15 research outputs found
Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes
Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma,
where the output entropy of a binary symmetric channel is bounded in terms of
the average entropy of the input projected on a random subset of coordinates.
Here, this result is applied for deriving novel lower bounds on the entropy
rate of binary hidden Markov processes. For symmetric underlying Markov
processes, our bound improves upon the best known bound in the very noisy
regime. The nonsymmetric case is also considered, and explicit bounds are
derived for Markov processes that satisfy the -RLL constraint
How to Quantize Outputs of a Binary Symmetric Channel to Bits?
Suppose that is obtained by observing a uniform Bernoulli random vector
through a binary symmetric channel with crossover probability .
The "most informative Boolean function" conjecture postulates that the maximal
mutual information between and any Boolean function is
attained by a dictator function. In this paper, we consider the "complementary"
case in which the Boolean function is replaced by
, namely, an bit
quantizer, and show that
for any such . Thus, in this case, the optimal function is of the form
.Comment: 5 pages, accepted ISIT 201
Boolean functions: noise stability, non-interactive correlation distillation, and mutual information
Let be the noise operator acting on Boolean functions , where is the noise parameter. Given
and fixed mean , which Boolean function has the
largest -th moment ? This question has
close connections with noise stability of Boolean functions, the problem of
non-interactive correlation distillation, and Courtade-Kumar's conjecture on
the most informative Boolean function. In this paper, we characterize
maximizers in some extremal settings, such as low noise (
is close to 0), high noise ( is close to 1/2), as well as
when is large. Analogous results are also established in
more general contexts, such as Boolean functions defined on discrete torus
and the problem of noise stability in a tree
model.Comment: Corrections of some inaccuracie