12 research outputs found
Syndrome-Based Encoding of Compressible Sources for M2M Communication
Data originating from many devices and sensors can be modeled as sparse signals. Hence, efficient compression techniques of such data are essential to reduce bandwidth and transmission power, especially for energy constrained devices within machine to machine communication scenarios. This paper provides accurate analysis of the operational distortion-rate function (ODR) for syndrome-based source encoders of noisy sparse sources. We derive the probability density function of error due to both quantization and pre- quantization noise for a type of mixed distributed source comprising Bernoulli and an arbitrary continuous distribution, e.g., Bernoulli- uniform sources. Then, we derive the ODR for two encoding schemes based on the syndromes of Reed-Solomon (RS) and Bose, Chaudhuri, and Hocquenghem (BCH) codes. The presented analysis allows designing a quantizer such that a target average distortion is achieved. As confirmed by numerical results, the closed-form expression for ODR perfectly coincides with the simulation. Also, the performance loss compared to an entropy based encoder is tolerable
Compression-Based Compressed Sensing
Modern compression algorithms exploit complex structures that are present in
signals to describe them very efficiently. On the other hand, the field of
compressed sensing is built upon the observation that "structured" signals can
be recovered from their under-determined set of linear projections. Currently,
there is a large gap between the complexity of the structures studied in the
area of compressed sensing and those employed by the state-of-the-art
compression codes. Recent results in the literature on deterministic signals
aim at bridging this gap through devising compressed sensing decoders that
employ compression codes. This paper focuses on structured stochastic processes
and studies the application of rate-distortion codes to compressed sensing of
such signals. The performance of the formerly-proposed compressible signal
pursuit (CSP) algorithm is studied in this stochastic setting. It is proved
that in the very low distortion regime, as the blocklength grows to infinity,
the CSP algorithm reliably and robustly recovers instances of a stationary
process from random linear projections as long as their count is slightly more
than times the rate-distortion dimension (RDD) of the source. It is also
shown that under some regularity conditions, the RDD of a stationary process is
equal to its information dimension (ID). This connection establishes the
optimality of the CSP algorithm at least for memoryless stationary sources, for
which the fundamental limits are known. Finally, it is shown that the CSP
algorithm combined by a family of universal variable-length fixed-distortion
compression codes yields a family of universal compressed sensing recovery
algorithms
Remote Source Coding under Gaussian Noise : Dueling Roles of Power and Entropy Power
The distributed remote source coding (so-called CEO) problem is studied in
the case where the underlying source, not necessarily Gaussian, has finite
differential entropy and the observation noise is Gaussian. The main result is
a new lower bound for the sum-rate-distortion function under arbitrary
distortion measures. When specialized to the case of mean-squared error, it is
shown that the bound exactly mirrors a corresponding upper bound, except that
the upper bound has the source power (variance) whereas the lower bound has the
source entropy power. Bounds exhibiting this pleasing duality of power and
entropy power have been well known for direct and centralized source coding
since Shannon's work. While the bounds hold generally, their value is most
pronounced when interpreted as a function of the number of agents in the CEO
problem
Rate Distortion Behavior of Sparse Sources
This paper studies the rate distortion behavior of sparse memoryless sources that serve as models of sparse signal representations. For the Hamming distortion criterion, is shown to be essentially linear. For the mean squared error measure, two models are analyzed: the mixed discrete/continuous spike processes and Gaussian mixtures. The latter are shown to be a better model for ``natural'' data such as sparse wavelet coefficients. Finally, the geometric mean of a continuous random variable is introduced as a sparseness measure. It yields upper and lower bounds on the entropy and thus characterizes high-rate
Optimal Phase Transitions in Compressed Sensing
Compressed sensing deals with efficient recovery of analog signals from
linear encodings. This paper presents a statistical study of compressed sensing
by modeling the input signal as an i.i.d. process with known distribution.
Three classes of encoders are considered, namely optimal nonlinear, optimal
linear and random linear encoders. Focusing on optimal decoders, we investigate
the fundamental tradeoff between measurement rate and reconstruction fidelity
gauged by error probability and noise sensitivity in the absence and presence
of measurement noise, respectively. The optimal phase transition threshold is
determined as a functional of the input distribution and compared to suboptimal
thresholds achieved by popular reconstruction algorithms. In particular, we
show that Gaussian sensing matrices incur no penalty on the phase transition
threshold with respect to optimal nonlinear encoding. Our results also provide
a rigorous justification of previous results based on replica heuristics in the
weak-noise regime.Comment: to appear in IEEE Transactions of Information Theor
Distributed Scalar Quantization for Computing: High-Resolution Analysis and Extensions
Communication of quantized information is frequently followed by a
computation. We consider situations of \emph{distributed functional scalar
quantization}: distributed scalar quantization of (possibly correlated) sources
followed by centralized computation of a function. Under smoothness conditions
on the sources and function, companding scalar quantizer designs are developed
to minimize mean-squared error (MSE) of the computed function as the quantizer
resolution is allowed to grow. Striking improvements over quantizers designed
without consideration of the function are possible and are larger in the
entropy-constrained setting than in the fixed-rate setting. As extensions to
the basic analysis, we characterize a large class of functions for which
regular quantization suffices, consider certain functions for which asymptotic
optimality is achieved without arbitrarily fine quantization, and allow limited
collaboration between source encoders. In the entropy-constrained setting, a
single bit per sample communicated between encoders can have an
arbitrarily-large effect on functional distortion. In contrast, such
communication has very little effect in the fixed-rate setting.Comment: 36 pages, 10 figure
On the Rate-Distortion Function of Random Vectors and Stationary Sources with Mixed Distributions
The asymptotic (small distortion) behavior of the ratedistortion function of an n-dimensional source vector with mixed distribution is derived. The source distribution is a finite mixture of components such that under each component distribution a certain subset of the coordinates have a discrete distribution while the remaining coordinates have a joint density. The expected number of coordinates with a joint density is shown to equal the rate-distortion dimension of the source vector. Also, the exact small distortion asymptotic behavior of the rate-distortion function of a special but interesting class of stationary information sources is determined