Algebraic Variety Models for High-Rank Matrix Completion

TL;DR

This paper extends low-rank matrix completion to algebraic variety models, enabling recovery of high-rank data by mapping to higher-dimensional feature spaces, with an efficient kernel-based algorithm demonstrating theoretical and empirical success.

Contribution

It introduces a novel algebraic variety framework for matrix completion, especially for union of affine subspaces, and proposes a kernel trick-based algorithm that outperforms existing methods.

Findings

Successfully recovers synthetic data within theoretical sampling bounds.

Outperforms standard low-rank and subspace clustering methods on real data.

Provides sampling complexity analysis for variety-based matrix completion.

Abstract

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In addition, varieties can be used to model a richer class of nonlinear quadratic and higher degree curves and surfaces. We study the sampling requirements for matrix completion under a variety model with a focus on a union of affine subspaces. We also propose an efficient matrix completion algorithm that minimizes a convex or non-convex surrogate of the rank of the matrix of monomial features. Our algorithm uses the well-known "kernel trick" to avoid working directly with the high-dimensional monomial matrix. We show the proposed algorithm is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low rank matrix completion and subspace clustering techniques in experiments with real data.