A Rigorous Approach to High-Resolution Entropy-Constrained Vector Quantization

The nonnegativity of relative entropy implies that the differential entropy of a random vector X with probability density function (pdf) f is upper-bounded by -E[log g(X)]for any arbitrary pdf g. Using this inequality with a cleverly chosen g, we derive a lower bound on the asymptotic excess rate of entropy-constrained vector quantization for d-dimensional sources and rth-power distortion, where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate-distortion function in the limit as the distortion tends to zero. Specialized to the one-dimensional case, this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer, thereby recovering the result by Gish and Pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero. Furthermore, in the one-dimensional case, the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal. This condition implies that any sequence of asymptotically optimal almost-regular quantizers must converge to a uniform quantizer as the distortion tends to zero. While the obtained lower bound itself is not novel, to the best of our knowledge, we present the first rigorous derivation that follows the direct approach by Gish and Pierce without resorting to heuristic high-resolution approximations commonly found in the quantization literature. Furthermore, our derivation holds for all d-dimensional sources having finite differential entropy and whose integer part has finite entropy. In contrast to Gish and Pierce, we do not require additional constraints on the continuity or decay of the source pdf.This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 714161), from the 7th European Union Framework Programme under Grant 333680, from the Ministerio de Economía y Competitividad of Spain under Grants TEC2013-41718-R, RYC-2014-16332, IJCI-2015-27020, TEC2015-69648-REDC, and TEC2016-78434-C3-3-R (AEI/FEDER, EU), and from the Comunidad de Madrid under Grant S2103/ICE-2845. The material in this paper was presented in part at the 2016 IEEE International Symposium on Information Theory, Barcelona, Spain, July 2016

Quantization of Midisuperspace Models

We give a comprehensive review of the quantization of midisuperspace models. Though the main focus of the paper is on quantum aspects, we also provide an introduction to several classical points related to the definition of these models. We cover some important issues, in particular, the use of the principle of symmetric criticality as a very useful tool to obtain the required Hamiltonian formulations. Two main types of reductions are discussed: those involving metrics with two Killing vector fields and spherically symmetric models. We also review the more general models obtained by coupling matter fields to these systems. Throughout the paper we give separate discussions for standard quantizations using geometrodynamical variables and those relying on loop quantum gravity inspired methods.Comment: To appear in Living Review in Relativit

Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint

Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$)mismatch results to general values of the entropy order parameter $\alpha

Quantum Geometry and Gravity: Recent Advances

Over the last three years, a number of fundamental physical issues were addressed in loop quantum gravity. These include: A statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons; a natural resolution of the big-bang singularity; the development of spin-foam models which provide background independent path integral formulations of quantum gravity and `finiteness proofs' of some of these models; and, the introduction of semi-classical techniques to make contact between the background independent, non-perturbative theory and the perturbative, low energy physics in Minkowski space. These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field. The goal of this article is to communicate these advances in general physical terms, accessible to researchers in all areas of gravitational physics represented in this conference.Comment: 24 pages, 2 figures; report of the plenary talk at the 16th International Conference on General Relativity and Gravitation, held at Durban, S. Africa in July 200

Multiple Description Quantization via Gram-Schmidt Orthogonalization

The multiple description (MD) problem has received considerable attention as a model of information transmission over unreliable channels. A general framework for designing efficient multiple description quantization schemes is proposed in this paper. We provide a systematic treatment of the El Gamal-Cover (EGC) achievable MD rate-distortion region, and show that any point in the EGC region can be achieved via a successive quantization scheme along with quantization splitting. For the quadratic Gaussian case, the proposed scheme has an intrinsic connection with the Gram-Schmidt orthogonalization, which implies that the whole Gaussian MD rate-distortion region is achievable with a sequential dithered lattice-based quantization scheme as the dimension of the (optimal) lattice quantizers becomes large. Moreover, this scheme is shown to be universal for all i.i.d. smooth sources with performance no worse than that for an i.i.d. Gaussian source with the same variance and asymptotically optimal at high resolution. A class of low-complexity MD scalar quantizers in the proposed general framework also is constructed and is illustrated geometrically; the performance is analyzed in the high resolution regime, which exhibits a noticeable improvement over the existing MD scalar quantization schemes.Comment: 48 pages; submitted to IEEE Transactions on Information Theor