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We prove that the inverse of a positive-definite matrix can be approximatedby a weighted-sum of a small number of matrix exponentials. Combining this witha previous result [OSV12], we establish an equivalence between matrix inversionand exponentiation up to polylogarithmic factors. In particular, thisconnection justifies the use of Laplacian solvers for designing fastsemi-definite programming based algorithms for certain graph problems. Theproof relies on the Euler-Maclaurin formula and certain bounds derived from theRiemann zeta function.

Matrix Inversion Is As Easy As Exponentiation