Abstract
This chapter introduces Shannon-inspired performance limits associated with the classification of low-dimensional subspaces embedded in a high-dimensional ambient space from compressive and noisy measurements. In particular, it introduces the diversity-discrimination tradeoff that describes the interplay between the number of classes that can be separated by a compressive classifier-measured via the discrimination gain-and the performance of such a classifier-measured via the diversity gain-and the relation of such an interplay to the underlying problem geometry, including the ambient space dimension, the subspaces dimension, and the number of compressive measurements. Such a fundamental limit on performance is derived from a syntactic equivalence between the compressive classification problem and certain wireless communications problems. This equivalence provides an opportunity to cross-pollinate ideas between the wireless information theory domain and the compressive classification domain. This chapter also demonstrates how theory aligns with practice in a concrete application: face recognition from a set of noisy compressive measurements.

Abstract

Abstract
In this paper, we study the problem of projection kernel design for the reconstruction of high-dimensional signals from low-dimensional measurements in the presence of side information, assuming that the signal of interest and the side information signal are described by a joint Gaussian mixture model (GMM). In particular, we consider the case where the projection kernel for the signal of interest is random, whereas the projection kernel associated to the side information is designed. We then derive sufficient conditions on the number of measurements needed to guarantee that the minimum mean-squared error (MMSE) tends to zero in the low-noise regime. Our results demonstrate that the use of a designed kernel to capture side information can lead to substantial gains in relation to a random one, in terms of the number of linear projections required for reliable reconstruction. © 2016 IEEE.

Abstract
We develop a general framework for compressive linear-projection measurements with side information. Side information is an additional signal correlated with the signal of interest. We investigate the impact of side information on classification and signal recovery from low-dimensional measurements. Motivated by real applications, two special cases of the general model are studied. In the first, a joint Gaussian mixture model is manifested on the signal and side information. The second example again employs a Gaussian mixture model for the signal, with side information drawn from a mixture in the exponential family. Theoretical results on recovery and classification accuracy are derived. The presence of side information is shown to yield improved performance, both theoretically and experimentally. © 2016 IEEE.

Abstract
This paper investigates the impact of projection design on the reconstruction of high-dimensional signals from low-dimensional measurements in the presence of side information. In particular, we assume that both the signal of interest and the side information are described by a joint Gaussian mixture model (GMM) distribution. Sharp necessary and sufficient conditions on the number of measurements needed to guarantee that the average reconstruction error approaches zero in the low-noise regime are derived, for both cases when the side information is available at the decoder or at the decoder and encoder. Numerical results are also presented to showcase the impact of projection design on applications with real imaging data in the presence of side information. © 2016 IEEE.

Abstract
Achievable secrecy rates over a multiple-input multiple-output multipleeavesdropper (MIMOME) wiretap channel are considered, when the legitimate users have perfect knowledge only of the legitimate channel state and the eavesdropper channel is drawn from a (possibly unknown) continuous probability density. Legitimate users are assumed to deploy more antennas than the eavesdropper. A signaling transmission based on K-class Gaussian mixture model (GMM) distributions is proposed, which can be considered as an artificial-noise augmented signal, where the noise statistics are data-dependent. The proposed scheme is shown to achieve the secrecy capacity, log K, in the high signal-to-noise ratio (SNR) regime. Moreover, the tradeoff between secrecy and reliability at finite SNR is explored via the characterization of an upper bound to the error probability at the legitimate receiver, an upper bound to themutual information leakage to the eavesdropper and via numerical simulations. © Springer International Publishing Switzerland 2016.