We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples $y^{1},y^{2},\ldots, y^{n}$ into an appropriate basis (dictionary) $A^*$ and sparse vectors $x^{1*},\ldots,x^{n*}$. Our algorithm is a simple alternating minimization procedure that switches between $\ell_1$ minimization and gradient descent in alternate steps. Dictionary learning and specifically alternating minimization algorithms for dictionary learning are well studied both theoretically and empirically. However, in contrast to previous theoretical analyses for this problem, we replace the condition on the operator norm (that is, the largest magnitude singular value) of the true underlying dictionary $A^*$ with a condition on the matrix infinity norm (that is, the largest magnitude term). This not only allows us to get convergence rates for the error of the estimated dictionary measured in the matrix infinity norm, but also ensures that a random initialization will provably converge to the global optimum. Our guarantees are under a reasonable generative model that allows for dictionaries with growing operator norms, and can handle an arbitrary level of overcompleteness, while having sparsity that is information theoretically optimal. We also establish upper bounds on the sample complexity of our algorithm.

利用替代最小化算法，本篇论文提出了一种字典学习/ 稀疏编码问题的理论保证算法。与以往的理论分析不同，该算法在真实字典的运算符范数条件上转换为矩阵无穷范数条件，并具有信息理论上的优化稀疏性和适应不同过完备度的特点，同时也确立了算法的样本复杂性上界。

交替极小化字典学习：局部收敛保证