Tensor Completion via Integer Optimization

TL;DR

This work tackles tensor completion by balancing computation and information-theoretic sample efficiency. It introduces a gauge norm, defined via the convex hull of rank-1 tensors with ±1 entries, to form a convex relaxation that can certify low-rank structure while enabling scalable optimization. A Blended Conditional Gradients algorithm with a weak-separation oracle—implemented through a mix of alternating maximization and integer programming—solves the resulting convex program and achieves linear convergence, matching the information-theoretic estimation rate $\sqrt{ k\cdot\sum_i r_i / n }$. Empirical results show the method scales to tensors with up to $10^7$ entries and outperforms several established tensor completion algorithms on diverse test cases.

Abstract

The main challenge with the tensor completion problem is a fundamental tension between computation power and the information-theoretic sample complexity rate. Past approaches either achieve the information-theoretic rate but lack practical algorithms to compute the corresponding solution, or have polynomial-time algorithms that require an exponentially-larger number of samples for low estimation error. This paper develops a novel tensor completion algorithm that resolves this tension by achieving both provable convergence (in numerical tolerance) in a linear number of oracle steps and the information-theoretic rate. Our approach formulates tensor completion as a convex optimization problem constrained using a gauge-based tensor norm, which is defined in a way that allows the use of integer linear optimization to solve linear separation problems over the unit-ball in this new norm. Adaptations based on this insight are incorporated into a Frank-Wolfe variant to build our algorithm. We show our algorithm scales-well using numerical experiments on tensors with up to ten million entries.

Figures (4)

• Figure 1: NMSE and computation time (in s) for order-3 tensors with size $r \times r \times r$ and $n = 1000$ samples.
• Figure 2: NMSE and computation time (in s) for increasing order tensors with size $10^{\times p}$ and $n=10,000$ samples.
• Figure 3: NMSE and computation time (in s) for tensors with size $10^{\times 6}$ and increasing $n$ samples.
• Figure 4: NMSE and computation time (in s) for tensors with size $10^{\times 7}$ and increasing $n$ samples.

Theorems & Definitions (22)

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