102 research outputs found
At Every Corner: Determining Corner Points of Two-User Gaussian Interference Channels
The corner points of the capacity region of the two-user Gaussian
interference channel under strong or weak interference are determined using the
notions of almost Gaussian random vectors, almost lossless addition of random
vectors, and almost linearly dependent random vectors. In particular, the
"missing" corner point problem is solved in a manner that differs from previous
works in that it avoids the use of integration over a continuum of SNR values
or of Monge-Kantorovitch transportation problems
A Lower Bound on the Entropy Rate for a Large Class of Stationary Processes and its Relation to the Hyperplane Conjecture
We present a new lower bound on the differential entropy rate of stationary
processes whose sequences of probability density functions fulfill certain
regularity conditions. This bound is obtained by showing that the gap between
the differential entropy rate of such a process and the differential entropy
rate of a Gaussian process with the same autocovariance function is bounded.
This result is based on a recent result on bounding the Kullback-Leibler
divergence by the Wasserstein distance given by Polyanskiy and Wu. Moreover, it
is related to the famous hyperplane conjecture, also known as slicing problem,
in convex geometry originally stated by J. Bourgain. Based on an entropic
formulation of the hyperplane conjecture given by Bobkov and Madiman we discuss
the relation of our result to the hyperplane conjecture.Comment: presented at the 2016 IEEE Information Theory Workshop (ITW),
Cambridge, U
Maximal power output of a stochastic thermodynamic engine
Classical thermodynamics aimed to quantify the efficiency of thermodynamic engines, by bounding the maximal amount of mechanical energy produced, compared to the amount of heat required. While this was accomplished early on, by Carnot and Clausius, the more practical problem to quantify limits of power that can be delivered, remained elusive due to the fact that quasistatic processes require infinitely slow cycling, resulting in a vanishing power output. Recent insights, drawn from stochastic models, appear to bridge the gap between theory and practice in that they lead to physically meaningful expressions for the dissipation cost in operating a thermodynamic engine over a finite time window. Indeed, the problem to optimize power can be expressed as a stochastic control problem. Building on this framework of stochastic thermodynamics we derive bounds on the maximal power that can be drawn by cycling an overdamped ensemble of particles via a time-varying potential while alternating contact with heat baths of different temperature (Tc cold, and Th hot). Specifically, assuming a suitable bound M on the spatial gradient of the controlling potential, we show that the maximal achievable power is bounded by [Formula presented]. Moreover, we show that this bound can be reached to within a factor of [Formula presented] by operating the cyclic thermodynamic process with a quadratic potential
The Quantum Wasserstein Distance of Order 1
We propose a generalization of the Wasserstein distance of order 1 to the
quantum states of qudits. The proposal recovers the Hamming distance for
the vectors of the canonical basis, and more generally the classical
Wasserstein distance for quantum states diagonal in the canonical basis. The
proposed distance is invariant with respect to permutations of the qudits and
unitary operations acting on one qudit and is additive with respect to the
tensor product. Our main result is a continuity bound for the von Neumann
entropy with respect to the proposed distance, which significantly strengthens
the best continuity bound with respect to the trace distance. We also propose a
generalization of the Lipschitz constant to quantum observables. The notion of
quantum Lipschitz constant allows us to compute the proposed distance with a
semidefinite program. We prove a quantum version of Marton's transportation
inequality and a quantum Gaussian concentration inequality for the spectrum of
quantum Lipschitz Moreover, we derive bounds on the contraction coefficients of
shallow quantum circuits and of the tensor product of one-qudit quantum
channels with respect to the proposed distance. We discuss other possible
applications in quantum machine learning, quantum Shannon theory, and quantum
many-body systems
Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation
This paper studies convergence of empirical measures smoothed by a Gaussian
kernel. Specifically, consider approximating , for
, by
, where is the empirical measure,
under different statistical distances. The convergence is examined in terms of
the Wasserstein distance, total variation (TV), Kullback-Leibler (KL)
divergence, and -divergence. We show that the approximation error under
the TV distance and 1-Wasserstein distance () converges at rate
in remarkable contrast to a typical
rate for unsmoothed (and ). For the
KL divergence, squared 2-Wasserstein distance (), and
-divergence, the convergence rate is , but only if
achieves finite input-output mutual information across the additive
white Gaussian noise channel. If the latter condition is not met, the rate
changes to for the KL divergence and , while
the -divergence becomes infinite - a curious dichotomy. As a main
application we consider estimating the differential entropy
in the high-dimensional regime. The distribution
is unknown but i.i.d samples from it are available. We first show that
any good estimator of must have sample complexity
that is exponential in . Using the empirical approximation results we then
show that the absolute-error risk of the plug-in estimator converges at the
parametric rate , thus establishing the minimax
rate-optimality of the plug-in. Numerical results that demonstrate a
significant empirical superiority of the plug-in approach to general-purpose
differential entropy estimators are provided.Comment: arXiv admin note: substantial text overlap with arXiv:1810.1158
On the High-SNR Capacity of the Gaussian Interference Channel and New Capacity Bounds
The best outer bound on the capacity region of the two-user Gaussian
Interference Channel (GIC) is known to be the intersection of regions of
various bounds including genie-aided outer bounds, in which a genie provides
noisy input signals to the intended receiver. The Han and Kobayashi (HK) scheme
provides the best known inner bound. The rate difference between the best known
lower and upper bounds on the sum capacity remains as large as 1 bit per
channel use especially around , where is the symmetric power
constraint and is the symmetric real cross-channel coefficient. In this
paper, we pay attention to the \emph{moderate interference regime} where
. We propose a new upper-bounding technique
that utilizes noisy observation of interfering signals as genie signals and
applies time sharing to the genie signals at the receivers. A conditional
version of the worst additive noise lemma is also introduced to derive new
capacity bounds. The resulting upper (outer) bounds on the sum capacity
(capacity region) are shown to be tighter than the existing bounds in a certain
range of the moderate interference regime. Using the new upper bounds and the
HK lower bound, we show that characterizes the capacity of the symmetric real
GIC to within bit per channel use in the moderate interference regime
at any signal-to-noise ratio (SNR). We further establish a high-SNR
characterization of the symmetric real GIC, where the proposed upper bound is
at most bit far from a certain HK achievable scheme with Gaussian
signaling and time sharing for . In particular,
is achievable at high SNR by the proposed HK scheme and turns out to be the
high-SNR capacity at least at .Comment: Submitted to IEEE Transactions on Information Theory on June 2015,
revised on November 2016, and accepted for publication on Feb. 28, 201
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