Tensor Methods and Emerging Applications to the Physical and Data Sciences

Linear algebra is an essential tool in mathematics, science, and engineering, as almost all natural processes are linear in small increments. The most natural generalization of linear algebra is multilinear algebra where matrices are replaced by tensors.

In recent years researchers have actively been working on tensor related problems in multiple fields, ranging from many-body quantum problems to analysis of large data sets in high dimension. Tensor representation, analysis and algorithms have found tremendous applications in almost every discipline of science and engineering including applied mathematics, statistics, physics, chemistry, machine learning, engineering, and others. On the physical sciences side, tensor network formats have been widely used to represent ground and thermal states for many-body quantum systems. Tensor-based numerical methods, such as the density matrix renormalization group (DMRG) method, have become the method of choice for one-dimensional physical systems and are beginning to overtake previous methods of choice such as the coupled-cluster method in quantum chemistry. On the data science side, tensor decompositions have been used successfully for learning latent variable models, training neural networks, reinforcement learning, and others. In mathematics, tensor decompositions have been connected to algebraic geometry, and have been shown to have a direct relationship with some of the long-standing problems in computational complexity: P vs NP and matrix multiplication.

While exciting results have emerged from various research communities, there has not been much exchange and collaboration between theoreticians and developers of practical algorithms. The aim of this long term program is to bring together experts and junior participants from different fields and experiences, to exchange ideas, tackle challenges, collaborate, and advances the general field of tensor methods. 

Professor Joseph Landsberg (TAMU) has been appointed as a Clay Senior Scholar to particpate in this program.