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. It describes natural phenomena where the variation is linear if we keep all but one factor constant. Furthermore, tensors and multilinear algebra emerge from discretization of multivariate functions – one can simply view the grid values as coefficients of a multivariate tensor.
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. We foresee this program to be a milestone platform for the future development of the research area and to have a long standing impact.
Victor Batista (Yale University)
Fernando Brandao (California Institute of Technology)
Gero Friesecke (Technische Universtitat München)
Lek-Heng Lim (University of Chicago)
Jianfeng Lu (Duke University)
Elina Robeva (University of British Columbia)
Ming Yuan (Columbia University)