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This paper considers the problem of minimizing a differentiable function withlocally Lipschitz continuous gradient on the algebraic variety of real matricesof upper-bounded rank. This problem is known to enable the formulation ofseveral machine learning and signal processing tasks such as collaborativefiltering and signal recovery. Several definitions of stationarity exist forthis nonconvex problem. Among them, Bouligand stationarity is the strongestfirst-order necessary condition for local optimality. This paper proposes afirst-order algorithm that combines the well-known projected-projected gradientdescent map with a rank reduction mechanism and generates a sequence in thevariety whose accumulation points are Bouligand stationary. This algorithmcompares favorably with the three other algorithms known in the literature toenjoy this stationarity property, regarding both the typical computational costper iteration and empirically observed numerical performance. A framework todesign hybrid algorithms enjoying the same property is proposed and illustratedthrough an example.

Low-rank optimization methods based on projected-projected gradient descent that accumulate at Bouligand stationary points