5,795 research outputs found

### Verifiable conditions of $\ell_1$-recovery of sparse signals with sign restrictions

We propose necessary and sufficient conditions for a sensing matrix to be
"s-semigood" -- to allow for exact $\ell_1$-recovery of sparse signals with at
most $s$ nonzero entries under sign restrictions on part of the entries. We
express the error bounds for imperfect $\ell_1$-recovery in terms of the
characteristics underlying these conditions. Furthermore, we demonstrate that
these characteristics, although difficult to evaluate, lead to verifiable
sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently
computable upper bounds on those $s$ for which a given sensing matrix is
$s$-semigood. We concentrate on the properties of proposed verifiable
sufficient conditions of $s$-semigoodness and describe their limits of
performance

### An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

We consider linear inverse problems where the solution is assumed to have a
sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that
replacing the usual quadratic regularizing penalties by weighted l^p-penalties
on the coefficients of such expansions, with 1 < or = p < or =2, still
regularizes the problem. If p < 2, regularized solutions of such l^p-penalized
problems will have sparser expansions, with respect to the basis under
consideration. To compute the corresponding regularized solutions we propose an
iterative algorithm that amounts to a Landweber iteration with thresholding (or
nonlinear shrinkage) applied at each iteration step. We prove that this
algorithm converges in norm. We also review some potential applications of this
method.Comment: 30 pages, 3 figures; this is version 2 - changes with respect to v1:
small correction in proof (but not statement of) lemma 3.15; description of
Besov spaces in intro and app A clarified (and corrected); smaller pointsize
(making 30 instead of 38 pages

### Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data

We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i,
i=1,..., n, are independent exponential with individual means $\mu_i$. The
vector $\mu =(\mu_i)$ is thought to be sparse, with most coordinates 1 but a
small fraction significantly larger than 1; roughly, most coordinates are
simply `noise,' but a small fraction contain `signal.' We measure risk by
per-coordinate mean-squared error in recovering $\log(\mu_i)$, and study
minimax estimation over parameter spaces defined by constraints on the
per-coordinate p-norm of $\log(\mu_i)$:
$\frac{1}{n}\sum_{i=1}^n\log^p(\mu_i)\leq \eta^p$. We show for large n and
small $\eta$ that FDR thresholding can be nearly Minimax. The FDR control
parameter 0<q<1 plays an important role: when $q\leq 1/2$, the FDR estimator is
nearly minimax, while choosing a fixed q>1/2 prevents near minimaxity. These
conclusions mirror those found in the Gaussian case in Abramovich et al. [Ann.
Statist. 34 (2006) 584--653]. The techniques developed here seem applicable to
a wide range of other distributional assumptions, other loss measures and
non-i.i.d. dependency structures.Comment: Published at http://dx.doi.org/10.1214/009053606000000920 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Higher criticism for detecting sparse heterogeneous mixtures

Higher criticism, or second-level significance testing, is a
multiple-comparisons concept mentioned in passing by Tukey. It concerns a
situation where there are many independent tests of significance and one is
interested in rejecting the joint null hypothesis. Tukey suggested comparing
the fraction of observed significances at a given \alpha-level to the expected
fraction under the joint null. In fact, he suggested standardizing the
difference of the two quantities and forming a z-score; the resulting z-score
tests the significance of the body of significance tests. We consider a
generalization, where we maximize this z-score over a range of significance
levels 0<\alpha\leq\alpha_0.
We are able to show that the resulting higher criticism statistic is
effective at resolving a very subtle testing problem: testing whether n normal
means are all zero versus the alternative that a small fraction is nonzero. The
subtlety of this ``sparse normal means'' testing problem can be seen from work
of Ingster and Jin, who studied such problems in great detail. In their
studies, they identified an interesting range of cases where the small fraction
of nonzero means is so small that the alternative hypothesis exhibits little
noticeable effect on the distribution of the p-values either for the bulk of
the tests or for the few most highly significant tests.
In this range, when the amplitude of nonzero means is calibrated with the
fraction of nonzero means, the likelihood ratio test for a precisely specified
alternative would still succeed in separating the two hypotheses.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000026

### Counting faces of randomly-projected polytopes when the projection radically lowers dimension

This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page

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