基于投影投影梯度下降的低秩优化方法，能够收敛到Bouligand驻点
Low-rank optimization methods based on projected projected-gradient descent that accumulate at Bouligand stationary points
AI总结：
本文提出两种低秩优化方法，通过投影投影梯度下降与秩减少机制或投影梯度下降与投影投影梯度下降的混合，生成序列收敛到Bouligand驻点，具有高效性和实用性。
摘要：
本文考虑在代数流形上最小化可微函数的问题，该函数具有局部Lipschitz连续梯度，且矩阵的秩有上界。此问题可应用于机器学习和信号处理任务，如降维、协同过滤和信号恢复。本文提出两种一阶方法，生成序列的极限点为Bouligand驻点。第一种方法结合已知的投影投影梯度下降映射与秩减少机制，第二种方法是投影梯度下降与投影投影梯度下降的混合。这两种方法在低秩优化领域因其收敛性、设计简洁性、计算成本和实测性能而突出。用于分析方法的理论框架具有独立价值。
英文摘要：
This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of real matrices of upper-bounded rank. This problem is known to enable the formulation of various machine learning or signal processing tasks such as dimensionality reduction, collaborative filtering, and signal recovery. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest necessary condition for local optimality. This paper proposes two first-order methods that generate a sequence in the variety whose accumulation points are Bouligand stationary. The first method combines the well-known projected projected-gradient descent map with a rank reduction mechanism. The second method is a hybrid of projected gradient descent and projected projected-gradient descent. Both methods stand out in the field of low-rank optimization methods when considering their convergence properties, their streamlined design, their typical computational cost per iteration, and their empirically observed numerical performance. The theoretical framework used to analyze the proposed methods is of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
