2\u00d7Naive\n0.837724\n0.849935\n0.846294\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.095838\n1.099828\n1.081216\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.087066\n1.11567\n1.10654\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.469454\n1.983888\n1.47692\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.658592\n4.599235\n1.683913\n\n301.696\nout of RAM\n> 2\u00d7Naive\n2.802112\n29.231309\n2.855083\n\n301.696\n3.892312\n> 2\u00d7Naive\n6.018158\n72.435177\n6.265131\n\n301.696\n> 2\u00d7Naive\n> 2\u00d7Naive\n30.294007\n285.719266\n24.598749\n\n301.696\n2.01846\n> 2\u00d7Naive\n62.077669\n18.450387\n5.063365\n\n301.696\n> 2\u00d7Naive\n> 2\u00d7Naive\n280.633106\n12.886239\n7.142465\n\ncolors50k (astronomy: colors), D = 2, N = 50000, h\u2217 = 0.0016911\n\nNaive\nFGT\nIFGT\nDFD\nDFGT\nDFGTH\n\nNaive\nFGT\nIFGT\nDFD\nDFGT\nDFGTH\n\nNaive\nFGT\nIFGT\nDFD\nDFGT\nDFGTH\n\nNaive\nFGT\nIFGT\nDFD\nDFGT\nDFGTH\n\nNaive\nFGT\nIFGT\nDFD\nDFGT\nDFGTH\n\n301.696\n0.319538\n> 2\u00d7Naive\n151.590062\n2.777454\n1.036626\n\n301.696\n0.475281\n> 2\u00d7Naive\n81.373053\n5.336602\n1.78648\n\n301.696\n0.210799\n> 2\u00d7Naive\n357.099354\n3.424304\n1.883216\n\n354.868751\n> 2\u00d7Naive\n> 2\u00d7Naive\n42.022605\n125.059911\n22.6106\n\n1000\n\n301.696\n0.183616\n7.576783\n1.551019\n2.532401\n0.68471\n\n301.696\n0.114430\n7.55986\n3.604753\n3.5638\n0.627554\n\n301.696\n0.059664\n7.585585\n0.743045\n1.977302\n0.436596\n\n354.868751\n> 2\u00d7Naive\n> 2\u00d7Naive\n383.12048\n109.353701\n87.488392\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n107.675935\n> 2\u00d7Naive\n> 2\u00d7Naive\n\nedsgc-radec-rnd (astronomy: angles), D = 2, N = 50000, h\u2217 = 0.00466204\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.682261\n4.346061\n1.737799\n\n301.696\n2.859245\n> 2\u00d7Naive\n5.860172\n73.036687\n6.037217\n\n301.696\nout of RAM\n> 2\u00d7Naive\n1.083528\n1.120015\n1.104545\n\n301.696\nout of RAM\n> 2\u00d7Naive\n0.812462\n0.84023\n0.821672\nmockgalaxy-D-1M-rnd (cosmology: positions), D = 3, N = 50000, h\u2217 = 0.000768201\n354.868751\nout of RAM\n> 2\u00d7Naive\n0.70054\n0.73007\n0.724004\n\n354.868751\n> 2\u00d7Naive\n> 2\u00d7Naive\n1.086608\n50.619588\n1.265064\n\n301.696\n1.768738\n> 2\u00d7Naive\n63.849361\n21.652047\n5.7398\n\n354.868751\nout of RAM\n> 2\u00d7Naive\n0.701547\n0.733638\n0.719951\n\n354.868751\nout of RAM\n> 2\u00d7Naive\n0.843451\n0.999316\n0.877564\n\n354.868751\nout of RAM\n> 2\u00d7Naive\n0.761524\n0.799711\n0.789002\n\nbio5-rnd (biology: drug activity), D = 5, N = 50000, h\u2217 = 0.000567161\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n2.249868\n> 2\u00d7Naive\n> 2\u00d7Naive\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n2.4958865\n> 2\u00d7Naive\n> 2\u00d7Naive\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n4.70948\n> 2\u00d7Naive\n> 2\u00d7Naive\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n12.065697\n> 2\u00d7Naive\n> 2\u00d7Naive\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n94.345003\n> 2\u00d7Naive\n> 2\u00d7Naive\n\n364.439228\nout of RAM\n> 2\u00d7Naive\n412.39142\n> 2\u00d7Naive\n> 2\u00d7Naive\n\nDiscussion. The experiments indicate that the DFGTH method is able to achieve rea-\nsonable performance across all bandwidth scales. Unfortunately none of the series\napproximation-based methods do well on the 5-dimensional data, as expected, highlight-\ning the main weakness of the approach presented. Pursuing corrections to the error bounds\nnecessary to use the intriguing series form of [14] may allow an increase in dimensionality.\n\nReferences\n[1] A. W. Appel. An Ef\ufb01cient Program for Many-Body Simulations. SIAM Journal on Scienti\ufb01c and Statistical Computing,\n\n6(1):85\u2013103, 1985.\n\n[2] J. Barnes and P. Hut. A Hierarchical O(N logN ) Force-Calculation Algorithm. Nature, 324, 1986.\n[3] B. Baxter and G. Roussos. A new error estimate of the fast gauss transform. SIAM Journal on Scienti\ufb01c Computing,\n\n24(1):257\u2013259, 2002.\n\n[4] P. Callahan and S. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and\n\nn-body potential \ufb01elds. Journal of the ACM, 62(1):67\u201390, January 1995.\n\n[5] A. Gray and A. W. Moore. N-Body Problems in Statistical Learning. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors,\n\nAdvances in Neural Information Processing Systems 13 (December 2000). MIT Press, 2001.\n\n[6] A. G. Gray. Bringing Tractability to Generalized N-Body Problems in Statistical and Scienti\ufb01c Computation. PhD thesis,\n\nCarnegie Mellon University, 2003.\n\n[7] A. G. Gray and A. W. Moore. Rapid Evaluation of Multiple Density Models. In Arti\ufb01cial Intelligence and Statistics 2003,\n\n2003.\n\n[8] L. Greengard and V. Rokhlin. A Fast Algorithm for Particle Simulations. Journal of Computational Physics, 73, 1987.\n[9] L. Greengard and J. Strain. The fast gauss transform. SIAM Journal on Scienti\ufb01c and Statistical Computing, 12(1):79\u201394,\n\n1991.\n\n[10] L. Greengard and X. Sun. A new version of the fast gauss transform. Documenta Mathematica, Extra Volume ICM(III):575\u2013\n\n584, 1998.\n\n[11] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and Hall, 1986.\n[12] J. Strain. The fast gauss transform with variable scales. SIAM Journal on Scienti\ufb01c and Statistical Computing, 12:1131\u2013\n\n1139, 1991.\n\n[13] O. Sz\u00b4asz. On the relative extrema of the hermite orthogonal functions. J. Indian Math. Soc., 15:129\u2013134, 1951.\n[14] C. Yang, R. Duraiswami, N. A. Gumerov, and L. Davis. Improved fast gauss transform and ef\ufb01cient kernel density estima-\n\ntion. International Conference on Computer Vision, 2003.\n\n\f", "award": [], "sourceid": 2928, "authors": [{"given_name": "Dongryeol", "family_name": "Lee", "institution": null}, {"given_name": "Andrew", "family_name": "Moore", "institution": null}, {"given_name": "Alexander", "family_name": "Gray", "institution": null}]}