We study the vector Gaussian CEO problem under logarithmic loss distortion measure. Specifically, @math agents observe independent noisy versions of a remote vector Gaussian source, and communicate independently with a decoder over rate-constrained noise-free links. The CEO also has its own Gaussian noisy observation of the source and wants to reconstruct the remote source to within some prescribed distortion level where the incurred distortion is measured under the logarithmic loss penalty criterion. We find an explicit characterization of the rate-distortion region of this model. For the proof of this result, we first extend Courtade-Weissman's result on the rate-distortion region of the DM @math -encoder CEO problem to the case in which the CEO has access to a correlated side information stream which is such that the agents' observations are independent conditionally given the side information and remote source. Next, we obtain an outer bound on the region of the vector Gaussian CEO problem by evaluating the outer bound of the DM model by means of a technique that relies on the de Bruijn identity and the properties of Fisher information. The approach is similar to Ekrem-Ulukus outer bounding technique for the vector Gaussian CEO problem under quadratic distortion measure, for which it was there found generally non-tight; but it is shown here to yield a complete characterization of the region for the case of logarithmic loss measure. Also, we show that Gaussian test channels with time-sharing exhaust the Berger-Tung inner bound, which is optimal. Furthermore, application of our results allows us to find the complete solutions of three related problems: the vector Gaussian distributed hypothesis testing against conditional independence problem, a quadratic vector Gaussian CEO problem with determinant constraint, and the vector Gaussian distributed Information Bottleneck problem.
In this paper, we study the problem of two-encoder multiterminal source coding with side information of Figure in the case of logarithmic loss distortion measure. That is, our model generalizes that of @cite_20 to the setting in which the decoder observes side information sequence that is statistically dependent on the sources to be compressed. We develop a single-letter characterization of the rate-distortion region of this model in the discrete memoryless case. In doing so, we show that a slight generalization of the Gastpar inner bound [Theorem 2] G04 for general distortion measures that accounts for time-sharing is optimal. The proof of the converse follows that of [Theorem 12] CW14 , and extends it to the case of correlated side information at decoder. It also involves a redefinition of the required auxiliary random variables appropriately. Furthermore, specializing the results to the case in which only Encoder 2 communicates with the decoder, i.e., @math , and the decoder is interested in reproducing 'soft' estimates of only the source @math , we characterize the trade-off among the complexity and accuracy of the system, i.e., a generalization of the so-called Information Bottleneck Method @cite_50 @cite_23 to the setting with side information at decoder.
@cite_50 @cite_23 @cite_20
2109053700 2150412388 2173267959 2021889595
Consider a pair of correlated Gaussian sources (X 1,X 2). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of X 1 and X 2 to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components X 1 and X 2 first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory. We then consider the case of linear reconstruction of K sources and provide an inner bound to the optimal rate-distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by ldquocorrelatedrdquo lattice-structured binning. Let (X_ k , Y_ k ) ^ _ k=1 be a sequence of independent drawings of a pair of dependent random variables X, Y . Let us say that X takes values in the finite set X . It is desired to encode the sequence X_ k in blocks of length n into a binary stream of rate R , which can in turn be decoded as a sequence X _ k , where X _ k X , the reproduction alphabet. The average distortion level is (1 n) ^ n _ k=1 E[D(X_ k , X _ k )] , where D(x, x ) 0, x X , x X , is a preassigned distortion measure. The special assumption made here is that the decoder has access to the side information Y_ k . In this paper we determine the quantity R (d) , defined as the infimum ofrates R such that (with > 0 arbitrarily small and with suitably large n )communication is possible in the above setting at an average distortion level (as defined above) not exceeding d + . The main result is that R (d) = [I(X;Z) - I(Y;Z)] , where the infimum is with respect to all auxiliary random variables Z (which take values in a finite set Z ) that satisfy: i) Y,Z conditionally independent given X ; ii) there exists a function f: Y Z X , such that E[D(X,f(Y,Z))] d . Let R_ X | Y (d) be the rate-distortion function which results when the encoder as well as the decoder has access to the side information Y_ k . In nearly all cases it is shown that when d > 0 then R (d) > R_ X|Y (d) , so that knowledge of the side information at the encoder permits transmission of the X_ k at a given distortion level using a smaller transmission rate. This is in contrast to the situation treated by Slepian and Wolf [5] where, for arbitrarily accurate reproduction of X_ k , i.e., d = for any >0 , knowledge of the side information at the encoder does not allow a reduction of the transmission rate. We consider the classical two-encoder multiterminal source coding problem where distortion is measured under logarithmic loss. We provide a single-letter description of the achievable rate distortion region for all discrete memoryless sources with finite alphabets. By 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. In this paper, we consider the problem of multifunctional compression with side information. The problem is how we can compress a source X so that the receiver is able to compute some deterministic functions f1(X, Y1), ..., fm(X, Ym), where Yi, 1 ≤ i ≤ m, are available at the receiver as side information. In [1], Wyner and Ziv considered this problem for the special case of m = 1 and f1(X, Y1) = X and derived a rate-distortion function. Yamamoto extended this result in [2] to the case of having one general function f1(X, Y1). Both of these results were in terms of an auxiliary random variable. For the case of zero distortion, in [3], Orlitsky and Roche gave an interpretation of this variable in terms of properties of the characteristic graph which led to a particular coding scheme. This result was extended in [4] by providing an achievable scheme based on colorings of the characteristic graph. In a recent work, reference [5] has considered this problem for a general tree network where intermediate nodes are allowed to perform some computations. These previous works only considered the case where the receiver only wants to compute one function (m=1). Here, we want to consider the case in which the receiver wants to compute several functions with different side information random variables and zero distortion. Our results do not depend on the fact that all functions are desired in one receiver and one can apply them to the case of having several receivers with different desired functions (i.e., functions are separable). We define a new concept named the multi-functional graph entropy which is an extension of the graph entropy defined by Korner in [6]. We show that the minimum achievable rate for this problem is equal to the conditional multi-functional graph entropy of random variable X given side informations. We also propose a coding scheme based on graph colorings to achieve this rate.