MA 617 Tensor Methods for Data Analysis
The goal of course is to introduce tensor numerical methods designed for the solution of the multidimensional problems in scientific computing and data analysis. These methods are based on the rank-structured approximation of multivariate functions and operators by using appropriate tensor decompositions (formats). The old and new rank-structured tensor formats are presented, as: canonical, Tucker, hierarchical, tensor train, quantized tensor train formats and their generalizations, which leads to tensor networks, i.e. representation of high dimensional tensors as interconnected low dimensional tensors in variety of ways. Under suitable conditions these formats allow a stable representation and a reduction of the data size from exponential complexity (with respect to the dimension of the space) to a linear complexity, which kills the curse of dimensionality. Another goal of the course is to present variety of relatively novel unsupervised machine learning methods using matrix and tensor decompositions, as latent variable analysis based on (1) statistical independence and (2) sparsity assumptions, kernel and generalized principal component analysis, and non-negative tensor decomposition, all of which reveal hidden patterns in the data.