Tree-Independent Dual-Tree Algorithms

Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split: the tree, the traversal, the point-to-point base case, and the pruning rule. We provide a meta-algorithm which allows development of dual-tree algorithms in a tree-independent manner and easy extension to entirely new types of trees. Representations are provided for five common algorithms; for k-nearest neighbor search, this leads to a novel, tighter pruning bound. The meta-algorithm also allows straightforward extensions to massively parallel settings.Comment: accepted in ICML 201

The Bregman Variational Dual-Tree Framework

Graph-based methods provide a powerful tool set for many non-parametric frameworks in Machine Learning. In general, the memory and computational complexity of these methods is quadratic in the number of examples in the data which makes them quickly infeasible for moderate to large scale datasets. A significant effort to find more efficient solutions to the problem has been made in the literature. One of the state-of-the-art methods that has been recently introduced is the Variational Dual-Tree (VDT) framework. Despite some of its unique features, VDT is currently restricted only to Euclidean spaces where the Euclidean distance quantifies the similarity. In this paper, we extend the VDT framework beyond the Euclidean distance to more general Bregman divergences that include the Euclidean distance as a special case. By exploiting the properties of the general Bregman divergence, we show how the new framework can maintain all the pivotal features of the VDT framework and yet significantly improve its performance in non-Euclidean domains. We apply the proposed framework to different text categorization problems and demonstrate its benefits over the original VDT.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013

Multibody Multipole Methods

A three-body potential function can account for interactions among triples of particles which are uncaptured by pairwise interaction functions such as Coulombic or Lennard-Jones potentials. Likewise, a multibody potential of order $n$ can account for interactions among $n$-tuples of particles uncaptured by interaction functions of lower orders. To date, the computation of multibody potential functions for a large number of particles has not been possible due to its $O(N^n)$ scaling cost. In this paper we describe a fast tree-code for efficiently approximating multibody potentials that can be factorized as products of functions of pairwise distances. For the first time, we show how to derive a Barnes-Hut type algorithm for handling interactions among more than two particles. Our algorithm uses two approximation schemes: 1) a deterministic series expansion-based method; 2) a Monte Carlo-based approximation based on the central limit theorem. Our approach guarantees a user-specified bound on the absolute or relative error in the computed potential with an asymptotic probability guarantee. We provide speedup results on a three-body dispersion potential, the Axilrod-Teller potential.Comment: To appear in Journal of Computational Physic

A Fast Adaptive Method for the Evaluation of Heat Potentials in One Dimension

We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a sum-of-exponential based fast Gauss transform that evaluates the convolution of a Gaussian with either discrete or continuous volume distributions. The latest implementation of the algorithm allows for both periodic and free space boundary conditions. The history dependence is overcome by splitting the heat potentials into a smooth history part and a singular local part. We discuss the resolution of the history part on an adaptive volume grid in detail, providing sharp estimates that allow for the construction of an optimal grid, justifying the efficiency of the bootstrapping scheme. While the discussion in this paper is restricted to one spatial dimension, the generalization to two and three dimensions is straightforward. The performance of the algorithm is illustrated via several numerical examples.Comment: 14 pages, 3 table

A CUDA-based implementation of an improved SPH method on GPU

We present a CUDA-based parallel implementation on GPU architecture of a modified version of the Smoothed Particle Hydrodynamics (SPH) method. This modified formulation exploits a strategy based on the Taylor series expansion, which simultaneously improves the approximation of a function and its derivatives with respect to the standard formulation. The improvement in accuracy comes at the cost of an additional computational effort. The computational demand becomes increasingly crucial as problem size increases but can be addressed by employing fast summations in a parallel computational scheme. The experimental analysis showed that our parallel implementation significantly reduces the runtime, with speed-ups of up to 90,when compared to the CPU-based implementation