Universal Consistency of Support Tensor Machine Tensor (multidimensional array) classification problem has become popular in modern applications such as computer vision and spatialtemporal data analysis. The Support Tensor Machine (STM) classifier, which is extended from support vector machine, takes tensor type data as predictors to predict the labels of the data. The distributionfree property of STM highlights its potential in handling different types of data applications. In this work, we provide a theoretical result for the universal consistency of STM. This result guarantees the solid generalization ability of STM with universal tensor based kernel functions. In addition, we give out a way of constructing universal kernel functions for tensor data, which may be helpful for other types of tensor based kernel methods.

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.