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In this paper we show how to recover a spectral approximations to broadclasses of structured matrices using only a polylogarithmic number of adaptivelinear measurements to either the matrix or its inverse. Leveraging this resultwe obtain faster algorithms for variety of linear algebraic problems. Keyresults 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 $n \times n$ linearsystems that are constant spectral approximations of Laplacians or moregenerally, SDD matrices.  $\bullet$ An $\tilde{O}(n^2)$ algorithm to recover a spectral approximationof a $n$-vertex graph using only $\tilde{O}(1)$ matrix-vector multiplies withits Laplacian matrix.  The previous best results for each problem either used a trivial number ofqueries 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 programmingbased linear sized sparsifier results of Lee and Sun (2017) and providingiterative methods inspired by the semistreaming sparsification results ofKapralov, Lee, Musco, Musco and Sidford (2014) and input sparsity time linearsystem solving results of Li, Miller, and Peng (2013). We hope that byinitiating study of these natural problems, expanding the robustness and scopeof recent nearly linear time linear system solving research, and providinggeneral matrix recovery machinery this work may serve as a stepping stone forfaster algorithms.

Efficient Structured Matrix Recovery and Nearly-Linear Time Algorithms for Solving Inverse Symmetric $M$-Matrices