  In this paper we show how to recover a spectral approximations to broad
classes of structured matrices using only a polylogarithmic number of adaptive
linear measurements to either the matrix or its inverse. Leveraging this result
we obtain faster algorithms for variety of linear algebraic problems. Key
results include:
  $\bullet$ A nearly linear time algorithm for solving the inverse of symmetric
$M$-matrices, a strict superset of Laplacians and SDD matrices.
  $\bullet$ An $\tilde{O}(n^2)$ time algorithm for solving $n \times n$ linear
systems that are constant spectral approximations of Laplacians or more
generally, SDD matrices.
  $\bullet$ An $\tilde{O}(n^2)$ algorithm to recover a spectral approximation
of a $n$-vertex graph using only $\tilde{O}(1)$ matrix-vector multiplies with
its Laplacian matrix.
  The previous best results for each problem either used a trivial number of
queries to exactly recover the matrix or a trivial $O(n^\omega)$ running time,
where $\omega$ is the matrix multiplication constant.
  We achieve these results by generalizing recent semidefinite programming
based linear sized sparsifier results of Lee and Sun (2017) and providing
iterative methods inspired by the semistreaming sparsification results of
Kapralov, Lee, Musco, Musco and Sidford (2014) and input sparsity time linear
system solving results of Li, Miller, and Peng (2013). We hope that by
initiating study of these natural problems, expanding the robustness and scope
of recent nearly linear time linear system solving research, and providing
general matrix recovery machinery this work may serve as a stepping stone for
faster algorithms.
