882 research outputs found
A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing
Subspace-based signal processing traditionally focuses on problems involving
a few subspaces. Recently, a number of problems in different application areas
have emerged that involve a significantly larger number of subspaces relative
to the ambient dimension. It becomes imperative in such settings to first
identify a smaller set of active subspaces that contribute to the observation
before further processing can be carried out. This problem of identification of
a small set of active subspaces among a huge collection of subspaces from a
single (noisy) observation in the ambient space is termed subspace unmixing.
This paper formally poses the subspace unmixing problem under the parsimonious
subspace-sum (PS3) model, discusses connections of the PS3 model to problems in
wireless communications, hyperspectral imaging, high-dimensional statistics and
compressed sensing, and proposes a low-complexity algorithm, termed marginal
subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the
subspace unmixing problem for the PS3 model into a multiple hypothesis testing
(MHT) problem and its analysis in the paper helps control the family-wise error
rate of this MHT problem at any level under two random
signal generation models. Some other highlights of the analysis of the MSD
algorithm include: (i) it is applicable to an arbitrary collection of subspaces
on the Grassmann manifold; (ii) it relies on properties of the collection of
subspaces that are computable in polynomial time; and () it allows for
linear scaling of the number of active subspaces as a function of the ambient
dimension. Finally, numerical results are presented in the paper to better
understand the performance of the MSD algorithm.Comment: Submitted for journal publication; 33 pages, 14 figure
Model Selection: Two Fundamental Measures of Coherence and Their Algorithmic Significance
The problem of model selection arises in a number of contexts, such as
compressed sensing, subset selection in linear regression, estimation of
structures in graphical models, and signal denoising. This paper generalizes
the notion of \emph{incoherence} in the existing literature on model selection
and introduces two fundamental measures of coherence---termed as the worst-case
coherence and the average coherence---among the columns of a design matrix. In
particular, it utilizes these two measures of coherence to provide an in-depth
analysis of a simple one-step thresholding (OST) algorithm for model selection.
One of the key insights offered by the ensuing analysis is that OST is feasible
for model selection as long as the design matrix obeys an easily verifiable
property. In addition, the paper also characterizes the model-selection
performance of OST in terms of the worst-case coherence, \mu, and establishes
that OST performs near-optimally in the low signal-to-noise ratio regime for N
x C design matrices with \mu = O(N^{-1/2}). Finally, in contrast to some of the
existing literature on model selection, the analysis in the paper is
nonasymptotic in nature, it does not require knowledge of the true model order,
it is applicable to generic (random or deterministic) design matrices, and it
neither requires submatrices of the design matrix to have full rank, nor does
it assume a statistical prior on the values of the nonzero entries of the data
vector.Comment: 5 pages; Accepted for Proc. 2010 IEEE International Symposium on
Information Theory (ISIT 2010
Frame Coherence and Sparse Signal Processing
The sparse signal processing literature often uses random sensing matrices to
obtain performance guarantees. Unfortunately, in the real world, sensing
matrices do not always come from random processes. It is therefore desirable to
evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse
signals. To this end, the present paper investigates two parameters that
measure the coherence of a frame: worst-case and average coherence. We first
provide several examples of frames that have small spectral norm, worst-case
coherence, and average coherence. Next, we present a new lower bound on
worst-case coherence and compare it to the Welch bound. Later, we propose an
algorithm that decreases the average coherence of a frame without changing its
spectral norm or worst-case coherence. Finally, we use worst-case and average
coherence, as opposed to the Restricted Isometry Property, to garner
near-optimal probabilistic guarantees on both sparse signal detection and
reconstruction in the presence of noise. This contrasts with recent results
that only guarantee noiseless signal recovery from arbitrary frames, and which
further assume independence across the nonzero entries of the signal---in a
sense, requiring small average coherence replaces the need for such an
assumption
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