1,365 research outputs found
Improved Upper Bounds to the Causal Quadratic Rate-Distortion Function for Gaussian Stationary Sources
We improve the existing achievable rate regions for causal and for zero-delay
source coding of stationary Gaussian sources under an average mean squared
error (MSE) distortion measure. To begin with, we find a closed-form expression
for the information-theoretic causal rate-distortion function (RDF) under such
distortion measure, denoted by , for first-order Gauss-Markov
processes. Rc^{it}(D) is a lower bound to the optimal performance theoretically
attainable (OPTA) by any causal source code, namely Rc^{op}(D). We show that,
for Gaussian sources, the latter can also be upper bounded as Rc^{op}(D)\leq
Rc^{it}(D) + 0.5 log_{2}(2\pi e) bits/sample. In order to analyze
for arbitrary zero-mean Gaussian stationary sources, we
introduce \bar{Rc^{it}}(D), the information-theoretic causal RDF when the
reconstruction error is jointly stationary with the source. Based upon
\bar{Rc^{it}}(D), we derive three closed-form upper bounds to the additive rate
loss defined as \bar{Rc^{it}}(D) - R(D), where R(D) denotes Shannon's RDF. Two
of these bounds are strictly smaller than 0.5 bits/sample at all rates. These
bounds differ from one another in their tightness and ease of evaluation; the
tighter the bound, the more involved its evaluation. We then show that, for any
source spectral density and any positive distortion D\leq \sigma_{x}^{2},
\bar{Rc^{it}}(D) can be realized by an AWGN channel surrounded by a unique set
of causal pre-, post-, and feedback filters. We show that finding such filters
constitutes a convex optimization problem. In order to solve the latter, we
propose an iterative optimization procedure that yields the optimal filters and
is guaranteed to converge to \bar{Rc^{it}}(D). Finally, by establishing a
connection to feedback quantization we design a causal and a zero-delay coding
scheme which, for Gaussian sources, achieves...Comment: 47 pages, revised version submitted to IEEE Trans. Information Theor
Applications of Information Nonanticipative Rate Distortion Function
The objective of this paper is to further investigate various applications of
information Nonanticipative Rate Distortion Function (NRDF) by discussing two
working examples, the Binary Symmetric Markov Source with parameter
(BSMS()) with Hamming distance distortion, and the multidimensional
partially observed Gaussian-Markov source. For the BSMS(), we give the
solution to the NRDF, and we use it to compute the Rate Loss (RL) of causal
codes with respect to noncausal codes. For the multidimensional Gaussian-Markov
source, we give the solution to the NRDF, we show its operational meaning via
joint source-channel matching over a vector of parallel Gaussian channels, and
we compute the RL of causal and zero-delay codes with respect to noncausal
codes.Comment: 5 pages, 3 figures, accepted for publication in IEEE International
Symposium on Information Theory (ISIT) proceedings, 201
An Upper Bound to Zero-Delay Rate Distortion via Kalman Filtering for Vector Gaussian Sources
We deal with zero-delay source coding of a vector Gaussian autoregressive
(AR) source subject to an average mean squared error (MSE) fidelity criterion.
Toward this end, we consider the nonanticipative rate distortion function
(NRDF) which is a lower bound to the causal and zero-delay rate distortion
function (RDF). We use the realization scheme with feedback proposed in [1] to
model the corresponding optimal "test-channel" of the NRDF, when considering
vector Gaussian AR(1) sources subject to an average MSE distortion. We give
conditions on the vector Gaussian AR(1) source to ensure asymptotic
stationarity of the realization scheme (bounded performance). Then, we encode
the vector innovations due to Kalman filtering via lattice quantization with
subtractive dither and memoryless entropy coding. This coding scheme provides a
tight upper bound to the zero-delay Gaussian RDF. We extend this result to
vector Gaussian AR sources of any finite order. Further, we show that for
infinite dimensional vector Gaussian AR sources of any finite order, the NRDF
coincides with the zero-delay RDF. Our theoretical framework is corroborated
with a simulation example.Comment: 7 pages, 6 figures, accepted for publication in IEEE Information
Theory Workshop (ITW
Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources
We deal with zero-delay source coding of a vector-valued Gauss-Markov source
subject to a mean-squared error (MSE) fidelity criterion characterized by the
operational zero-delay vector-valued Gaussian rate distortion function (RDF).
We address this problem by considering the nonanticipative RDF (NRDF) which is
a lower bound to the causal optimal performance theoretically attainable (OPTA)
function and operational zero-delay RDF. We recall the realization that
corresponds to the optimal "test-channel" of the Gaussian NRDF, when
considering a vector Gauss-Markov source subject to a MSE distortion in the
finite time horizon. Then, we introduce sufficient conditions to show existence
of solution for this problem in the infinite time horizon. For the asymptotic
regime, we use the asymptotic characterization of the Gaussian NRDF to provide
a new equivalent realization scheme with feedback which is characterized by a
resource allocation (reverse-waterfilling) problem across the dimension of the
vector source. We leverage the new realization to derive a predictive coding
scheme via lattice quantization with subtractive dither and joint memoryless
entropy coding. This coding scheme offers an upper bound to the operational
zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then
for "r" active dimensions of the vector Gauss-Markov source the gap between the
obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1
bits/vector. We further show that it is possible when we use vector
quantization, and assume infinite dimensional Gauss-Markov sources to make the
previous gap to be negligible, i.e., Gaussian NRDF approximates the operational
zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian
sources of any finite memory under mild conditions. Our theoretical framework
is demonstrated with illustrative numerical experiments.Comment: 32 pages, 9 figures, published in IEEE Journal of Selected Topics in
Signal Processin
Information Nonanticipative Rate Distortion Function and Its Applications
This paper investigates applications of nonanticipative Rate Distortion
Function (RDF) in a) zero-delay Joint Source-Channel Coding (JSCC) design based
on average and excess distortion probability, b) in bounding the Optimal
Performance Theoretically Attainable (OPTA) by noncausal and causal codes, and
computing the Rate Loss (RL) of zero-delay and causal codes with respect to
noncausal codes. These applications are described using two running examples,
the Binary Symmetric Markov Source with parameter p, (BSMS(p)) and the
multidimensional partially observed Gaussian-Markov source. For the
multidimensional Gaussian-Markov source with square error distortion, the
solution of the nonanticipative RDF is derived, its operational meaning using
JSCC design via a noisy coding theorem is shown by providing the optimal
encoding-decoding scheme over a vector Gaussian channel, and the RL of causal
and zero-delay codes with respect to noncausal codes is computed.
For the BSMS(p) with Hamming distortion, the solution of the nonanticipative
RDF is derived, the RL of causal codes with respect to noncausal codes is
computed, and an uncoded noisy coding theorem based on excess distortion
probability is shown. The information nonanticipative RDF is shown to be
equivalent to the nonanticipatory epsilon-entropy, which corresponds to the
classical RDF with an additional causality or nonanticipative condition imposed
on the optimal reproduction conditional distribution.Comment: 34 pages, 12 figures, part of this paper was accepted for publication
in IEEE International Symposium on Information Theory (ISIT), 2014 and in
book Coordination Control of Distributed Systems of series Lecture Notes in
Control and Information Sciences, 201
- …