Convolutional Imputation of Matrix Networks
This addresses matrix completion in networked data for applications like medical imaging and social networks, representing an incremental advance with theoretical guarantees.
The paper tackles the problem of completing partially observed matrix networks where some matrices might be completely unobserved, proposing a convex optimization method with exact recovery guarantees and discovering a phase transition phenomenon. It demonstrates applications in MRI and Facebook networks with numerical characterization of recovery regimes.
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.