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 several machine learning and signal processing tasks such as collaborative filtering and signal recovery. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest first-order necessary condition for local optimality. This paper proposes a first-order algorithm that combines the well-known projected-projected gradient descent map with a rank reduction mechanism and generates a sequence in the variety whose accumulation points are Bouligand stationary. This algorithm compares favorably with the three other algorithms known in the literature to enjoy this stationarity property, regarding both the typical computational cost per iteration and empirically observed numerical performance. A framework to design hybrid algorithms enjoying the same property is proposed and illustrated through an example.

翻译：本文研究在实矩阵上界秩代数簇上最小化具有局部Lipschitz连续梯度的可微函数问题。该问题已知可表述多种机器学习与信号处理任务，如协同过滤和信号恢复。针对这一非凸问题存在多种稳定性定义，其中Bouligand稳定性是局部最优性最强的的一阶必要条件。本文提出一种一阶算法，将经典的投影-投影梯度下降映射与降秩机制相结合，在代数簇上生成收敛于Bouligand稳定点的序列。该算法在迭代计算复杂度和实际数值性能方面均优于文献中已知的三种具有相同稳定性特征的算法。本文进一步提出构建具有相同特性的混合算法框架，并通过实例加以说明。