Abstract:
:Spectral methods are of fundamental importance in statistics and machine learning, because they underlie algorithms from classical principal components analysis to more recent approaches that exploit manifold structure. In most cases, the core technical problem can be reduced to computing a low-rank approximation to a positive-definite kernel. For the growing number of applications dealing with very large or high-dimensional datasets, however, the optimal approximation afforded by an exact spectral decomposition is too costly, because its complexity scales as the cube of either the number of training examples or their dimensionality. Motivated by such applications, we present here 2 new algorithms for the approximation of positive-semidefinite kernels, together with error bounds that improve on results in the literature. We approach this problem by seeking to determine, in an efficient manner, the most informative subset of our data relative to the kernel approximation task at hand. This leads to two new strategies based on the Nyström method that are directly applicable to massive datasets. The first of these-based on sampling-leads to a randomized algorithm whereupon the kernel induces a probability distribution on its set of partitions, whereas the latter approach-based on sorting-provides for the selection of a partition in a deterministic way. We detail their numerical implementation and provide simulation results for a variety of representative problems in statistical data analysis, each of which demonstrates the improved performance of our approach relative to existing methods.
journal_name
Proc Natl Acad Sci U S Aauthors
Belabbas MA,Wolfe PJdoi
10.1073/pnas.0810600105subject
Has Abstractpub_date
2009-01-13 00:00:00pages
369-74issue
2eissn
0027-8424issn
1091-6490pii
0810600105journal_volume
106pub_type
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