2022/09/29 by Yuetian Luo, Nicolas Garcia Trillos, Nicolás García Trillos +2
Medicine · #Corneal surgery and disorders #Ophthalmology and Eye Disorders #Scoliosis diagnosis and treatment
paper · pdf · doi:10.1007/s10107-026-02400-x
Abstract In this paper we study the landscape of a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization, i.e., factorize a PSD matrix X <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> as YY^\top <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:msup> <mml:mrow> <mml:mi>Y</mml:mi> </mml:mrow> <mml:mi>⊤</mml:mi> </mml:msup> </mml:mrow> </mml:math> , and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. In particular, we show that the entire search space can be divided into three regions: ( R1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; ( R2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> ) the region containing neighborhoods of all strict saddle points; ( R3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ) the remaining regions, where the factorized objective has a large gradient. Our results cover both noisy and noiseless settings in applications of interest. To the best of our knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our main results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge in establishing the results in the general setting. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> with a radius one-third of the least singular value of Y <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> is a geodesically convex set under the Riemannian quotient geometry, a result that, as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained in this paper is sharp up to constants.