Convolutional Imputation of Matrix Networks

TL;DR

This paper introduces a novel method for completing matrix networks with incomplete data by leveraging low-rank graph Fourier transforms, providing theoretical guarantees and practical algorithms for applications like MRI and social networks.

Contribution

It proposes a new low-rank graph Fourier transform assumption, formulates a convex optimization for matrix network completion, and develops an efficient iterative imputation algorithm.

Findings

Exact recovery guarantees under certain conditions

Discovery of a phase transition in recovery success

Successful application to MRI and social network data

Abstract

A matrix network is a family of matrices, with relatedness modeled by a weighted graph. We consider the task of completing a partially observed matrix network. We assume a novel sampling scheme where a fraction of matrices might be completely unobserved. How can we recover the entire matrix network from incomplete observations? This mathematical problem arises in many applications including medical imaging and social networks. To recover the matrix network, we propose a structural assumption that the matrices have a graph Fourier transform which is low-rank. We formulate a convex optimization problem and prove an exact recovery guarantee for the optimization problem. Furthermore, we numerically characterize the exact recovery regime for varying rank and sampling rate and discover a new phase transition phenomenon. Then we give an iterative imputation algorithm to efficiently solve the optimization problem and complete large scale matrix networks. We demonstrate the algorithm with a variety of applications such as MRI and Facebook user network.