Learning tensors from partial binary measurements

TL;DR

This paper extends 1-bit matrix completion to higher-order tensors, demonstrating efficient recovery from binary measurements with theoretical guarantees and practical advantages over matricization, especially for recommender systems.

Contribution

It introduces a method for low-rank tensor recovery from binary measurements using max-qnorm and M-norm regularization, with theoretical sample complexity results.

Findings

Low-rank tensors can be recovered from O(Nd) binary measurements.

Sample complexity matches unquantized measurement recovery for tensors.

1-bit tensor completion outperforms matricization in theory and practice.

Abstract

In this paper we generalize the 1-bit matrix completion problem to higher order tensors. We prove that when $r=O(1)$ a bounded rank-$r$, order-$d$ tensor $T$ in $\mathbb{R}^{N} \times \mathbb{R}^{N} \times \cdots \times \mathbb{R}^{N}$ can be estimated efficiently by only $m=O(Nd)$ binary measurements by regularizing its max-qnorm and M-norm as surrogates for its rank. We prove that similar to the matrix case, i.e., when $d=2$, the sample complexity of recovering a low-rank tensor from 1-bit measurements of a subset of its entries is the same as recovering it from unquantized measurements. Moreover, we show the advantage of using 1-bit tensor completion over matricization both theoretically and numerically. Specifically, we show how the 1-bit measurement model can be used for context-aware recommender systems.