Committee: 

Dr. Davenport, Advisor  

Dr. Romberg, Chair

Dr. Koltchinskii

Abstract:

The objective of the proposed research is to develop and analyze algorithms for high-dimensional statistical inference and machine learning that exploit intrinsic low-dimensional structure. We want to show that the sample complexity and susceptibility to noise of such algorithms is determined by the relatively low intrinsic dimension rather than the high ambient dimension. In addition to our prior low-rank matrix completion and denoising work, we specifically consider the application of reproducing kernel Hilbert space methods to function estimation on a manifold. We want to show that the complexity of estimating functions on a manifold scales with the manifold dimension; we furthermore want to show that, for manifolds embedded in high-dimensional space, this can be achieved with practical, manifold-agnostic kernel methods.