Johannes J. Brust

Johannes J. Brust, Tamara G. Kolda

Tensor decompositions are a fundamental tool 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. 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 to fit reproducing Kernel Hilbert space (RKHS) representations to the data, 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, exploiting the problem's structure for fast matrix-vector products and efficient preconditioning. In our experiments, the proposed methods speed up the solution up to 500x compared to the prior naive direct methods, in line with the reduction in the theoretical computational complexity.

Johannes J. Brust, Michael A. Saunders

For a datastream, the change over a short interval is often of low rank. For high throughput information arranged in matrix format, recomputing an optimal SVD approximation after each step is typically prohibitive. Instead, incremental and truncated updating strategies are used, which may not scale for large truncation ranks. Therefore, we propose a set of efficient new algorithms that update a bidiagonal factorization, and which are similarly accurate as the SVD methods. In particular, we develop a compact Householder-type algorithm that decouples a sparse part from a low-rank update and has about half the memory requirements of standard bidiagonalization methods. A second algorithm based on Givens rotations has only about 10 flops per rotation and scales quadratically with the problem size, compared to a typical cubic scaling. The algorithm is therefore effective for processing high-throughput updates, as we demonstrate in tracking large subspaces of recommendation systems and networks, and when compared to well known software such as LAPACK or the incremental SVD.

Johannes J. Brust

For minimization problems without 2nd derivative information, methods that estimate Hessian matrices can be very effective. However, conventional techniques generate dense matrices that are prohibitive for large problems. Limited-memory compact representations express the dense arrays in terms of a low rank representation and have become the state-of-the-art for software implementations on large deterministic problems. We develop new compact representations that are parameterized by a choice of vectors and that reduce to existing well known formulas for special choices. We demonstrate effectiveness of the compact representations for large eigenvalue computations, tensor factorizations and nonlinear regressions.

Johannes J. Brust

For large nonlinear least squares loss functions in machine learning we exploit the property that the number of model parameters typically exceeds the data in one batch. This implies a low-rank structure in the Hessian of the loss, which enables effective means to compute search directions. Using this property, we develop two algorithms that estimate Jacobian matrices and perform well when compared to state-of-the-art methods.