标题： 严格低秩约束优化 -- 一种渐近的$\mathcal{O}(\frac{1}{t^2})$方法 显示英文标题

标题： Strictly Low Rank Constraint Optimization -- An Asymptotically $\mathcal{O}(\frac{1}{t^2})$ Method

摘要： 我们研究一类带有\textit{排名}正则化的非凸和非光滑问题，以在最优解中促进稀疏性。 我们提出应用近端梯度下降方法来解决这个问题，并通过在中间更新的奇异值上应用一种新颖的支持集投影操作来加速该过程。 我们证明我们的算法能够达到$O(\frac{1}{t^2})$的收敛速度，这与Nesterov在光滑和凸问题上的一阶方法的最优收敛速度完全相同。 可以期望严格的稀疏性，并且每次更新期间奇异值的支持集是单调缩减的，据我们所知，这在基于动量的算法中是新颖的。 显示英文摘要

摘要： We study a class of non-convex and non-smooth problems with \textit{rank} regularization to promote sparsity in optimal solution. We propose to apply the proximal gradient descent method to solve the problem and accelerate the process with a novel support set projection operation on the singular values of the intermediate update. We show that our algorithms are able to achieve a convergence rate of $O(\frac{1}{t^2})$, which is exactly same as Nesterov's optimal convergence rate for first-order methods on smooth and convex problems. Strict sparsity can be expected and the support set of singular values during each update is monotonically shrinking, which to our best knowledge, is novel in momentum-based algorithms.