1 On the Covariance Matrix Distortion Constraint for the Gaussian Wyner-Ziv Problem Adel Zahedi, Jan Østergaard, Senior Member, IEEE, Søren Holdt Jensen, Senior Member, IEEE, Patrick A. Naylor, Senior Member, IEEE, and Søren Bech Abstract—We first present an explicit formula R(D) for the rate-distortion function (RDF) of the vector Gaussian re- mote Wyner-Ziv problem with covariance matrix distortion constraints. To prove the lower bound, we use a particular variant of joint matrix diagonalization to establish a notion of the minimum of two symmetric positive-definite matrices. We then show that from the resulting RDF, it is possible to derive RDFs with different distortion constraints. Specifically, we rederive the RDF for the vector Gaussian remote Wyner-Ziv problem with the mean-squared error distortion constraint, and a rate-mutual information function. This is done by minimizing R(D) subject to appropriate constraints on the distortion matrix D. The key idea to solve the resulting minimization problems is to lower- bound them with simpler optimization problems and show that they lead to identical solutions. We thus illustrate the generality of the covariance matrix distortion constraint in the Wyner-Ziv setup. Index Terms—Covariance matrix distortions, Wyner-Ziv prob- lem, Gaussian multiterminal source coding, Rate-distortion func- tions, Joint matrix diagonalization I. I NTRODUCTION A. Notation We denote matrices and vectors by boldface uppercase and lowercase letters, respectively. We consider zero-mean sta- tionary Gaussian sources, which generate independent vectors x ∈ R nx , y ∈ R ny , and z ∈ R nz , being the remote source, the observation at the encoder, and the side information available to the decoder, respectively. The block diagram of the problem is shown in Fig.1. At the encoder, the sequence of observations y is encoded into another sequence u. An optimal estimation ˆ x of x is obtained from u and z at the decoder. For a given positive- definite distortion matrix D, the problem is to find the min- imum achievable rate for the encoded sequence u such that the reconstruction error e = x − ˆ x at the decoder satisfies the following: Σ e = E  (x − ˆ x)(x − ˆ x) T   D, (1) The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n ◦ ITN-GA-2012-316969. The work of J. Østergaard is financially supported by VILLUM FONDEN Young Investigator Programme, Project No. 10095. This work was presented in part at the 2014 IEEE International Symposium on Information Theory [1]. A. Zahedi, J. Østergaard, and S. H. Jensen (email: {adz, jo, shj}@es.aau.dk) are with the Department of Electronic Systems, Aalborg University, Denmark. P. Naylor (email: p.naylor@imperial.ac.uk) is with the Electrical and Elec- tronic Engineering Department, London Imperial College, UK, and S. Bech (email: sbe@es.aau.dk) is with Bang & Olufsen, Denmark and the Department of Electronic Systems, Aalborg University, Denmark. Encoder Decoder y u z E[x|u,z] Fig. 1. Block diagram of the remote source coding problem where E[·] denotes the expectation operation and Σ e is the covariance matrix of e. For random vectors x, y and z, the conditional cross- covariance of x and y given z is denoted by Σ xy|z . The n × n identity matrix is denoted by I n . We use diag{λ i ,i = 1,...,n} to show an n × n diagonal matrix having the ele- ments λ 1 ,...,λ n on its main diagonal. The set of symmetric positive-definite matrices is denoted by S + . The statement A  B (A ≻ B) means that A − B is positive semidefinite (definite). Markov chains are denoted by two-headed arrows as in x ↔ u ↔ y. We denote the determinant and trace operations by |·| and tr(·), respectively. We also make use of the following notations for briefness: (a) + Δ = max (a, 1) , (a) − Δ = min (a, 1) . Finally, if Σ x|yz ≺ D does not hold for the problem described above, the rate will become infinite. We thus assume that it holds. B. Motivation During the recent years, there has been a shift from the tra- ditional mean-squared error distortion constraint to covariance matrix distortions in the area of multiterminal source coding [3]–[9]. In general with this kind of distortion constraint, the matrix form of the target distortion gives rise to new issues compared to the scalar target distortions, which make the problem harder to solve. Consider the problem introduced in Section I-A as an example. The covariance matrix of the unknown part of the source at the decoder is Σ x|z . With a scalar target distortion d, the following two cases might arise: • tr(Σ x|z ) ≤ n x d, for which the required rate for u is zero. • tr(Σ x|z ) >n x d, which means that some nonzero rate has to be spent on u. In this case, with an optimal coding arXiv:1504.01090v1 [cs.IT] 5 Apr 2015