Memory-efficient Kernel PCA via Partial Matrix Sampling and Nonconvex Optimization: a Model-free Analysis of Local Minima

Kernel PCA is a widely used nonlinear dimension reduction technique in machine learning, but storing the kernel matrix is notoriously challenging when the sample size is large. Inspired by Yi et al. [2016], where the idea of partial matrix sampling followed by nonconvex optimization is proposed for matrix completion and robust PCA, we apply a similar approach to memory-efficient Kernel PCA. In theory, with no assumptions on the kernel matrix in terms of eigenvalues or eigenvectors, we established a model-free theory for the low-rank approximation based on any local minimum of the proposed objective function. As interesting byproducts, when the underlying positive semidefinite matrix is assumed to be low-rank and highly structured, corollaries of our main theorem improve the state-of-the-art results of Ge et al. [2016, 2017] for nonconvex matrix completion with no spurious local minima. Numerical experiments also show that our approach is competitive in terms of approximation accuracy compared to the well-known Nystr\"{o}m algorithm for Kernel PCA.