论文标题
非convex矩阵分解是地球上的凸：从riemannian的角度来看固定级矩阵优化的全局景观分析
Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
论文作者
论文摘要
我们研究了固定级阳性半芬矿（PSD）约束的一般矩阵优化问题。我们执行Burer-Monteiro分解，并在搜索空间中考虑特定的Riemannian商的几何形状，该搜索空间中配备了欧几里得度量的总空间。当原始目标F满足标准限制的强凸度和平滑性属性时，我们表征了Riemannian商几何形状下的分解目标的全球景观。我们显示整个搜索空间可以分为三个区域：（R1）感兴趣的目标参数附近的区域，其中分解的物镜在地理上是强烈凸面和光滑的； （R2）包含所有严格马鞍点的街区的区域； （R3）其余区域，分解目标具有较大的梯度。据我们所知，这是Riemannian商几何形状下的Burer-Monteiro分解目标的首次全球景观分析。我们的结果为burer-monteiro分解下的香草梯度下降的出色性能提供了完全几何解释。当F满足较弱的严格凸性属性时，我们表明存在局部最小化器附近的邻域，因此分解的目标是地理上的凸面。为了证明我们的结果，我们提供了最小二乘目标的矩阵分解问题的全面景观分析，这是一个关键的桥梁。我们的结论还基于独立兴趣的结果，表明以y为中心的地半径为y的半径为y的半径为1/3，这是基于riemannian商的几何形状的地质性凸面，作为延纹，这也意味着在bures-the-bures-the-bures-wasseinseactein space中的convexity半径限制。获得的凸度半径是锐利的常数。
We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization 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. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best 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 results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.
