1-Bit Tensor Completion

Abstract

Higher-order tensor structured data arise in many imaging scenarios, including hyperspectral imaging and color video. The recovery of a tensor from an incomplete set of its entries, known as tensor completion, is crucial in applications like compression. Furthermore, in many cases observations are not only incomplete, but also highly quantized. Quantization is a critical step for high dimensional data transmission and storage in order to reduce storage requirements and power consumption, especially for energy-limited systems. In this paper, we propose a novel approach for the recovery of low-rank tensors from a small number of binary (1-bit) measurements. The proposed method, called 1bit Tensor Completion, relies on the application of 1-bit matrix completion over different matricizations of the underlying tensor. Experimental results on hyperspectral images demonstrate that directly operating with the binary measurements, rather than treating them as real values, results in lower recovery error.

Tensor Recovery from Noisy and Multi-Level Quantized Measurements

“…n k ) log(4K/3) + log(2/δ)). (41) Proof The proof is completed by combining Lemma 1 and Theorem 1 in [59].…”

“…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.…”

“…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.…”

“…This paper for the first time studies the problem of low-rank tensor recovery from multi-level quantized measurements, while the existing works [41,43,42] only consider one-bit measurements. This paper is also the first one to study the recovery problem for SVDtensors for quantized measurements.…”

Tensor Recovery from Noisy and Multi-Level Quantized Measurements

“…Existing works on low-rank tensor recovery mainly consider random noise or sparse noise [9], [28], [38], while only a few works [1], [17], [22] consider tensor recovery from one-bit measurements. [1] unfolds the tensors to matrices and applies matrix recovery techniques. Ref.…”