Speaker
Description
Kernel matrices arising in applications such as Gaussian processes may not always admit a low-rank approximation. Important examples are kernel matrices induced by certain members of the Matérn family of covariance kernels, with smaller length scales and values of $\nu$. Still, they can often be approximated by a hierarchical matrix ($\mathcal{H}$-matrix or $\mathcal{H}^{2}$-matrix), which consists of a hierarchy of small near-field blocks (sub-matrices) stored in a dense format and large low-rank far-field blocks that are efficiently stored in factored form. A hierarchical matrix approximation of a kernel matrix can be constructed, stored, and used to perform matrix-vector multiplication in log-linear or linear complexity with respect to $n$. Standard methods for approximating kernel matrices with hierarchical matrices do not account for the following: kernel matrices often depend on certain hyperparameters that must be optimized over a fixed parameter space. For example, in Gaussian processes and Bayesian inverse problems, estimating the hyperparameters from the data involves solving an optimization problem, which requires repeatedly forming or approximating the kernel matrices for a range of parameters. To address this computational challenge, we introduce a new class of hierarchical matrices, namely, parametric (parameter-dependent) hierarchical matrices. The construction of a parametric hierarchical matrix follows an offline-online paradigm. In the offline stage, the near-field and far-field blocks are approximated by using polynomial approximation and tensor compression. In the fast online stage, for a particular hyperparameter, the parametric hierarchical matrix is instantiated efficiently as a standard hierarchical matrix. Numerical experiments show speedups of over 100 times compared with existing techniques.