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Sampling points from the uniform distribution on a polytope is a well-studied problem, and is an important ingredient in several computational tasks involving polytopes, such as volume estimation. This is achieved by setting up a random walk inside the polytope, with its stationary distribution being uniform in the interior of the polytope. Kannan-Narayanan and Narayanan proposed the Dikin walk based on interior point methods, where the next point is sampled, roughly, from the Dikin ellipsoid at the current point. In this paper, we give a simple proof of the mixing time of the Dikin walk, using well-known properties of Gaussians, and concentration of Gaussian polynomials. Crack gestionale 1 zucchetti rubinetteria new york.

We show how to perform sparse approximate Gaussian elimination for Laplacian matrices. We present a simple, nearly linear time algorithm that approximates a Laplacian by a matrix with a sparse Cholesky factorization, the version of Gaussian elimination for symmetric matrices. This is the first nearly linear time solver for Laplacian systems that is based purely on random sampling, and does not use any graph theoretic constructions such as low-stretch trees, sparsifiers, or expanders. The crux of our analysis is a novel concentration bound for matrix martingales where the differences are sums of conditionally independent variables. We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations.

These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process. We use these new algorithms to derive the first nearly linear time algorithms for solving systems of equations in connection Laplacians—a generalization of Laplacian matrices that arise in many problems in image and signal processing. We also prove that every connection Laplacian has a linear sized approximate inverse. This is an LU factorization with a linear number of nonzero entries that is a strong approximation of the original matrix. Using such a factorization one can solve systems of equations in a connection Laplacian in linear time. Semejnij nudizm torrent. Such a factorization was unknown even for ordinary graph Laplacians.

We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [OSV12], we establish an equivalence between matrix inversion and exponentiation up to polylogarithmic factors. In particular, this connection justifies the use of Laplacian solvers for designing fast semi-definite programming based algorithms for certain graph problems. The proof relies on the Euler-Maclaurin formula and certain bounds derived from the Riemann zeta function. We present a new approach for graph based semi-supervised learning based on a multi- component extension to the Gaussian MRF model. This approach models the observa- tions on the vertices as a Gaussian ensemble with an inverse covariance matrix that is a weighted linear combination of multiple ma- trices. Building on randomized matrix trace estimation and fast Laplacian solvers, we de- velop fast and efficient algorithms for comput- ing the best-fit (maximum likelihood) model and the predicted labels using gradient de- scent.