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 → R p 1 , X 2 : Ω → R p 2 with two square-error fidelity criteria, and we derive additional structural properties of the optimal realizations, and use these to characterize the RDF as a convex optimization problem with respect to the parameters of the realizations. We show that the computation of the joint RDF can be performed by semidefinite programming. Further, we derive closed-form expressions of the joint RDF, such that Gray's [1] lower bounds hold with equality, and verify their consistency with the semidefinite programming computations. I. LITERATURE REVIEW, PROBLEM FORMULATION, AND MAIN CONTRIBUTIONS A. Literature ReviewGray [1, Theorem 3.1, Corollary 3.1] derived lower bounds on the joint rate distortion functions (RDFs), of a tuple of Random Variables (RVs) taking values in arbitrary, abstract spaces, X 1 : Ω → X 1 , X 2 : Ω → X 2 , with a weighted distortion, expressed in terms of conditional RDFs, and marginal RDFs. Gray and Wyner in [2], characterized the rate distortion region of a tuple of correlated RVs, using the joint, conditional and marginal RDFs. Xiao and Luo [3, Theorem 6] derived the closed-form expression of the joint RDF for a tuple of scalar-valued correlated Gaussian RVs, with two square-error distortion criteria, while Lapidoth and Tinguely [4] re-derived Xiao's and Luo's joint RDF using an alternative method. Xu, Liu and Chen [5] and Viswanatha, Akyol and Rose [6], generalized Wyner's common information [7] to its lossy counterpart, as the minimum common message rate on the Gray and Wyner rate region with sum rate equal to the joint RDF with two individual distortion functions. The analysis in [5], [6], includes the application of a tuple of scalar-valued, jointly Gaussian RVs. More recent work on rates that lie on the Gray and Wyner rate region are found in [8]. SourceEncoder: f E (•)

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We consider the problem of correcting insertion and deletion errors in the d-dimensional space. This problem is well understood for vectors (one-dimensional space) and was recently studied for arrays (two-dimensional space). For vectors and arrays, the problem is motivated by several practical applications such as DNA-based storage and racetrack memories. From a theoretical perspective, it is interesting to know whether the same properties of insertion/deletion correcting codes generalize to the d-dimensional space. In this work, we show that the equivalence between insertion and deletion correcting codes generalizes to the d-dimensional space. As a particular result, we show the following missing equivalence for arrays: a code that can correct tr and tc row/column deletions can correct any combination of t ins r + t del r = tr and t ins c + t del c = tc row/column insertions and deletions. The fundamental limit on the redundancy and a construction of insertion/deletion correcting codes in the ddimensional space remain open for future work.

The joint nonanticipative rate distortion function (NRDF) for a tuple of random processes with individual fidelity criteria is considered. Structural properties of optimal test channel distributions are derived. Further, for the application example of the joint NRDF of a tuple of jointly multivariate Gaussian Markov processes with individual square-error fidelity criteria, a realization of the reproduction processes which induces the optimal test channel distribution is derived, and the corresponding joint NRDF is characterized. The analysis of the simplest example, of a tuple of scalar correlated Markov processes, illustrates many of the challenging aspects of such problems.