Tensor optimal transport, distance between sets of measures and tensor scaling

We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization term, which gives rise to a scaling of tensors. We give a variation of the celebrated Sinkhorn scaling algorithm. We show that this algorithm can be viewed as a partial minimization algorithm of a strictly convex function. Under appropriate conditions the rate of convergence is geometric and we estimate the rate. Our results are generalizations of known results for the classical case of two discrete measures.

Compressive Sensing of Sparse Tensors

Compressive sensing (CS) has triggered 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 non-adaptive 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 multi-sensor 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 compressive sensing 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 (GTCS-S) and a parallelizable method (GTCS-P). We then compare the performance of the proposed method with Kronecker compressive sensing (KCS) and multi way 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.