Rasmus Kyng

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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(More)
Given a directed acyclic graph G, and a set of values y on the vertices, the Isotonic Regression of y is a vector x that respects the partial order described by G, and minimizes x − y , for a specified norm. This paper gives improved algorithms for computing the Isotonic Regression for all weighted p-norms with rigorous performance guarantees. Our(More)
We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large p of p-Laplacian regularization. We present an algorithm that(More)
We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultra-sparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC'10]. Combining this with the recursive preconditioning(More)
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(More)
Given a directed acyclic graph G, and a set of values y on the vertices, the Isotonic Regression of y is a vector x that respects the partial order described by G, and minimizes x − y , for a specified norm. This paper gives improved algorithms for computing the Isotonic Regression for all weighted ℓp-norms with rigorous performance guarantees. Our(More)
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