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.
Recently, the achievable rate-distortion region of the two-encoder multiterminal source coding problem was completely characterized by Courtade and Weissman [Theorem 6] CW14 for another important special case, that of a logarithmic distortion measure. More precisely, in @cite_20 Courtade and Weissman study the model of Figure side information, i.e., @math , and with the decoder restricted to generate 'soft' estimates of the sources sequences. Namely, @math , @math , is a probability distribution on the alphabet @math of source @math and @math is the relative entropy (i.e., Kullback-Leibler divergence) between the empirical distribution of the event @math and the estimate @math . Using a particularly useful property of the logarithmic loss distortion measure, which states that the expected distortion is lower bounded by conditional entropy, the authors characterize the rate-distortion region of both the CEO problem [Theorem 3] CW14 and the two-encoder distributed source coding problem [Theorem 6] CW14 . Their results, which are valid when there is side information at the decoder, i.e., @math , require no specific assumptions on the sources, other than that they have finite alphabets.
@cite_20
2173267959 2109053700 2043312033 2021889595
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. 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. We consider the CEO problem for non-regular source distributions (such as uniform or truncated Gaussian). A group of agents observe independently corrupted versions of data and transmit coded versions over rate-limited links to a CEO. The CEO then estimates the underlying data based on the received coded observations. Agents are not allowed to convene before transmitting their observations. This formulation is motivated by the practical problem of a firm’s CEO estimating (non-regular) beliefs about a sequence of events, before acting on them. Agents’ observations are modeled as jointly distributed with the underlying data through a given conditional probability density function. We study the asymptotic behavior of the minimum achievable mean squared error distortion at the CEO in the limit when the number of agents @math and the sum rate @math tend to infinity. We establish a @math convergence of the distortion, an intermediate regime of performance between the exponential behavior in discrete CEO problems [Berger, Zhang, and Viswanathan (1996)], and the @math behavior in Gaussian CEO problems [Viswanathan and Berger (1997)]. Achievability is proved by a layered architecture with scalar quantization, distributed entropy coding, and midrange estimation. The converse is proved using the Bayesian Chazan–Zakai–Ziv bound. 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.