A perturbation proof of the vector Gaussian one-help-one problem

Rate Region of the Vector Gaussian CEO Problem With the Trace Distortion Constraint

“…which means that the subspace spanned by the column vectors of V 1 belongs to the image space of Ψ 1 , i.e., V 1 ⊆ Im(Ψ 1 ). Thus by the complementary slackness conditions (14) in KKT conditions, we have B * 1 Ψ 1 = 0; as a consequence, the kernel space of B * 1 contains the image space of Ψ 1 , i.e., Ker(B * 1 ) ⊇ Im(Ψ 1 ), which implies…”

“…Following [10], [14], we use the covariance preserved transform proposed by Dembo et al in [16]. Specifically, for any γ ∈ (0, 1), define…”

“…where (a) is from B i V i = 0 of Proposition 3 in Theorem 2, (b) is from complementary slackness conditions in KKT conditions (14):…”

“…For this reason, we develop a spectral decomposition method, which can be effectively incorporated into the perturbation argument to obtain the desired inequality. It is worth mentioning that our spectral decomposition method is partly motivated by the distortion projection technique developed by Rahman and Wagner [12], [13] for the vector Gaussian onehelp-one problem (see also [14] for a direct proof based on the perturbation method).…”

New Proofs of Extremal Inequalities With Applications

“…Inspired by the recent work of [29], [31], we construct the Gaussian perturbation variable via a continuously parameterized tensorization process. Although the proposed method works in the product probability space as in [30], it is monotone path centric, and enables to leverage standard perturbation techniques [2], [21] to prove the optimality of the Gaussian solution. To show flexibility and informatics of our proposed construction, we will recover some existing extremal inequalities under the monotone path arguments, including Liu-Viswanath extremal inequality [2] and vector generalization of Costa's EPI [10].…”