1-Bit Tensor Completion

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. Introduction Massive multiway data emerge in many fields. Multidimensional arrays, known as Tensors, are higher-order generalizations of matrices and vectors, and provide a natural way to represent data objects whose entries are indexed by several variables. For instance, a hyperspectral image can be modeled as a third-order tensor defined by two indices for spatial variables and one index for the spectral dimension while a color video corresponds a fourth-order tensor with an additional index for the temporal dimension. Employing therefore tensors to process high dimensional observations has become increasingly popular [1, 2]. One issue which arises frequently is that a tensor may have a significant number of entries missing. Fortunately, in highdimensional spaces, the corresponding tensors can be well approximated by lower dimensional structures. In such case, we would like to fill in those missing entries based on the available observations. The problem of recovering a low-rank tensor from an incomplete set of its entries, is known as low-rank tensor completion [18]. In many cases, not only are many observations missing, but the available also quantized to a predefined number of bits. Quantization is an integral part of data acquisition, especially for remote sensing scenario like communiting measurements from airborne and space-borne platform where channel bandwidth is very limited, as well as application involving energy-limited systems such as Wireless Sensor Networks and Internet of Things platforms, where full data transmission is directly related to an increase in power consumption and a subsequent reduction in the network lifetime. As a result, quantization of a signal is a critical step for high dimensional data transmission but also for data storage. The question we address in this work is whether it is possible to recover the real-valued entries of tensors from a small number of highly quantized (binary) observations. This problem significantly differs from currently available tensor completion algorithms which are challenged when the observations are highly quantized, since they treat them as real values instead of discrete. More specifically, we introduce a novel approach for the recovery of a low-rank tensor from a small number of binary measurements, where given a 3-order tensor where only a small number of extremely quantized entries are available, we unfold it into 3 matrices and we apply the 1-bit matrix completion algorithm to the all-mode matricizations of the tensor as illustrated in Figure 1. To the best of our knowledge, this is the first work that examines the interaction between quantization and sampling in high-order structured data. Furthermore, by considering binary encoding, the proposed scheme is directly applicable to tensor data compression, a new and underexplored research topic. In short, the key novelties of this work are • Explore the recovery of missing and real-valued entries of highly quantized high-order tensors. • Propose a formally approach for the recovery of a tensor from a small number of binary observations. • Investigate the performance of the proposed method on publicly available hyperspectral images. Related Work The problem of low-rank tensor completion can be regarded as an extension of low rank Matrix Completion. According to MC, the recovery is possible provided the matrix is characterized by a small rank (compared to its dimensions) and enough randomly selected entries of the matrix are acquired [3]. Unfortunately, rank minimization is an NP-hard problem therefore cannot be used for reasonably size data. Since the rank of a matrix corresponds to the number of nonzero singular values, the rank minimization can be replaced by the minimization of the nuclear norm, which is the sum of the singular values of the matrix [4]. The nuclear norm is a convex function which makes the above problem easy to solve, while under mild conditions, the nuclear norm minimization can estimate the same matrix as the rank minimization with high probability [5]. To solve the nuclear norm minimization problem, various approaches have been proposed including the Singular Value Thresholding [7] and the Augmented Lagrange Multiplier Method [8]. There are many methods which extent the problem of LRTC to low-rank matrix completion. Specifically, in [17, 18] is employed matrix nuclear-norm minimization and is used the singular value decomposition (SVD) in the algorithms. A non-convex approach of this problem presented in [20], where the authors apply low-rank matrix factorization [9] to each mode unfolding of IS&T International Symposium on Electronic Imaging 2018 Image Processing: Algorithms and Systems XVI 261-1 https://doi.org/10.2352/ISSN.2470-1173.2018. 13.IPAS-261 © 2018, Society for Imaging Science and Technology