Randomized algorithms play a central role in low rank approximations of large matrices. In this paper, the scheme of the randomized SVD is extended to a randomized LU algorithm. Several error bounds are introduced, that are based on recent results from random matrix theory related to subgassian matrices. The bounds also improve the existing bounds of already known randomized algorithm for low rank approximation. The algorithm is fully parallelized and thus can utilize efficiently GPUs without any CPU-GPU data transfer. Numerical examples, which illustrate the performance of the algorithm and compare it to other decomposition methods, are presented.Lemma 3.7 ([46]). For any m × n matrix A, let R be the n × l SRFT matrix. Then, Y = AR can be computed in O(mn log l) floating point operations. Interpolative decomposition (ID)Let A be an m × n of rank r. A ≈ A (:,J) X is the ID of rank r of A if:1. J is a subset of r indices from 1, . . . , n. 2. The r × n matrix A (:,J) is a subset of J columns from A.

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Abstract-Particle filter is a powerful method that tracks the state of a target based on non-linear observations. We present a multiscale based method that accelerates the computation of particle filters. Unlike the conventional way, which calculates weights over all particles in each cycle of the algorithm, we sample a small subset from the source particles using matrix decomposition methods. Then, we apply a function extension algorithm that uses a particle subset to recover the density function for all the rest of the particles not included in the chosen subset. The computational effort is substantial especially when multiple objects are tracked concurrently. The proposed algorithm reduces significantly the computational load. By using the Fast Gaussian Transform, the complexity of the particle selection step is reduced to a linear time in n and k , where n is the number of particles and k is the number of particles in the selected subset. We demonstrate our method on both simulated and on real data such as objects tracking in videos sequences.

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