
Krylov Iterative Methods for the Geometric Mean of Two Matrices Times a Vector
In this work, we are presenting an efficient way to compute the geometri...
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Optimal Approximation of Doubly Stochastic Matrices
We consider the leastsquares approximation of a matrix C in the set of ...
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Complexity Analysis of a Fast Directional MatrixVector Multiplication
We consider a fast, datasparse directional method to realize matrixvec...
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Fast and stable deterministic approximation of general symmetric kernel matrices in high dimensions
Kernel methods are used frequently in various applications of machine le...
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The Fast and Free Memory Method for the efficient computation of convolution kernels
We introduce the Fast Free Memory method (FFM), a new fast method for th...
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Simple, fast and accurate evaluation of the action of the exponential of a rate matrix on a probability vector
Given a timehomogeneous, finitestatespace Markov chain with a rate mat...
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Positivity and Transportation
We prove in this paper that the weighted volume of the set of integral t...
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Efficient construction of an HSS preconditioner for symmetric positive definite ℋ^2 matrices
In an iterative approach for solving linear systems with illconditioned, symmetric positive definite (SPD) kernel matrices, both fast matrixvector products and fast preconditioning operations are required. Fast (linearscaling) matrixvector products are available by expressing the kernel matrix in an ℋ^2 representation or an equivalent fast multipole method representation. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the ℋ^2 representation, such as the hierarchically semiseparable (HSS) matrix representation, which provides fast solve operations. Previously, an algorithm was presented to construct an HSS approximation to an SPD kernel matrix that is guaranteed to be SPD. However, this algorithm has quadratic cost and was only designed for recursive binary partitionings of the points defining the kernel matrix. This paper presents a general algorithm for constructing an SPD HSS approximation. Importantly, the algorithm uses the ℋ^2 representation of the SPD matrix to reduce its computational complexity from quadratic to quasilinear. Numerical experiments illustrate how this SPD HSS approximation performs as a preconditioner for solving linear systems arising from a range of kernel functions.
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