Compressive sensing of sparse tensors.

Shmuel Friedland,Qun Li,Dan Schonfeld

doi:10.1109/tip.2014.2348796

Shmuel Friedland, Qun Li + Show 1 more

Open Access

https://doi.org/10.1109/tip.2014.2348796

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Abstract

Compressive sensing (CS) has triggered an enormous research activity since its first appearance. CS exploits the signal's sparsity or compressibility in a particular domain and integrates data compression and acquisition, thus allowing exact reconstruction through relatively few nonadaptive linear measurements. While conventional CS theory relies on data representation in the form of vectors, many data types in various applications, such as color imaging, video sequences, and multisensor networks, are intrinsically represented by higher order tensors. Application of CS to higher order data representation is typically performed by conversion of the data to very long vectors that must be measured using very large sampling matrices, thus imposing a huge computational and memory burden. In this paper, we propose generalized tensor compressive sensing (GTCS)-a unified framework for CS of higher order tensors, which preserves the intrinsic structure of tensor data with reduced computational complexity at reconstruction. GTCS offers an efficient means for representation of multidimensional data by providing simultaneous acquisition and compression from all tensor modes. In addition, we propound two reconstruction procedures, a serial method and a parallelizable method. We then compare the performance of the proposed method with Kronecker compressive sensing (KCS) and multiway compressive sensing (MWCS). We demonstrate experimentally that GTCS outperforms KCS and MWCS in terms of both reconstruction accuracy (within a range of compression ratios) and processing speed. The major disadvantage of our methods (and of MWCS as well) is that the compression ratios may be worse than that offered by KCS.

Highlights

Recent literature has witnessed an explosion of interest in sensing that exploits structured prior knowledge in the general form of sparsity, meaning that signals can be represented by only a few coefficients in some domain
We propose Generalized Tensor Compressive Sensing (GTCS)– a unified framework for compressive sensing of higher-order tensors which preserves the intrinsic structure of tensor data with reduced computational complexity at reconstruction
We compare the performance of the proposed method with Kronecker compressive sensing (KCS) and multi-way compressive sensing (MWCS)

Summary

Introduction

Recent literature has witnessed an explosion of interest in sensing that exploits structured prior knowledge in the general form of sparsity, meaning that signals can be represented by only a few coefficients in some domain. CS theory permits relatively few linear measurements of the signal while still allowing exact reconstruction via nonlinear recovery process. The first intuitive approach to a reconstruction algorithm consists in searching for the sparsest vector that is consistent with the linear measurements. This l0minimization problem is NP-hard in general and computationally infeasible. . .) and calligraphic capitals represent tensors The order of a tensor is the number of modes. Tensor X ∈ RN1×...×Nd has order d and the dimension of its ith mode ( called mode i directly) is Ni

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