>2,p \in N$) maximization approaches for the complete dictionary learning problem from theoretical and algorithmic aspects. Specifically, we prove that the global maximizers of these formulations are very close to the true dictionary with high probability, even when Gaussian noise is present. Based on the generalized power method (GPM), an efficient algorithm is then developed for the $\ell_p$-based formulations. We further show the efficacy of the developed algorithm: for the population GPM algorithm over the sphere constraint, it first quickly enters the neighborhood of a global maximizer, and then converges linearly in this region. Extensive experiments will demonstrate that the $\ell_p$-based approaches enjoy a higher computational efficiency and better robustness than conventional approaches and $p=3$ performs the best.
BibTeX
@InProceedings{pmlr-v124-shen20a,
title = {Complete Dictionary Learning via $\ell_p$-norm Maximization},
author = {Shen, Yifei and Xue, Ye and Zhang, Jun and Letaief, Khaled and Lau, Vincent},
booktitle = {Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)},
pages = {280--289},
year = {2020},
editor = {Peters, Jonas and Sontag, David},
volume = {124},
series = {Proceedings of Machine Learning Research},
month = {03--06 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v124/shen20a/shen20a.pdf},
url = {https://proceedings.mlr.press/v124/shen20a.html},
abstract = {Dictionary learning is a classic representation learning method that has been widely applied in signal processing and data analytics. In this paper, we investigate a family of $\ell_p$-norm ($p>2,p \in N$) maximization approaches for the complete dictionary learning problem from theoretical and algorithmic aspects. Specifically, we prove that the global maximizers of these formulations are very close to the true dictionary with high probability, even when Gaussian noise is present. Based on the generalized power method (GPM), an efficient algorithm is then developed for the $\ell_p$-based formulations. We further show the efficacy of the developed algorithm: for the population GPM algorithm over the sphere constraint, it first quickly enters the neighborhood of a global maximizer, and then converges linearly in this region. Extensive experiments will demonstrate that the $\ell_p$-based approaches enjoy a higher computational efficiency and better robustness than conventional approaches and $p=3$ performs the best.}
}