80 research outputs found
Dimensionality reduction with subgaussian matrices: a unified theory
We present a theory for Euclidean dimensionality reduction with subgaussian
matrices which unifies several restricted isometry property and
Johnson-Lindenstrauss type results obtained earlier for specific data sets. In
particular, we recover and, in several cases, improve results for sets of
sparse and structured sparse vectors, low-rank matrices and tensors, and smooth
manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for
data sets taking the form of an infinite union of subspaces of a Hilbert space
Tail bounds via generic chaining
We modify Talagrand's generic chaining method to obtain upper bounds for all
p-th moments of the supremum of a stochastic process. These bounds lead to an
estimate for the upper tail of the supremum with optimal deviation parameters.
We apply our procedure to improve and extend some known deviation inequalities
for suprema of unbounded empirical processes and chaos processes. As an
application we give a significantly simplified proof of the restricted isometry
property of the subsampled discrete Fourier transform.Comment: Added detailed proof of Theorem 3.5; Application to dimensionality
reduction expanded and moved to separate note arXiv:1402.397
It\^{o} isomorphisms for -valued Poisson stochastic integrals
Motivated by the study of existence, uniqueness and regularity of solutions
to stochastic partial differential equations driven by jump noise, we prove
It\^{o} isomorphisms for -valued stochastic integrals with respect to a
compensated Poisson random measure. The principal ingredients for the proof are
novel Rosenthal type inequalities for independent random variables taking
values in a (noncommutative) -space, which may be of independent interest.
As a by-product of our proof, we observe some moment estimates for the operator
norm of a sum of independent random matrices.Comment: Published in at http://dx.doi.org/10.1214/13-AOP906 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Robust one-bit compressed sensing with partial circulant matrices
We present optimal sample complexity estimates for one-bit compressed sensing
problems in a realistic scenario: the procedure uses a structured matrix (a
randomly sub-sampled circulant matrix) and is robust to analog pre-quantization
noise as well as to adversarial bit corruptions in the quantization process.
Our results imply that quantization is not a statistically expensive procedure
in the presence of nontrivial analog noise: recovery requires the same sample
size one would have needed had the measurement matrix been Gaussian and the
noisy analog measurements been given as data
Some remarks on noncommutative Khintchine inequalities
Normalized free semi-circular random variables satisfy an upper Khintchine
inequality in . We show that this implies the corresponding upper
Khintchine inequality in any noncommutative Banach function space. As
applications, we obtain a very simple proof of a well-known interpolation
result for row and column operator spaces and, moreover, answer an open
question on noncommutative moment inequalities concerning a paper by Bekjan and
Chen
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