A Causal rate distortion function with a general fidelity criterion is formulated on abstract alphabets and the optimal reconstruction kernel is derived, which consists of a product of causal kernels. In the process, general abstract spaces are introduced to show existence of the minimizing kernel using weak*-convergence. Certain properties of the causal rate distortion function are presented.
In this paper, we consider the mismatched rate-distortion problem, in which the encoding is done using a codebook, and the encoder chooses the minimum-distortion codeword according to a mismatched distortion function that differs from the true one. For the case of discrete memoryless sources, we establish achievable rate-distortion bounds using multi-user coding techniques, namely, superposition coding and expurgated parallel coding. We study examples where these attain the matched rate-distortion trade-off but a standard ensemble with independent codewords fails to do so. On the other hand, i…
A causal rate distortion function with a general fidelity criterion is formulated on abstract alphabets and a coding theorem is derived. Existence of the minimizing kernel is shown using the topology of weak convergence of probability measures. The optimal reconstruction kernel is derived, which is causal, and certain properties of the causal rate distortion function are presented.
A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including conditional typicality lemma, joint typicality lemma, and packing and covering lemmas, which are fundamental tools for deriving many inner bounds of various multi-terminal coding problems, are obtained in terms of the proposed notion. This enables us to directly generalize lots of results on finite alphabet problems to general problems involving abstract alph…
Let G be a finite strongly connected aperiodic directed graph in which each edge carries a label from a finite alphabet A. Then G induces a trellis coded quantizer for encoding an alphabet A memoryless source. A source sequence of long finite length is encoded by finding a path in G of that length whose sequence of labels is closest in Hamming distance to the source sequence; finding the minimum distance path is a dynamic programming problem that is solved using the Viterbi algorithm. We show how a Markov chain can be used to obtain a closed form expression for the asymptotic expected Hamming …
Motivated from the fact that universal source coding on countably infinite alphabets is not feasible, this work introduces the notion of almost lossless source coding. Analog to the weak variable-length source coding problem studied by Han (IEEE TIT, 2000, 46, 1217-1226), almost lossless source coding aims at relaxing the lossless block-wise assumption to allow an average per-letter distortion that vanishes asymptotically as the block-length tends to infinity. In this setup, we show on one hand that Shannon entropy characterizes the minimum achievable rate (similarly to the case of finite alph…
In successive refinement of information, the decoder refines its representation of the source progressively as it receives more encoded bits. The rate-distortion region of successive refinement describes the minimum rates required to attain the target distortions at each decoding stage. In this paper, we derive a parametric characterization of the rate-distortion region for successive refinement of abstract sources. Our characterization extends Csiszar's result to successive refinement, and generalizes a result by Tuncel and Rose, applicable for finite alphabet sources, to abstract sources. Th…
The problem of joint universal source coding and identification is considered in the setting of fixed-rate lossy coding of continuous-alphabet memoryless sources. For a wide class of bounded distortion measures, it is shown that any compactly parametrized family of $\R^d$-valued i.i.d. sources with absolutely continuous distributions satisfying appropriate smoothness and Vapnik--Chervonenkis learnability conditions, admits a joint scheme for universal lossy block coding and parameter estimation, such that when the block length $n$ tends to infinity, the overhead per-letter rate and the distort…
We consider the classical two-encoder multiterminal source coding problem where distortion is measured under logarithmic loss. We provide a single-letter characterization of the achievable rate distortion region for arbitrarily correlated sources with finite alphabets. In doing so, we also give the rate distortion region for the $m$-encoder CEO problem (also under logarithmic loss). Several applications and examples are given.
Motivated by the lossy compression of an active-vision video stream, we consider the problem of finding the rate-distortion function of an arbitrarily varying source (AVS) composed of a finite number of subsources with known distributions. Berger's paper `The Source Coding Game', \emph{IEEE Trans. Inform. Theory}, 1971, solves this problem under the condition that the adversary is allowed only strictly causal access to the subsource realizations. We consider the case when the adversary has access to the subsource realizations non-causally. Using the type-covering lemma, this new rate-distortio…
A lossy source code $\mathcal{C}$ with rate $R$ for a discrete memoryless source $S$ is called subset-universal if for every $0<R'< R$, almost every subset of $2^{nR'}$ of its codewords achieves average distortion close to the source's distortion-rate function $D(R')$. In this paper we prove the asymptotic existence of such codes. Moreover, we show the asymptotic existence of a code that is subset-universal with respect to all sources with the same alphabet.
This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable "test functions." In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to a…
In this paper we analyze the joint rate distortion function (RDF), for a tuple of correlated sources taking values in abstract alphabet spaces (i.e., continuous) subject to two individual distortion criteria. First, we derive structural properties of the realizations of the reproduction Random Variables (RVs), which induce the corresponding optimal test channel distributions of the joint RDF. Second, we consider a tuple of correlated multivariate jointly Gaussian RVs, $X_1 : Ω\rightarrow {\mathbb R}^{p_1}, X_2 : Ω\rightarrow {\mathbb R}^{p_2}$ with two square-error fidelity criteria, and we de…
This paper studies fixed-rate randomized vector quantization under the constraint that the quantizer's output has a given fixed probability distribution. A general representation of randomized quantizers that includes the common models in the literature is introduced via appropriate mixtures of joint probability measures on the product of the source and reproduction alphabets. Using this representation and results from optimal transport theory, the existence of an optimal (minimum distortion) randomized quantizer having a given output distribution is shown under various conditions. For sources…
A privacy-utility tradeoff is developed for an arbitrary set of finite-alphabet source distributions. Privacy is quantified using differential privacy (DP), and utility is quantified using expected Hamming distortion maximized over the set of distributions. The family of source distribution sets (source sets) is categorized into three classes, based on different levels of prior knowledge they capture. For source sets whose convex hull includes the uniform distribution, symmetric DP mechanisms are optimal. For source sets whose probability values have a fixed monotonic ordering, asymmetric DP m…
Today's networks are controlled assuming pre-compressed and packetized data. For video, this assumption of data packets abstracts out one of the key aspects - the lossy compression problem. Therefore, first, this paper develops a framework for network control that incorporates both source-rate and source-distortion. Next, it decomposes the network control problem into an application-layer compression control, a transport-layer congestion control and a network-layer scheduling. It is shown that this decomposition is optimal for concave utility functions. Finally, this paper derives further insi…
Beckman Institute for Advanced Science and Technology, University of Illinois（伊利诺伊大学贝克曼先进科学技术研究所）；Duke University（杜克大学）
We consider the problem of joint universal variable-rate lossy coding and identification for parametric classes of stationary $β$-mixing sources with general (Polish) alphabets. Compression performance is measured in terms of Lagrangians, while identification performance is measured by the variational distance between the true source and the estimated source. Provided that the sources are mixing at a sufficiently fast rate and satisfy certain smoothness and Vapnik-Chervonenkis learnability conditions, it is shown that, for bounded metric distortions, there exist universal schemes for joint los…
This paper considers lossy source coding of $n$-dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion $d$ no greater than $ε$, which is simpler than known dispersion-based approximations. Our approach takes inspiration in the celebrated classical result stating that the Shannon lower bound to rate-distortion function becomes tight in the limit $d \to 0$. We formulate an abstract version of the Shannon lower bound that recovers both the classical Shannon lower bound and the rate-distortion…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Max Planck Institute for Biolog. Cybernetics（马克斯·普朗克生物控制论研究所）
A distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise. From an information theoretic point of view abstractions are desirable because they allow for very efficient information processing. In artificial systems abstractions are often implemented through computationally costly formations of groups or clusters. In this work we establish the relation between the free-energy framework for decision making and rate-distortion theory and demonstrate how the application of rate-disto…
Wyner's common information was originally defined for a pair of dependent discrete random variables. Its significance is largely reflected in, hence also confined to, several existing interpretations in various source coding problems. This paper attempts to both generalize its definition and to expand its practical significance by providing a new operational interpretation. The generalization is two-folded: the number of dependent variables can be arbitrary, so are the alphabet of those random variables. New properties are determined for the generalized Wyner's common information of N dependen…
An effective 'on-the-fly' mechanism for stochastic lossy coding of Markov sources using string matching techniques is proposed in this paper. Earlier work has shown that the rate-distortion bound can be asymptotically achieved by a 'natural type selection' (NTS) mechanism which iteratively encodes asymptotically long source strings (from an unknown source distribution P) and regenerates the codebook according to a maximum likelihood distribution framework, after observing a set of K codewords to 'd-match' (i.e., satisfy the distortion constraint for) a respective set of K source words. This re…
The relation between nonanticipative Rate Distortion Function (RDF) and filtering theory is discussed on abstract spaces. The relation is established by imposing a realizability constraint on the reconstruction conditional distribution of the classical RDF. Existence of the extremum solution of the nonanticipative RDF is shown using weak$^*$-convergence on appropriate topology. The extremum reconstruction conditional distribution is derived in closed form, for the case of stationary processes. The realization of the reconstruction conditional distribution which achieves the infimum of the nona…
A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions
Tsinghua University（清华大学）；Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China（电子科技大学长三角研究院（湖州））；Huawei Technologies Co. Ltd.（华为技术有限公司）；University of Science and Technology of China（中国科学技术大学）
本文改进 Blahut-Arimoto 算法，通过 Newton 求根动态更新乘子，直接计算给定失真下的 RD 及 DR 函数，并证明其 O(1/n) 收敛率与运算复杂度，实验显示显著加速。
Consider a memoryless degraded broadcast channel (DBC) in which the channel output is a single-letter function of the channel input and the channel noise. As examples, for the Gaussian broadcast channel (BC) this single-letter function is regular Euclidian addition and for the binary-symmetric BC this single-letter function is Galois-Field-two addition. This paper identifies several classes of discrete memoryless DBCs for which a relatively simple encoding scheme, which we call natural encoding, achieves capacity. Natural Encoding (NE) combines symbols from independent codebooks (one for each …
