Many modern machine learning applications-from online principal component analysis to
covariance matrix identification and dictionary learning-can be formulated as minimization …

In this paper, we propose a new global analysis framework for a class of low-rank matrix
recovery problems on the Riemannian manifold. We analyze the global behavior for the …

In this paper, we present a novel penalty model called ExPen for optimization over the
Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen …

We propose to use the {\L} ojasiewicz inequality as a general tool for analyzing the
convergence rate of gradient descent on a Hilbert manifold, without resorting to the …

We show that on the manifold of fixed-rank and symmetric positive semi-definite matrices,
the Riemannian gradient descent algorithm almost surely escapes some spurious critical …

Low-rank matrix recovery problems are prevalent in modern data science, machine learning,
and artificial intelligence, and the low-rank property of matrices is widely exploited to extract …

In this thesis, we will talk about some non-convex analysis applied on the low-rank matrix
related problems via the Riemannian optimization. In Chapter 2, we will develop tools to …