Structured matrix recovery and operator learning

Can one recover a structured matrix $A$ from only matrix-vector product queries $x \mapsto Ax$ and $y \mapsto A^\top y$? If so, how many are needed? We will discuss the matrix recovery problem for common matrix families, with an emphasis on hierarchical rank-structured matrices. This problem arises in the emerging field of operator learning, where one approximates the solution operator of a PDE from only input-output pairs of forcing terms and solutions. We will conclude with some open problems resulting from this connection between matrix recovery and operator learning.