Abstract: Sparse principal component analysis (PCA) is an important technique for dimensionality reduction and variable selection of high-dimensional data. Existing sparse PCA algorithms face new challenges in modern applications such as genetics data, as non-convex optimization methods lack global convergence and stability guarantee, whereas convex relaxation methods become computationally prohibitive when the dimensionality is tens of thousands. In this work we develop efficient algorithms for sparse PCA with global convergence guarantee. Our approach starts from combining some recent theory on convex optimization with the particular geometry of sparse PCA, and converts the convexified sparse PCA problem to an unconstrained problem, making it compatible with many existing gradient-based methods and online algorithms. The resulting algorithms greatly extend the applicability of sparse PCA, as demonstrated in a high-dimensional gene expression data for the detection of functional gene groups.