Nested Sampling and its Applications in Stable Compressive Covariance Estimation and Phase Retrieval with Near-Minimal Measurements

Compressed covariance sensing using quadratic samplers is gaining increasing interest in recent literature. Covariance matrix often plays the role of a sufficient statistic in many signal and information processing tasks. However, owing to the large dimension of the data, it may become necessary to obtain a compressed sketch of the high dimensional covariance matrix to reduce the associated storage and communication costs. Nested sampling has been proposed in the past as an efficient sub-Nyquist sampling strategy that enables perfect reconstruction of the autocorrelation sequence of Wide-Sense Stationary (WSS) signals, as though it was sampled at the Nyquist rate. The key idea behind nested sampling is to exploit properties of the difference set that naturally arises in quadratic measurement model associated with covariance compression. In this thesis, we will focus on developing novel versions of nested sampling for low rank Toeplitz covariance estimation, and phase retrieval, where the latter problem finds many applications in high resolution optical imaging, X-ray crystallography and molecular imaging. The problem of low rank compressive Toeplitz covariance estimation is first shown to be fundamentally related to that of line spectrum recovery. In absence if noise, this connection can be exploited to develop a particular kind of sampler called the Generalized Nested Sampler (GNS), that can achieve optimal compression rates. In presence of bounded noise, we develop a regularization-free algorithm that provably leads to stable recovery of the high dimensional Toeplitz matrix from its order-wise minimal sketch acquired using a GNS. Contrary to existing TV-norm and nuclear norm based reconstruction algorithms, our technique does not use any tuning parameters, which can be of great practical value. The idea of nested sampling idea also finds a surprising use in the problem of phase retrieval, which has been of great interest in recent times for its convex formulation via PhaseLift, By using another modified version of nested sampling, namely the Partial Nested Fourier Sampler (PNFS), we show that with probability one, it is possible to achieve a certain conjectured lower bound on the necessary measurement size. Moreover, for sparse data, an l1 minimization based algorithm is proposed that can lead to stable phase retrieval using order-wise minimal number of measurements.

CREATING ORGANIZATIONAL CITIZENS: A QUASI-EXPERIMENTAL AND QUALITATIVE EXAMINATION OF SUPERVSIOR- AND PEER-BASED INTERVENTIONS

Research suggests that supervisors and peers can help employees make sense of what is important or expected from them at work and, thereby, shape their behaviors. In this dissertation, I examine how employees’ organizational citizenship behaviors (OCB), such as helping and voice, are differentially affected by these two sources of influence over time. In particular, I compare the relative and joint effectiveness of two field interventions to enhance OCB: (a) a role clarification intervention in which supervisors are trained to set expectations for OCB for their employees and encourage them to engage in OCB and (b) a norm establishment intervention in which peers are trained to set expectations for each other and encourage each other to perform OCB. I utilize a mixed methods approach involving a quasi-field experiment to test for changes in OCB and qualitative data to explore the theoretical mechanisms over the course of three months in a large food processing plant. I find that role clarification interventions alone have immediate positive effects on OCB, whereas norm establishment interventions alone take a longer period of time to increase OCB. In addition, in the condition where both interventions were combined, norm establishment interventions weaken the effects of role clarification earlier on; however, at later stages in time, this pattern reverses as norm establishment enhances the effects of role clarification on OCB. Through these findings, I highlight how (a) organizations seeking quick increases in citizenship might be better off focusing on supervisors as sources of influence; (b) organizations need to persist with peer-focused interventions to see positive gains; and (c) despite initial hurdles with peer-focused interventions, over time, they can lead to the highest increases in OCB when combined with supervisor-focused interventions.