Asymptotics of Nonlinear LSE Precoders with Applications to Transmit Antenna Selection

This paper studies the large-system performance of Least Square Error (LSE) precoders which~minimize~the~input-output distortion over an arbitrary support subject to a general penalty function. The asymptotics are determined via the replica method in a general form which encloses the Replica Symmetric (RS) and Replica Symmetry Breaking (RSB) ans\"atze. As a result, the "marginal decoupling property" of LSE precoders for $b$-steps of RSB is derived. The generality of the studied setup enables us to address special cases in which the number of active transmit antennas are constrained. Our numerical investigations depict that the computationally efficient forms of LSE precoders based on "$\ell_1$-norm" minimization perform close to the cases with "zero-norm" penalty function which have a considerable improvements compared to the random antenna selection. For the case with BPSK signals and restricted number of active antennas, the results show that RS fails to predict the performance while the RSB ansatz is consistent with theoretical bounds.Comment: 5 pages; 2 figures; to be presented at ISIT 201

Replica Symmetry Breaking in Compressive Sensing

For noisy compressive sensing systems, the asymptotic distortion with respect to an arbitrary distortion function is determined when a general class of least-square based reconstruction schemes is employed. The sampling matrix is considered to belong to a large ensemble of random matrices including i.i.d. and projector matrices, and the source vector is assumed to be i.i.d. with a desired distribution. We take a statistical mechanical approach by representing the asymptotic distortion as a macroscopic parameter of a spin glass and employing the replica method for the large-system analysis. In contrast to earlier studies, we evaluate the general replica ansatz which includes the RS ansatz as well as RSB. The generality of the solution enables us to study the impact of symmetry breaking. Our numerical investigations depict that for the reconstruction scheme with the "zero-norm" penalty function, the RS fails to predict the asymptotic distortion for relatively large compression rates; however, the one-step RSB ansatz gives a valid prediction of the performance within a larger regime of compression rates.Comment: 7 pages, 3 figures, presented at ITA 201

RSB Decoupling Property of MAP Estimators

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector $\boldsymbol{x}$ from the observation $\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z}$ with $\mathbf{A}$ being chosen from a wide range of matrix ensembles, and the noise vector $\boldsymbol{z}$ being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the $b$RSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and $b$ correlated interference terms. As the $b$RSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.Comment: 5 pages, presented in Information Theory Workshop 201

Empirical Coordination in a Triangular Multiterminal Network

In this paper, we investigate the problem of the empirical coordination in a triangular multiterminal network. A triangular multiterminal network consists of three terminals where two terminals observe two external i.i.d correlated sequences. The third terminal wishes to generate a sequence with desired empirical joint distribution. For this problem, we derive inner and outer bounds on the empirical coordination capacity region. It is shown that the capacity region of the degraded source network and the inner and outer bounds on the capacity region of the cascade multiterminal network can be directly obtained from our inner and outer bounds. For a cipher system, we establish key distribution over a network with a reliable terminal, using the results of the empirical coordination. As another example, the problem of rate distortion in the triangular multiterminal network is investigated in which a distributed doubly symmetric binary source is available.Comment: Accepted in ISIT 201