Matrix Completion Under Monotonic Single Index Models
This addresses a real-world challenge in matrix completion for data analysis where nonlinear distortions are common, representing an incremental advance by extending linear models to monotonic nonlinear cases.
The paper tackles the problem of matrix completion when the underlying low-rank structure is distorted by an unknown monotonic nonlinear transformation, proposing a method that alternates between low-rank matrix and monotonic function estimation to estimate missing entries, with empirical results showing competitiveness on synthetic and real-world datasets.
Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the challenge of matrix completion in the face of such nonlinearities. Given a few observations of a matrix that are obtained by applying a Lipschitz, monotonic function to a low rank matrix, our task is to estimate the remaining unobserved entries. We propose a novel matrix completion method that alternates between low-rank matrix estimation and monotonic function estimation to estimate the missing matrix elements. Mean squared error bounds provide insight into how well the matrix can be estimated based on the size, rank of the matrix and properties of the nonlinear transformation. Empirical results on synthetic and real-world datasets demonstrate the competitiveness of the proposed approach.