Tech Reports in Computer Science and Engineering

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Author: search.filters.author.Raykar, Vikas Chandrakant

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    Very fast optimal bandwidth selection for univariate kernel density estimation
    (2006-01-03T15:15:40Z) Raykar, Vikas Chandrakant; Duraiswami, Ramani
    Most automatic bandwidth selection procedures for kernel density estimates require estimation of quantities involving the density derivatives. Estimation of modes and inflexion points of densities also require derivative estimates. The computational complexity of evaluating the density derivative at M evaluation points given N sample points from the density is O(MN). In this paper we propose a computationally efficient $\epsilon$-exact approximation algorithm for univariate, Gaussian kernel based, density derivative estimation that reduces the computational complexity from O(MN) to linear order (O(N+M)). The constant depends on the desired arbitrary accuracy, $\epsilon$. We apply the density derivative evaluation procedure to estimate the optimal bandwidth for kernel density estimation, a process that is often intractable for large data sets. For example for N = M = 409,600 points while the direct evaluation of the density derivative takes around 12.76 hours the fast evaluation requires only 65 seconds with an error of around $10^{-12)$. Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented. We demonstrate the speedup achieved on the bandwidth selection using the ``solve-the-equation plug-in method'' [18]. We also demonstrate that the proposed procedure can be extremely useful for speeding up exploratory projection pursuit techniques.
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    Fast Computation of Sums of Gaussians in High Dimensions
    (2005-11-15T19:27:37Z) Raykar, Vikas Chandrakant; Yang, Changjaing; Duraiswami, Ramani; Gumerov, Nail
    Evaluating sums of multivariate Gaussian kernels is a key computational task in many problems in computational statistics and machine learning. The computational cost of the direct evaluation of such sums scales as the product of the number of kernel functions and the evaluation points. The fast Gauss transform proposed by Greengard and Strain (1991) is a $\epsilon$-exact approximation algorithm that reduces the computational complexity of the evaluation of the sum of $N$ Gaussians at $M$ points in $d$ dimensions from $\mathcal{O}(MN)$ to $\mathcal{O}(M+N)$. However, the constant factor in $\mathcal{O}(M+N)$ grows exponentially with increasing dimensionality $d$, which makes the algorithm impractical for dimensions greater than three. In this paper we present a new algorithm where the constant factor is reduced to asymptotically polynomial order. The reduction is based on a new multivariate Taylor's series expansion (which can act both as a local as well as a far field expansion) scheme combined with the efficient space subdivision using the $k$-center algorithm. The proposed method differs from the original fast Gauss transform in terms of a different factorization, efficient space subdivision, and the use of point-wise error bounds. Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented. As an example we shows how the proposed method can be used for very fast $\epsilon$-exact multivariate kernel density estimation.