Shape Constrained Tensor Decompositions We propose a new lowrank tensor factorization where one mode is coded as a sparse linear combination of elements from an overcomplete library. Our method, Shape Constrained Tensor Decomposition (SCTD) is based upon the CANDECOMPPARAFAC (CP) decomposition which produces rrank approximations of data tensors via outer products of vectors in each dimension of the data. The SCTD model can leverage prior knowledge about the shape of factors along a given mode, for example in tensor data where one mode represents time. By constraining the vector in the temporal dimension to known analytic forms which are selected from a large set of candidate functions, more readily interpretable decompositions are achieved and analytic time dependencies discovered. The SCTD method circumvents traditional flattening techniques where an Nway array is reshaped into a matrix in order to perform a singular value decomposition. A clear advantage of the SCTD algorithm is its ability to extract transient and intermittent phenomena which is often difficult for SVDbased methods. We motivate the SCTD method using several intuitively appealing results before applying it on a realworld data set to illustrate the efficiency of the algorithm in extracting interpretable spatiotemporal modes. With the rise of datadriven discovery methods, the decomposition proposed provides a viable technique for analyzing multitudes of data in a more comprehensible fashion.

Robust Schattenp Norm Based Approach for Tensor Completion The matrix nuclear norm has been widely applied to approximate the matrix rank for lowrank tensor completion because of its convexity. However, this relaxation may make the solution seriously deviate from the original solution for realworld data recovery. In this paper, using a nonconvex approximation of rank, i.e., the Schattenp norm, we propose a novel model for tensor completion. Itxe2x80x99s hard to solve this model directly because the objective function of the model is nonconvex. To solve the model, we develop a variant of this model via the classical quadric penalty method, and propose an algorithm, i.e., SpBCD, based on the block coordinate descent method. Although the objective function of the variant is nonconvex, we show that the sequence generated by SpBCD is convergent to a critical point. Our numerical experiments on realworld data show that SpBCD delivers stateofart performance in recovering missing data.