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We provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions on n elements. We present two algorithms that compute couplings between marginal distributions with an expected transportation cost that is within an additive ϵ of optimal in time O(n^2/eps); one algorithm is straightforward to parallelize and implementable in depth O(1/eps). Further, we show that additional improvements on our results must be coupled with breakthroughs in algorithmic graph theory. 

We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. \begin{itemize} \item \textbf{Diagonal preconditioning.} We give an algorithm which, given positive definite $\mathbf{K} \in \mathbb{R}^{d \times d}$ with $\mathrm{nnz}(\mathbf{K})$ nonzero entries, computes an $\epsilon$-optimal diagonal preconditioner in time $\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1}))$, where $\kappa^\star$ is the optimal condition number of the rescaled matrix. \item \textbf{Structured linear systems.} We give an algorithm which, given $\mathbf{M} \in \mathbb{R}^{d \times d}$ that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in $\mathbf{M}$ in $\widetilde{O}(d^2)$ time. \end{itemize} Our diagonal preconditioning results improve state-of-the-art runtimes of $\Omega(d^{3.5})$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $\Omega(d^{\omega})$ where $\omega > 2.3$ is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call \emph{matrix-dictionary approximation SDPs}, which we leverage to solve an associated problem we call \emph{matrix-dictionary recovery}.