We consider the block coordinate descent methods of Gauss-Seidel type with
proximal regularization (BCD-PR), which is a classical method of minimizing
general nonconvex objectives under constraints that has a wide range of
practical applications. We theoretically establish the worst-case complexity
bound for this algorithm. Namely, we show that for general nonconvex smooth
objectives with block-wise constraints, the classical BCD-PR algorithm
converges to an epsilon-stationary point within O(1/epsilon) iterations. Under
a mild condition, this result still holds even if the algorithm is executed
inexactly in each step. As an application, we propose a provable and efficient
algorithm for `Wasserstein CP-dictionary learning', which seeks a set of
elementary probability distributions that can well-approximate a given set of
d-dimensional joint probability distributions. Our algorithm is a version of
BCD-PR that operates in the dual space, where the primal problem is regularized
both entropically and proximally.

该论文主要介绍了基于 Gauss-Seidel 类型的块坐标下降算法（BCD-PR）的近端正则化方法，以及提出了一种双重正则化的原始问题的对偶算法，并将其应用于 Wasserstein CP - 字典学习中，从而实现了优化问题的求解。