Polynomial Matrix Completion for Missing Data Imputation and Transductive Learning

TL;DR

This paper introduces a novel polynomial matrix completion approach leveraging kernel methods to recover missing data in high-rank matrices with low intrinsic dimension, applicable to various real-world tasks.

Contribution

It proposes a new formulation using the kernel trick and a relaxation of the rank objective for efficient data imputation and transductive learning.

Findings

Outperforms state-of-the-art methods on synthetic and real datasets

Effectively completes data from multiple nonlinear manifolds

Improves accuracy in subspace clustering and motion capture recovery

Abstract

This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and wish to recover the missing entries under this assumption. We show that we can identify the complete matrix of minimum intrinsic dimension by minimizing the rank of the matrix in a high dimensional feature space. We develop a new formulation of the resulting problem using the kernel trick together with a new relaxation of the rank objective, and propose an efficient optimization method. We also show how to use our methods to complete data drawn from multiple nonlinear manifolds. Comparative studies on synthetic data, subspace clustering with missing data, motion capture data recovery, and transductive learning verify the superiority of our methods over the state-of-the-art.