10,459 research outputs found
Abstract alphabet distortion-rate functions
Two definitions have been given for the distortion-rate function of a sourceuser pair—one involving test channel induced pair probability measures and the other involving general pair probability measures. It is established that both definitions are equivalent for all source-user pairs. Examples are given which exhibit some kinds of the possible pathological behavior of the distortionrate function for general source-user pairs
Empirical processes, typical sequences and coordinated actions in standard Borel spaces
This paper proposes a new notion of typical sequences on a wide class of
abstract alphabets (so-called standard Borel spaces), which is based on
approximations of memoryless sources by empirical distributions uniformly over
a class of measurable "test functions." In the finite-alphabet case, we can
take all uniformly bounded functions and recover the usual notion of strong
typicality (or typicality under the total variation distance). For a general
alphabet, however, this function class turns out to be too large, and must be
restricted. With this in mind, we define typicality with respect to any
Glivenko-Cantelli function class (i.e., a function class that admits a Uniform
Law of Large Numbers) and demonstrate its power by giving simple derivations of
the fundamental limits on the achievable rates in several source coding
scenarios, in which the relevant operational criteria pertain to reproducing
empirical averages of a general-alphabet stationary memoryless source with
respect to a suitable function class.Comment: 14 pages, 3 pdf figures; accepted to IEEE Transactions on Information
Theor
Successive Refinement of Abstract Sources
In successive refinement of information, the decoder refines its
representation of the source progressively as it receives more encoded bits.
The rate-distortion region of successive refinement describes the minimum rates
required to attain the target distortions at each decoding stage. In this
paper, we derive a parametric characterization of the rate-distortion region
for successive refinement of abstract sources. Our characterization extends
Csiszar's result to successive refinement, and generalizes a result by Tuncel
and Rose, applicable for finite alphabet sources, to abstract sources. This
characterization spawns a family of outer bounds to the rate-distortion region.
It also enables an iterative algorithm for computing the rate-distortion
region, which generalizes Blahut's algorithm to successive refinement. Finally,
it leads a new nonasymptotic converse bound. In all the scenarios where the
dispersion is known, this bound is second-order optimal.
In our proof technique, we avoid Karush-Kuhn-Tucker conditions of optimality,
and we use basic tools of probability theory. We leverage the Donsker-Varadhan
lemma for the minimization of relative entropy on abstract probability spaces.Comment: Extended version of a paper presented at ISIT 201
Variable-rate source coding theorems for stationary nonergodic sources
For a stationary ergodic source, the source coding theorem and its converse imply that the optimal performance theoretically achievable by a fixed-rate or variable-rate block quantizer is equal to the distortion-rate function, which is defined as the infimum of an expected distortion subject to a mutual information constraint. For a stationary nonergodic source, however, the. Distortion-rate function cannot in general be achieved arbitrarily closely by a fixed-rate block code. We show, though, that for any stationary nonergodic source with a Polish alphabet, the distortion-rate function can be achieved arbitrarily closely by a variable-rate block code. We also show that the distortion-rate function of a stationary nonergodic source has a decomposition as the average of the distortion-rate functions of the source's stationary ergodic components, where the average is taken over points on the component distortion-rate functions having the same slope. These results extend previously known results for finite alphabets
Estimation of the Rate-Distortion Function
Motivated by questions in lossy data compression and by theoretical
considerations, we examine the problem of estimating the rate-distortion
function of an unknown (not necessarily discrete-valued) source from empirical
data. Our focus is the behavior of the so-called "plug-in" estimator, which is
simply the rate-distortion function of the empirical distribution of the
observed data. Sufficient conditions are given for its consistency, and
examples are provided to demonstrate that in certain cases it fails to converge
to the true rate-distortion function. The analysis of its performance is
complicated by the fact that the rate-distortion function is not continuous in
the source distribution; the underlying mathematical problem is closely related
to the classical problem of establishing the consistency of maximum likelihood
estimators. General consistency results are given for the plug-in estimator
applied to a broad class of sources, including all stationary and ergodic ones.
A more general class of estimation problems is also considered, arising in the
context of lossy data compression when the allowed class of coding
distributions is restricted; analogous results are developed for the plug-in
estimator in that case. Finally, consistency theorems are formulated for
modified (e.g., penalized) versions of the plug-in, and for estimating the
optimal reproduction distribution.Comment: 18 pages, no figures [v2: removed an example with an error; corrected
typos; a shortened version will appear in IEEE Trans. Inform. Theory
- …