Tensor Completion via Integer Optimization

Xin Jennifer Chen, Sukanya Kudva, Yongzheng Dai, Anil Aswani, Chen Chen

Abstract:

The tensor completion problem is to fill-in unobserved entries of a partially observed tensor. However, past approaches to tensor completion either achieved the informationtheoretic rate but lacked practical algorithms, or proposed
polynomial-time algorithms that require an exponentially-larger
number of samples for low estimation error. In this paper,
we develop a novel tensor completion algorithm to tackle
this challenge by achieving both provable convergence (in
numerical tolerance) in a linear number of oracle steps and
the information-theoretic rate. We formulate tensor completion
as a convex optimization problem constrained using a gaugebased tensor norm and provide proofs of properties of this
norm including its computational complexity and tensor rank
surrogacy. We formulate this norm such that linear separation
problems over the gauge unit-ball can be solved using integer
optimization. This enables the use of Frank-Wolfe variant
to build our algorithm. We demonstrate effectiveness of our
method using experiments with simulated data and with an
application towards providing low-computation predictions of
battery storage flows that may be beneficial in billion-devicescale integration of electromobility systems with the grid.