549 research outputs found
On the Error in Phase Transition Computations for Compressed Sensing
Evaluating the statistical dimension is a common tool to determine the
asymptotic phase transition in compressed sensing problems with Gaussian
ensemble. Unfortunately, the exact evaluation of the statistical dimension is
very difficult and it has become standard to replace it with an upper-bound. To
ensure that this technique is suitable, [1] has introduced an upper-bound on
the gap between the statistical dimension and its approximation. In this work,
we first show that the error bound in [1] in some low-dimensional models such
as total variation and analysis minimization becomes poorly large.
Next, we develop a new error bound which significantly improves the estimation
gap compared to [1]. In particular, unlike the bound in [1] that is not
applicable to settings with overcomplete dictionaries, our bound exhibits a
decaying behavior in such cases
Estimation of Sparse MIMO Channels with Common Support
We consider the problem of estimating sparse communication channels in the
MIMO context. In small to medium bandwidth communications, as in the current
standards for OFDM and CDMA communication systems (with bandwidth up to 20
MHz), such channels are individually sparse and at the same time share a common
support set. Since the underlying physical channels are inherently
continuous-time, we propose a parametric sparse estimation technique based on
finite rate of innovation (FRI) principles. Parametric estimation is especially
relevant to MIMO communications as it allows for a robust estimation and
concise description of the channels. The core of the algorithm is a
generalization of conventional spectral estimation methods to multiple input
signals with common support. We show the application of our technique for
channel estimation in OFDM (uniformly/contiguous DFT pilots) and CDMA downlink
(Walsh-Hadamard coded schemes). In the presence of additive white Gaussian
noise, theoretical lower bounds on the estimation of SCS channel parameters in
Rayleigh fading conditions are derived. Finally, an analytical spatial channel
model is derived, and simulations on this model in the OFDM setting show the
symbol error rate (SER) is reduced by a factor 2 (0 dB of SNR) to 5 (high SNR)
compared to standard non-parametric methods - e.g. lowpass interpolation.Comment: 12 pages / 7 figures. Submitted to IEEE Transactions on Communicatio
Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation
We consider the estimation of a signal from the knowledge of its noisy linear
random Gaussian projections. A few examples where this problem is relevant are
compressed sensing, sparse superposition codes, and code division multiple
access. There has been a number of works considering the mutual information for
this problem using the replica method from statistical physics. Here we put
these considerations on a firm rigorous basis. First, we show, using a
Guerra-Toninelli type interpolation, that the replica formula yields an upper
bound to the exact mutual information. Secondly, for many relevant practical
cases, we present a converse lower bound via a method that uses spatial
coupling, state evolution analysis and the I-MMSE theorem. This yields a single
letter formula for the mutual information and the minimal-mean-square error for
random Gaussian linear estimation of all discrete bounded signals. In addition,
we prove that the low complexity approximate message-passing algorithm is
optimal outside of the so-called hard phase, in the sense that it
asymptotically reaches the minimal-mean-square error. In this work spatial
coupling is used primarily as a proof technique. However our results also prove
two important features of spatially coupled noisy linear random Gaussian
estimation. First there is no algorithmically hard phase. This means that for
such systems approximate message-passing always reaches the minimal-mean-square
error. Secondly, in a proper limit the mutual information associated to such
systems is the same as the one of uncoupled linear random Gaussian estimation
An Overview of Multi-Processor Approximate Message Passing
Approximate message passing (AMP) is an algorithmic framework for solving
linear inverse problems from noisy measurements, with exciting applications
such as reconstructing images, audio, hyper spectral images, and various other
signals, including those acquired in compressive signal acquisiton systems. The
growing prevalence of big data systems has increased interest in large-scale
problems, which may involve huge measurement matrices that are unsuitable for
conventional computing systems. To address the challenge of large-scale
processing, multiprocessor (MP) versions of AMP have been developed. We provide
an overview of two such MP-AMP variants. In row-MP-AMP, each computing node
stores a subset of the rows of the matrix and processes corresponding
measurements. In column- MP-AMP, each node stores a subset of columns, and is
solely responsible for reconstructing a portion of the signal. We will discuss
pros and cons of both approaches, summarize recent research results for each,
and explain when each one may be a viable approach. Aspects that are
highlighted include some recent results on state evolution for both MP-AMP
algorithms, and the use of data compression to reduce communication in the MP
network
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