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Estimating Mixture Entropy with Pairwise Distances
Mixture distributions arise in many parametric and non-parametric settings --
for example, in Gaussian mixture models and in non-parametric estimation. It is
often necessary to compute the entropy of a mixture, but, in most cases, this
quantity has no closed-form expression, making some form of approximation
necessary. We propose a family of estimators based on a pairwise distance
function between mixture components, and show that this estimator class has
many attractive properties. For many distributions of interest, the proposed
estimators are efficient to compute, differentiable in the mixture parameters,
and become exact when the mixture components are clustered. We prove this
family includes lower and upper bounds on the mixture entropy. The Chernoff
-divergence gives a lower bound when chosen as the distance function,
with the Bhattacharyya distance providing the tightest lower bound for
components that are symmetric and members of a location family. The
Kullback-Leibler divergence gives an upper bound when used as the distance
function. We provide closed-form expressions of these bounds for mixtures of
Gaussians, and discuss their applications to the estimation of mutual
information. We then demonstrate that our bounds are significantly tighter than
well-known existing bounds using numeric simulations. This estimator class is
very useful in optimization problems involving maximization/minimization of
entropy and mutual information, such as MaxEnt and rate distortion problems.Comment: Corrects several errata in published version, in particular in
Section V (bounds on mutual information
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
Alternative fidelity measure for quantum states
We propose an alternative fidelity measure (namely, a measure of the degree
of similarity) between quantum states and benchmark it against a number of
properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple
function of the linear entropy and the Hilbert-Schmidt inner product between
the given states and is thus, in comparison, not as computationally demanding.
It also features several remarkable properties such as being jointly concave
and satisfying all of "Jozsa's axioms". The trade-off, however, is that it is
supermultiplicative and does not behave monotonically under quantum operations.
In addition, new metrics for the space of density matrices are identified and
the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is
established.Comment: 12 pages, 3 figures. v2 includes minor changes, new references and
new numerical results (Sec. IV
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