6 research outputs found
Concentration without Independence via Information Measures
We propose a novel approach to concentration for non-independent random
variables. The main idea is to ``pretend'' that the random variables are
independent and pay a multiplicative price measuring how far they are from
actually being independent. This price is encapsulated in the Hellinger
integral between the joint and the product of the marginals, which is then
upper bounded leveraging tensorisation properties. Our bounds represent a
natural generalisation of concentration inequalities in the presence of
dependence: we recover exactly the classical bounds (McDiarmid's inequality)
when the random variables are independent. Furthermore, in a ``large
deviations'' regime, we obtain the same decay in the probability as for the
independent case, even when the random variables display non-trivial
dependencies. To show this, we consider a number of applications of interest.
First, we provide a bound for Markov chains with finite state space. Then, we
consider the Simple Symmetric Random Walk, which is a non-contracting Markov
chain, and a non-Markovian setting in which the stochastic process depends on
its entire past. To conclude, we propose an application to Markov Chain Monte
Carlo methods, where our approach leads to an improved lower bound on the
minimum burn-in period required to reach a certain accuracy. In all of these
settings, we provide a regime of parameters in which our bound fares better
than what the state of the art can provide
Information Extraction Under Privacy Constraints
A privacy-constrained information extraction problem is considered where for
a pair of correlated discrete random variables governed by a given
joint distribution, an agent observes and wants to convey to a potentially
public user as much information about as possible without compromising the
amount of information revealed about . To this end, the so-called {\em
rate-privacy function} is introduced to quantify the maximal amount of
information (measured in terms of mutual information) that can be extracted
from under a privacy constraint between and the extracted information,
where privacy is measured using either mutual information or maximal
correlation. Properties of the rate-privacy function are analyzed and
information-theoretic and estimation-theoretic interpretations of it are
presented for both the mutual information and maximal correlation privacy
measures. It is also shown that the rate-privacy function admits a closed-form
expression for a large family of joint distributions of . Finally, the
rate-privacy function under the mutual information privacy measure is
considered for the case where has a joint probability density function
by studying the problem where the extracted information is a uniform
quantization of corrupted by additive Gaussian noise. The asymptotic
behavior of the rate-privacy function is studied as the quantization resolution
grows without bound and it is observed that not all of the properties of the
rate-privacy function carry over from the discrete to the continuous case.Comment: 55 pages, 6 figures. Improved the organization and added detailed
literature revie
On contraction coefficients, partial orders and approximation of capacities for quantum channels
The data processing inequality is the most basic requirement for any
meaningful measure of information. It essentially states that
distinguishability measures between states decrease if we apply a quantum
channel. It is the centerpiece of many results in information theory and
justifies the operational interpretation of most entropic quantities. In this
work, we revisit the notion of contraction coefficients of quantum channels,
which provide sharper and specialized versions of the data processing
inequality. A concept closely related to data processing are partial orders on
quantum channels. We discuss several quantum extensions of the well known less
noisy ordering and then relate them to contraction coefficients. We further
define approximate versions of the partial orders and show how they can give
strengthened and conceptually simple proofs of several results on approximating
capacities. Moreover, we investigate the relation to other partial orders in
the literature and their properties, particularly with regards to
tensorization. We then investigate further properties of contraction
coefficients and their relation to other properties of quantum channels, such
as hypercontractivity. Next, we extend the framework of contraction
coefficients to general f-divergences and prove several structural results.
Finally, we consider two important classes of quantum channels, namely
Weyl-covariant and bosonic Gaussian channels. For those, we determine new
contraction coefficients and relations for various partial orders.Comment: 47 pages, 2 figure