[Paper Review] Exact tensor completion with sum-of-squares

Aaron Potechin, David Steurer|arXiv (Cornell University)|Feb 21, 2017

Tensor decomposition and applications1 references22 citations

TL;DR

This paper presents the first polynomial-time algorithm for exact tensor completion that improves upon the prior bound of $ r \cdot \tilde{O}(n^2) $ by reducing to matrix completion. It achieves exact recovery of a 3-tensor with $ r $ incoherent, orthogonal components from only $ r \cdot \tilde{O}(n^{1.5}) $ randomly observed entries using sum-of-squares certification of degree-4 polynomials over the sphere.

ABSTRACT

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r\cdot ilde O(n^{1.5})$ randomly observed entries of the tensor. This bound improves over the previous best one of $r\cdot ilde O(n^{2})$ by reduction to exact matrix completion. Our bound also matches the best known results for the easier problem of approximate tensor completion (Barak & Moitra, 2015). Our algorithm and analysis extends seminal results for exact matrix completion (Candes & Recht, 2009) to the tensor setting via the sum-of-squares method. The main technical challenge is to show that a small number of randomly chosen monomials are enough to construct a degree-3 polynomial with precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.

Motivation & Objective

• To develop a polynomial-time algorithm for exact tensor completion that surpasses the sample complexity bound implied by reduction to matrix completion.
• To extend the sum-of-squares method—previously successful in matrix completion—to the tensor setting for exact recovery.
• To construct and certify a degree-3 polynomial with precisely planted orthogonal global optima over the sphere using low-degree sum-of-squares proofs.
• To achieve a sample complexity matching the best-known results for approximate tensor completion, despite the greater difficulty of exact recovery.
• To establish that $ r \cdot \tilde{O}(n^{1.5}) $ randomly sampled entries are sufficient for exact recovery of 3-tensors with incoherent, orthogonal components.

Proposed method

• Leverages the sum-of-squares (SOS) proof system to construct a certificate that verifies exact recovery of a 3-tensor from a small number of observed entries.
• Constructs a degree-4 polynomial over the sphere whose global optima correspond exactly to the incoherent, orthogonal components of the tensor.
• Uses a novel decomposition of tensor components into products of vectors $ u_i, v_i, w_i $, where summing over shared indices forces global consistency and yields a factor of $ r $.
• Applies the trace moment method to bound the spectral norm of matrix representations derived from tensor entries, using intersection patterns and random partitioning.
• Employs a recursive trace power calculation for matrix blocks $ Y_j $, decomposed by hypergraph structure, to control the growth of moments.
• Introduces a key trick: summing over outer triangle indices forces global index agreement across components, reducing the number of free variables and enabling tighter norm bounds.

Experimental results

Research questions

• RQ1 Can exact tensor completion be achieved with sample complexity below $ r \cdot \tilde{O}(n^2) $, the bound implied by reduction to matrix completion?
• RQ2 Is it possible to use the sum-of-squares method to certify exact recovery of a 3-tensor with incoherent, orthogonal components?
• RQ3 What is the minimal number of randomly sampled entries required to guarantee exact recovery of a 3-tensor using polynomial-time algorithms?
• RQ4 Can degree-4 sum-of-squares certificates be constructed and verified efficiently to ensure exact recovery in the tensor setting?
• RQ5 Does the proposed method achieve sample complexity matching the best-known bounds for approximate tensor completion?

Key findings

• The algorithm achieves exact recovery of a 3-tensor with $ r $ incoherent, orthogonal components from $ r \cdot \tilde{O}(n^{1.5}) $ randomly observed entries.
• This sample complexity improves over the previous best bound of $ r \cdot \tilde{O}(n^2) $ obtained by reduction to matrix completion.
• The bound matches the best-known results for approximate tensor completion, demonstrating that exact recovery is possible with comparable sample efficiency.
• The sum-of-squares method successfully certifies exact recovery by constructing a degree-4 polynomial with precisely planted global optima over the sphere.
• The analysis shows that the expected trace power of matrix representations decays sufficiently fast under the new sampling regime, enabling tight spectral norm bounds.
• The method establishes that the number of required samples is optimal up to logarithmic factors among polynomial-time algorithms, under the given assumptions.