230 research outputs found
Optimal detection of sparse principal components in high dimension
We perform a finite sample analysis of the detection levels for sparse
principal components of a high-dimensional covariance matrix. Our minimax
optimal test is based on a sparse eigenvalue statistic. Alas, computing this
test is known to be NP-complete in general, and we describe a computationally
efficient alternative test using convex relaxations. Our relaxation is also
proved to detect sparse principal components at near optimal detection levels,
and it performs well on simulated datasets. Moreover, using polynomial time
reductions from theoretical computer science, we bring significant evidence
that our results cannot be improved, thus revealing an inherent trade off
between statistical and computational performance.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1127 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Revisiting two strong approximation results of Dudley and Philipp
We demonstrate the strength of a coupling derived from a Gaussian
approximation of Zaitsev (1987a) by revisiting two strong approximation results
for the empirical process of Dudley and Philipp (1983), and using the coupling
to derive extended and refined versions of them.Comment: Published at http://dx.doi.org/10.1214/074921706000000824 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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