11,706 research outputs found
Far-Field Compression for Fast Kernel Summation Methods in High Dimensions
We consider fast kernel summations in high dimensions: given a large set of
points in dimensions (with ) and a pair-potential function (the
{\em kernel} function), we compute a weighted sum of all pairwise kernel
interactions for each point in the set. Direct summation is equivalent to a
(dense) matrix-vector multiplication and scales quadratically with the number
of points. Fast kernel summation algorithms reduce this cost to log-linear or
linear complexity.
Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by
constructing approximate representations of interactions of points that are far
from each other. In algebraic terms, these representations correspond to
low-rank approximations of blocks of the overall interaction matrix. Existing
approaches require an excessive number of kernel evaluations with increasing
and number of points in the dataset.
To address this issue, we use a randomized algebraic approach in which we
first sample the rows of a block and then construct its approximate, low-rank
interpolative decomposition. We examine the feasibility of this approach
theoretically and experimentally. We provide a new theoretical result showing a
tighter bound on the reconstruction error from uniformly sampling rows than the
existing state-of-the-art. We demonstrate that our sampling approach is
competitive with existing (but prohibitively expensive) methods from the
literature. We also construct kernel matrices for the Laplacian, Gaussian, and
polynomial kernels -- all commonly used in physics and data analysis. We
explore the numerical properties of blocks of these matrices, and show that
they are amenable to our approach. Depending on the data set, our randomized
algorithm can successfully compute low rank approximations in high dimensions.
We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure
Scaling-up Empirical Risk Minimization: Optimization of Incomplete U-statistics
In a wide range of statistical learning problems such as ranking, clustering
or metric learning among others, the risk is accurately estimated by
-statistics of degree , i.e. functionals of the training data with
low variance that take the form of averages over -tuples. From a
computational perspective, the calculation of such statistics is highly
expensive even for a moderate sample size , as it requires averaging
terms. This makes learning procedures relying on the optimization of
such data functionals hardly feasible in practice. It is the major goal of this
paper to show that, strikingly, such empirical risks can be replaced by
drastically computationally simpler Monte-Carlo estimates based on terms
only, usually referred to as incomplete -statistics, without damaging the
learning rate of Empirical Risk Minimization (ERM)
procedures. For this purpose, we establish uniform deviation results describing
the error made when approximating a -process by its incomplete version under
appropriate complexity assumptions. Extensions to model selection, fast rate
situations and various sampling techniques are also considered, as well as an
application to stochastic gradient descent for ERM. Finally, numerical examples
are displayed in order to provide strong empirical evidence that the approach
we promote largely surpasses more naive subsampling techniques.Comment: To appear in Journal of Machine Learning Research. 34 pages. v2:
minor correction to Theorem 4 and its proof, added 1 reference. v3: typo
corrected in Proposition 3. v4: improved presentation, added experiments on
model selection for clustering, fixed minor typo
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
An ontology enhanced parallel SVM for scalable spam filter training
This is the post-print version of the final paper published in Neurocomputing. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2013 Elsevier B.V.Spam, under a variety of shapes and forms, continues to inflict increased damage. Varying approaches including Support Vector Machine (SVM) techniques have been proposed for spam filter training and classification. However, SVM training is a computationally intensive process. This paper presents a MapReduce based parallel SVM algorithm for scalable spam filter training. By distributing, processing and optimizing the subsets of the training data across multiple participating computer nodes, the parallel SVM reduces the training time significantly. Ontology semantics are employed to minimize the impact of accuracy degradation when distributing the training data among a number of SVM classifiers. Experimental results show that ontology based augmentation improves the accuracy level of the parallel SVM beyond the original sequential counterpart
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