Learning Tensors From Partial Binary Measurements

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 R N × R N × · · · × R N can be estimated efficiently by only m = O(N d) 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.

Tensor Recovery from Noisy and Multi-Level Quantized Measurements

“…, i K )-th entry of the quantized measurements Y ∈ [W ] n1×n2×···×n K . When W = 2, Y reduces to the one-bit case [17]. In general, Y is a log 2 W -bit tensor.…”

“…Most existing work on quantized data recovery consider the special case that the quantization boundaries are known [12], [15], [17] only except for [4]. In this case, (5) can be simplified tô…”

“…Ref. [17] recovers the tensor by solving a convex optimization problem and shows its recovery error is O(( r 3K−3 K n K−1 ) 1/4 ), where K is the number of tensor dimensions, and n is the size per dimension. [22] focuses on the case when a significant percentage of measurements are lost.…”

Tensor Recovery from Noisy and Multi-Level Quantized Measurements

“…Low-rank tensors with quantization noise exist in hyper-spectral data [41,42], rating systems [43], and the knowledge predicates [44]. Existing works on lowrank tensor recovery mainly consider random noise or sparse noise [45,46,47], while only a few works [41,43,42] consider tensor recovery from one-bit measurements, i.e., measurements are all in {0, 1}. Aidini et al [41] introduce a 1-bit tensor completion method that first unfolds the tensor measurements to matrices along all dimensions and then applies matrix recovery techniques to each matrix.…”

“…The final estimation is a real-valued tensor folded by the weighted sum of the recovered matrices. Ghadermarzy et al [43] use tensor M-norm constraint to replace the exact low-rank constraint and then recovers the tensor by solving the convex optimization problem. The recovery error is guaranteed to be O(( r 3K−3 K n K−1 ) 1/4 ), where K is the number of tensor dimensions, and n is the size per dimension.…”