A unified approach is presented for establishing
a broad class of Cram\'er-Rao inequalities for the location parameter,
including, as special cases,
the original inequality of Cram\'er and Rao, as well as an $L^p$ version recently
established by the authors. The new approach allows for
generalized moments and Fisher information measures to be defined by convex
functions that are not necessarily homogeneous.
In particular, it is shown that associated with any log-concave random
variable whose density satisfies certain boundary conditions is a
Cram\'er-Rao inequality for which the given log-concave random
variable is the extremal. Applications to specific instances are
We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.
We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log$n$, with$n$being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the$negative$solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.
Zhi-Chao ZhangState Key Laboratory of Networking and Switching Technology, Beijing University of Posts and TelecommunicationsKeqin FengDepartment of Mathematical Sciences, Tsinghua UniversityFei GaoState Key Laboratory of Networking and Switching Technology, Beijing University of Posts and TelecommunicationsQiao-Yan WenState Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications
entangled states. We construct sets of fewer than d orthogonal maximally entangled states which are not
distinguished by one-way local operations and classical communication (LOCC) in the Hilbert space of d ⊗ d.
The proof, based on the Fourier transform of an additive group, is very simple but quite effective. Simultaneously,
our results give a general unified upper bound for the minimum number of one-way LOCC indistinguishable
maximally entangled states. This improves previous results which only showed sets of N d − 2 such states.
Finally, our results also show that previous conjectures in Zhang et al. [Z.-C. Zhang, Q.-Y. Wen, F. Gao, G.-J.
Tian, and T.-Q. Cao, Quant. Info. Proc. 13, 795 (2014)] are indeed correct.
A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.