Speaker
Description
We describe a problem arising from operator learning for hyperbolic PDEs, in which one would like to recover an unknown, non-standard low-rank hierarchical partition of a linear operator using only input-output data, or, in the finite-dimensional case, matrix-vector products. We provide a solution to the operator learning problem by employing a continuous analogue of the randomized SVD (RSVD) to decide whether the operator, restricted to a given subdomain, is numerically low-rank or not. Doing so requires the RSVD to obtain good singular subspace estimates, which in theory depends on the sizes of gaps between singular values of the operator. Thus, in the second part of this talk, we derive exact descriptions for the angular error of the approximate singular subspaces returned by the RSVD, which helps explain why large singular value gaps are typically not required in practice.