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.

THE EFFECT OF A PROGRAM OF PORTABLE ELECTRONIC PIANO KEYBOARD EXPERIENCE ON THE ACQUISITION OF SIGHT-SINGING SKILL IN THE NOVICE HIGH SCHOOL CHORUS

The problem was to determine whether a method of aural and visual vocal training that included a program of portable electronic piano keyboard experience would be more effective in teaching sight-singing skills to novice high school chorus students than a method that included only aural and visual vocal training. A sub-problem was to determine whether novice chorus students enjoyed playing electronic keyboards in chorus as a reinforcement experience in sight-singing training. Students were randomly assigned to two treatment groups, tested with the Musical Aptitude Profile, Tonal Imagery, part A, and then trained separately. The experimental group sang repetitions of melodic patterns and utilized techniques associated with the Kodály Method while simultaneously playing keyboard. The comparison group received a similar treatment without using keyboards. The students were pre- and post-tested in sight-singing using the Vocal Sight-Reading Inventory. Results of the Analysis of Covariance using MAP scores as the covariate revealed no significant difference (p<.05) between post-test scores of the two groups. Improvement was noted in 96% of students from pre-test to post-test regardless of grouping. The repeated measures ANOVA revealed a significant relationship (p<.006) between aptitude group and post-test score. High aptitude students in both groups were found to benefit more from the training than low aptitude students. High aptitude keyboard group students achieved an average gain score that was 8.67 points higher than the comparison group. Of the total experimental group, 92% enjoyed playing keyboards in chorus. It is recommended that future research be undertaken to study the use of keyboards with advanced high school choruses and with uncertain singers in the high school chorus. Research is also needed to develop graded, valid, and reliable sight-singing tests for use in high school chorus. Techniques of the Kodály Method should be further investigated for use in high school sight-singing training.