Quartic First-Order Methods for Low-Rank Minimization

• Published in: Journal of optimization theory and applications Vol. 189; no. 2; pp. 341 - 363
• Main Authors: Dragomir, Radu-Alexandru, d’Aspremont, Alexandre, Bolte, Jérôme
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-05-2021; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Divergence; Engineering; Euclidean geometry; Kernel functions; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Operations Research/Decision Theory; Optimization; Theory of Computation; Burer–Monteiro; Euclidean distance matrix completion; Low-rank minimization; 90C06; Matrix factorization; 90C26; Bregman first-order methods
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-021-01820-3

We study a general nonconvex formulation for low-rank minimization problems. We use recent results on non-Euclidean first-order methods to provide efficient and scalable algorithms. Our approach uses the geometry induced by the Bregman divergence of well-chosen kernel functions; for unconstrained problems, we introduce a novel family of Gram quartic kernels that improve numerical performance. Numerical experiments on Euclidean distance matrix completion and symmetric nonnegative matrix factorization show that our algorithms scale well and reach state-of-the-art performance when compared to specialized methods.