Tutte Colloquium -Tammy Kolda -Fast and Accurate Tensor Decompositions on Infinite-Dimensional Function Spaces

Abstract: Tensor decompositions are fundamental tools in scientific computing and data analysis. In many applications — such as simulation data on irregular grids, surrogate modeling for parameterized PDEs, or spectroscopic measurements — the data has both discrete and continuous structure, and may only be observed at scattered sample points. The CP-HIFI (hybrid infinite-finite) decomposition generalizes the Canonical Polyadic (CP) tensor decomposition to settings where some factors are finite-dimensional vectors and others are functions drawn from infinite-dimensional spaces — a natural framework when the underlying data has continuous structure. The decomposition can be applied to a fully observed tensor (aligned) or, when only scattered observations are available, to a sparsely sampled tensor (unaligned). Current methods compute CP-HIFI factors by solving a sequence of dense linear systems arising from regularized least-squares problems, but these direct solves become computationally prohibitive as problem size grows. We propose new algorithms that achieve the same accuracy while being orders of magnitude faster. For aligned tensors, we exploit the Kronecker structure of the system to efficiently compute its eigendecomposition without ever forming the full system, reducing the solve to independent scalar equations. For unaligned tensors, we introduce a preconditioned conjugate gradient method applied to a reformulated system with favorable spectral properties. We analyze the computational complexity and memory requirements of the new methods and demonstrate their effectiveness on problems with smooth functional modes. I will also discuss the “First Proof” project, which aims to understand the capabilities of AI systems on problems that come up in math research, and the role that results from that experiment played in this project.