473 research outputs found
R\'enyi Entropy Power Inequalities via Normal Transport and Rotation
Following a recent proof of Shannon's entropy power inequality (EPI), a
comprehensive framework for deriving various EPIs for the R\'enyi entropy is
presented that uses transport arguments from normal densities and a change of
variable by rotation. Simple arguments are given to recover the previously
known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with
constant c and a modification with exponent {\alpha} of previous works. In
particular, for log-concave densities, we obtain a simple transportation proof
of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea
Two Measures of Dependence
Two families of dependence measures between random variables are introduced.
They are based on the R\'enyi divergence of order and the relative
-entropy, respectively, and both dependence measures reduce to
Shannon's mutual information when their order is one. The first
measure shares many properties with the mutual information, including the
data-processing inequality, and can be related to the optimal error exponents
in composite hypothesis testing. The second measure does not satisfy the
data-processing inequality, but appears naturally in the context of distributed
task encoding.Comment: 40 pages; 1 figure; published in Entrop
Strong converse for the quantum capacity of the erasure channel for almost all codes
A strong converse theorem for channel capacity establishes that the error
probability in any communication scheme for a given channel necessarily tends
to one if the rate of communication exceeds the channel's capacity.
Establishing such a theorem for the quantum capacity of degradable channels has
been an elusive task, with the strongest progress so far being a so-called
"pretty strong converse". In this work, Morgan and Winter proved that the
quantum error of any quantum communication scheme for a given degradable
channel converges to a value larger than in the limit of many
channel uses if the quantum rate of communication exceeds the channel's quantum
capacity. The present paper establishes a theorem that is a counterpart to this
"pretty strong converse". We prove that the large fraction of codes having a
rate exceeding the erasure channel's quantum capacity have a quantum error
tending to one in the limit of many channel uses. Thus, our work adds to the
body of evidence that a fully strong converse theorem should hold for the
quantum capacity of the erasure channel. As a side result, we prove that the
classical capacity of the quantum erasure channel obeys the strong converse
property.Comment: 15 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014
Samplers and Extractors for Unbounded Functions
Blasiok (SODA\u2718) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions f from {0,1}^m to the real numbers such that f(U_m) has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact averaging samplers for the broader class of subexponential functions) that match the best known constructions of averaging samplers for [0,1]-bounded functions in the regime of parameters where the approximation error epsilon and failure probability delta are subconstant. Our constructions are established via an extension of the standard notion of randomness extractor (Nisan and Zuckerman, JCSS\u2796) where the error is measured by an arbitrary divergence rather than total variation distance, and a generalization of Zuckerman\u27s equivalence (Random Struct. Alg.\u2797) between extractors and samplers. We believe that the framework we develop, and specifically the notion of an extractor for the Kullback-Leibler (KL) divergence, are of independent interest. In particular, KL-extractors are stronger than both standard extractors and subgaussian samplers, but we show that they exist with essentially the same parameters (constructively and non-constructively) as standard extractors
Hypothesis Testing Interpretations and Renyi Differential Privacy
Differential privacy is a de facto standard in data privacy, with
applications in the public and private sectors. A way to explain differential
privacy, which is particularly appealing to statistician and social scientists
is by means of its statistical hypothesis testing interpretation. Informally,
one cannot effectively test whether a specific individual has contributed her
data by observing the output of a private mechanism---any test cannot have both
high significance and high power.
In this paper, we identify some conditions under which a privacy definition
given in terms of a statistical divergence satisfies a similar interpretation.
These conditions are useful to analyze the distinguishability power of
divergences and we use them to study the hypothesis testing interpretation of
some relaxations of differential privacy based on Renyi divergence. This
analysis also results in an improved conversion rule between these definitions
and differential privacy
Generalized distribution based diversity measurement: Survey and unification
Social and natural sciences employ a number of different measures of diversity. The presents paper surveys those depending on the distribution of abundances among a given set of categories. Characteristic properties of the measures are generalized and a unifying notation is derived. It is argued that such unification enables scientists and decision makers to measure distribution based diversity in a new, more flexible manner, and represents a useful complement to models of generalized feature based diversity, such as Nehring and Puppeâs (2002) theory of diversity.Diversity measurement; Generalization; Nonâadditivity; Concavity; Numbers equivalence
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