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High-dimensional data analysis and estimation appear in many data science and machine learning applications. The underlying low-dimensional structure in these high-dimensional data inspires us to develop optimality guarantees as well as optimization-based techniques for the fundamental problems in data science and machine learning. In recent years, non-convex optimization widely appears in engineering and is solved by many heuristic local algorithms, but lacks global guarantees. The recent geometric/landscape analysis provides a way to determine whether an iterative algorithm can reach global optimality. The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. A favorable geometry guarantees that many algorithms can avoid saddle points and converge to local minima. In this talk, I will introduce some of our recent work on geometric analysis and stochastic algorithms for non-convex optimization problems.