Two algorithms for compressed sensing of sparse tensors

Compressed sensing (CS) exploits the sparsity of a signal in order to integrate acquisition and compression. CS theory enables exact reconstruction of a sparse signal from relatively few linear measurements via a suitable nonlinear minimization process. Conventional CS theory relies on vectorial data representation, which results in good compression ratios at the expense of increased computational complexity. In applications involving color images, video sequences, and multisensor networks, the data is intrinsically of high-order, and thus more suitably represented in tensorial form. Standard applications of CS to higher-order data typically involve representation of the data as long vectors that are in turn measured using large sampling matrices, thus imposing a huge computational and memory burden. In this chapter, we introduce Generalized Tensor Compressed Sensing (GTCS)–a unified framework for compressed sensing of higher-order tensors which preserves the intrinsic structure of tensorial data with reduced computational complexity at reconstruction. We demonstrate that 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 (GTCS-S) and a parallelizable method (GTCS-P), both capable Shmuel Friedland Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA. This work was supported by NSF grant DMS-1216393. email: [email protected] Qun Li PARC, Xerox Corporation, 800 Phillips Road, Webster, New York 14580, USA. e-mail: [email protected] Dan Schonfeld Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA. e-mail: [email protected] Edgar A. Bernal PARC, Xerox Corporation, 800 Phillips Road, Webster, New York 14580, USA. e-mail: [email protected]