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Sampling for Underdetermined Linear and Multilinear Inverse Problems: role of geometry and statistical priors


Estimating the underlying parameters of a statistical signal from noisy observations is a central problem in signal processing, with a wide variety of applications in many different fields such as machine learning, source localization, channel estimation for modern millimeter-wave (mmWave) communication systems, etc. Classical algorithms for source localization guarantee recovery of only $K=\mathcal{O}(M)$ sources using a Uniform Linear Array equipped with $M$ antennas. Recently, it has been shown that using certain non-uniform array designs, such as coprime and nested arrays, once certain correlation priors are assumed, it is possible to break this limit and go all the way up to $K=\mathcal{O}(M^2)$ sources. This thesis sheds more light on this phenomena, and more general cases of under-determined inverse problems in both linear, and non-linear settings. We show that for linear inverse problems, not only CRB exists for the case that $K=\mathcal{O}(M^2)$, for certain non-uniform arrays, but also it continues to exist even if the antenna locations are perturbed due to physical deformation of the device, or only a compressed version of the measurements are available. For a more general class of linear inverse problems, we show that in presence of certain correlation priors, one can recover sparse vectors of sparsity $K=\mathcal{O}(M^2)$, where the probability of detecting a wrong support for the sparse vector decays to zero exponentially fast as more and more temporal snapshots are obtained. We show these results hold for a variety of different statistical models, namely Gaussian sources, bounded sources, when the measurement matrices are equi-angular tight frames, and finally for the case that the measurements are obtained in adaptively. This thesis also considers multilinear inverse problems, namely tensor decompositions as well as certain non-convex problems with applications in millimeter-wave (mmWave) communication systems. We propose tensor decomposition algorithms for channel estimation for mmWave communication systems equipped with hybrid analog/digital beamforming, for cases such as multi-carrier Single-Input/Multiple-Output and single-carrier Multiple-Input/Multiple-Output, where we show the immense benefit gained by posing certain commonly considered statistical assumptions on the channel parameters, which leads to a provable increased identifiability compared to the existing algorithms for mmWave channel estimation.

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