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.

SOLO AND CHAMBER CLARINET WORKS BY AMERICAN- BORN WOMEN COMPOSERS: 1990-2005

During the period of 1990-2005, American-born women composers have contributed significantly to the standard clarinet repertoire. Pioneering composers such as Joan Tower, Margaret Brouwer, and Libby Larsen have created staples for clarinet literature. Yet, there is very little scholarly research on women composers of clarinet music, most being concentrated on Joan Tower. Through my research, I have discovered over seventy-five works by more than fifty composers in the following genres: solo clarinet; clarinet and piano; clarinet and voice, with or without piano; and small chamber pieces for up to five players. This performance dissertation project consists of three recitals featuring solo and chamber works by nine living women composers, and program notes containing pertinent biographical and compositional information. My intent is to increase recognition women composers, both prominent and lesser known, who are writing high-quality, accessible clarinet literature. Each woman selected is making a full or partial living from the sales of her compositions, has received recognition through awards, commissions, grants, and frequent performances, and has composed works that are both performer and audience accessible. Recital 1: Trios for Clarinet, Violin, and Piano Commissioned by the Verdehr Trio and Composed by American-Born Women Composers. Composers: Jennifer Higdon, Joan Tower, Margaret Brouwer, and Libby Larsen. Recital II: Programmatic Clarinet Works by American-Born Women Composers. Composers: Andrea Clearfield, Stella Sung, and Karen Amrhein. Recital III: Works for Solo Clarinet, Clarinet and Piano, and Clarinet Concerto Genres by American-Born Women Composers. Composers: Persis Parshall Vehar, Jenni Brandon, Margaret Brouwer, and Libby Larsen.

AMERICAN WOMEN COMPOSERS: SELECTED PUBLISHED WORKS FOR FLUTE AND PIANO AND FOR UNACCOMPANIED FLUTE COMPOSED BETWEEN 1930 AND 2008

The music of women composers often comprises only a small percentage of flutists‘ repertoire, yet there are actually many active women composers, many of whom have written for the flute. The aim of this dissertation is to chronicle a selection of works by several American women composers that have contributed to accessible flute repertoire. For the purpose of this dissertation, accessibility is described by the following parameters: works that limit the use of extended techniques, works that are suitable for performers from high school through a reasonably advanced level, works that are likely to elicit emotionally musical communication from the performer to the listener, and works that are reasonably available through music stores or outlets on the Internet that have a fairly comprehensive reach to the general public. My subjective judgment also played a role in the final selection of the 25 works included as part of this dissertation, and performed on three musically well-balanced recitals. A variety of resources were consulted for the repertoire, including Boenke‘s Flute Music by Women Composers: An Annotated Catalog, and the catalogs of publishers such as Arsis Press and Hildegard Publishing, both of which specialize in the music of women composers. The works performed and discussed are the following: Adrienne Albert – Sunswept; Marion Bauer – Prelude and Fugue, Op. 43.; Marilyn Bliss – Lament; Ann Callaway – Updraft; Ruth Crawford – Diaphonic Suite; Emma Lou Diemer – Sonata; Vivian Fine – Emily’s Images; Cynthia Folio – Arca Sacra; Nancy Galbraith – Atacama; Lita Grier – Sonata; Jennifer Higdon – The Jeffrey Mode; Edie Hill – This Floating World; Katherine Hoover – Masks; Mary Howe – Interlude between Two Pieces; Laura Kaminsky – Duo; Libby Larsen – Aubade; Alex Shapiro – Shiny Kiss; Judith Shatin – Coursing Through the Still Green; Faye-Ellen Silverman – Taming the Furies; Augusta Read Thomas – Euterpe’s Caprice; Joan Tower – Valentine Trills; Ludmila Ulehla – Capriccio; Elizabeth Vercoe – Kleemation; Gwyneth Walker – Sonata; and Judith Lang Zaimont – ‘Bubble-Up’ Rag. All of these works are worthy alternatives to the more frequently played flute repertoire, and they serve as a good starting point for anyone interested i n exploring the works of women composers.