74 research outputs found
Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values
Recently, the precise performance of the Generalized LASSO algorithm for
recovering structured signals from compressed noisy measurements, obtained via
i.i.d. Gaussian matrices, has been characterized. The analysis is based on a
framework introduced by Stojnic and heavily relies on the use of Gordon's
Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes.
As a result, corresponding characterizations for other ensembles of measurement
matrices have not been developed. In this work, we analyze the corresponding
performance of the ensemble of isotropically random orthogonal (i.r.o.)
measurements. We consider the constrained version of the Generalized LASSO and
derive a sharp characterization of its normalized squared error in the
large-system limit. When compared to its Gaussian counterpart, our result
analytically confirms the superiority in performance of the i.r.o. ensemble.
Our second result, derives an asymptotic lower bound on the minimum conic
singular values of i.r.o. matrices. This bound is larger than the corresponding
bound on Gaussian matrices. To prove our results we express i.r.o. matrices in
terms of Gaussians and show that, with some modifications, the GMT framework is
still applicable
Optimal Placement of Distributed Energy Storage in Power Networks
We formulate the optimal placement, sizing and control of storage devices in
a power network to minimize generation costs with the intent of load shifting.
We assume deterministic demand, a linearized DC approximated power flow model
and a fixed available storage budget. Our main result proves that when the
generation costs are convex and nondecreasing, there always exists an optimal
storage capacity allocation that places zero storage at generation-only buses
that connect to the rest of the network via single links. This holds regardless
of the demand profiles, generation capacities, line-flow limits and
characteristics of the storage technologies. Through a counterexample, we
illustrate that this result is not generally true for generation buses with
multiple connections. For specific network topologies, we also characterize the
dependence of the optimal generation cost on the available storage budget,
generation capacities and flow constraints.Comment: 15 pages, 9 figures, generalized result to include line losses in
Section 4
The Squared-Error of Generalized LASSO: A Precise Analysis
We consider the problem of estimating an unknown signal from noisy
linear observations . In many practical instances,
has a certain structure that can be captured by a structure inducing convex
function . For example, norm can be used to encourage a
sparse solution. To estimate with the aid of , we consider the
well-known LASSO method and provide sharp characterization of its performance.
We assume the entries of the measurement matrix and the noise vector
have zero-mean normal distributions with variances and
respectively. For the LASSO estimator , we attempt to calculate the
Normalized Square Error (NSE) defined as as
a function of the noise level , the number of observations and the
structure of the signal. We show that, the structure of the signal and
choice of the function enter the error formulae through the summary
parameters and , which are defined as the Gaussian
squared-distances to the subdifferential cone and to the -scaled
subdifferential, respectively. The first LASSO estimator assumes a-priori
knowledge of and is given by . We prove that its worst case NSE is achieved when
and concentrates around .
Secondly, we consider , for some
. This time the NSE formula depends on the choice of
and is given by . We then establish a mapping
between this and the third estimator . Finally, for a number of important structured signal classes,
we translate our abstract formulae to closed-form upper bounds on the NSE
Simple Error Bounds for Regularized Noisy Linear Inverse Problems
Consider estimating a structured signal from linear,
underdetermined and noisy measurements
, via solving a variant of the
lasso algorithm: . Here, is a
convex function aiming to promote the structure of , say
-norm to promote sparsity or nuclear norm to promote low-rankness. We
assume that the entries of are independent and normally
distributed and make no assumptions on the noise vector , other
than it being independent of . Under this generic setup, we derive
a general, non-asymptotic and rather tight upper bound on the -norm of
the estimation error . Our bound is
geometric in nature and obeys a simple formula; the roles of , and
are all captured by a single summary parameter
, termed the Gaussian squared
distance to the scaled subdifferential. We connect our result to the literature
and verify its validity through simulations.Comment: 6pages, 2 figur
Simple Bounds for Noisy Linear Inverse Problems with Exact Side Information
This paper considers the linear inverse problem where we wish to estimate a
structured signal from its corrupted observations. When the problem is
ill-posed, it is natural to make use of a convex function that
exploits the structure of the signal. For example, norm can be used
for sparse signals. To carry out the estimation, we consider two well-known
convex programs: 1) Second order cone program (SOCP), and, 2) Lasso. Assuming
Gaussian measurements, we show that, if precise information about the value
or the -norm of the noise is available, one can do a
particularly good job at estimation. In particular, the reconstruction error
becomes proportional to the "sparsity" of the signal rather than the ambient
dimension of the noise vector. We connect our results to existing works and
provide a discussion on the relation of our results to the standard
least-squares problem. Our error bounds are non-asymptotic and sharp, they
apply to arbitrary convex functions and do not assume any distribution on the
noise.Comment: 13 page
Analysis and Optimization of Aperture Design in Computational Imaging
There is growing interest in the use of coded aperture imaging systems for a
variety of applications. Using an analysis framework based on mutual
information, we examine the fundamental limits of such systems---and the
associated optimum aperture coding---under simple but meaningful propagation
and sensor models. Among other results, we show that when thermal noise
dominates, spectrally-flat masks, which have 50% transmissivity, are optimal,
but that when shot noise dominates, randomly generated masks with lower
transmissivity offer greater performance. We also provide comparisons to
classical pinhole cameras
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