论文标题
通过$ \ ell_p $ - 最大化完成词典学习
Complete Dictionary Learning via $\ell_p$-norm Maximization
论文作者
论文摘要
字典学习是一种经典的表示学习方法，已广泛应用于信号处理和数据分析。在本文中，我们研究了一个$ \ ell_p $ -norm的家庭（$ p> 2，p \ in \ mathbb {n} $）从理论和算法方面从理论和算法方面进行完整词典学习问题的最大化方法。具体而言，我们证明，即使存在高斯噪声，这些配方的全局最大化器也非常接近具有很高可能性的真实词典。基于广义功率方法（GPM），然后为基于$ \ ell_p $的公式开发有效的算法。我们进一步显示了开发算法的功效：对于在球体约束上的总体GPM算法，它首先迅速进入全球最大化器的邻域，然后在该区域线性收敛。广泛的实验将证明，基于$ \ ell_p $的方法比传统方法具有更高的计算效率和更好的鲁棒性，并且$ p = 3 $表现最好。
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 ($p>2,p \in \mathbb{N}$) 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.
