Download or read book Phase Retrieval in the High dimensional Regime written by Milad Bakhshizadeh and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose an iterative recovery method that can take advantage of any prior knowledge about the signal that is given as a compression code to efficiently solve the problem. We rigorously analyze the performance of our proposed method and provide extensive simulations to demonstrate its state-of-the-art performance.

Download or read book High dimensional Asymptotics for Phase Retrieval with Structured Sensing Matrices written by Rishabh Dudeja and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued ?-dimensional signal vector from the magnitudes of ? linear measurements. The linear measurements are specified using a ? × ? sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices.

Download or read book High Dimensional Probability written by Roman Vershynin and published by Cambridge University Press. This book was released on 2018-09-27 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Download or read book Concentration of Maxima and Fundamental Limits in High Dimensional Testing and Inference written by Zheng Gao and published by Springer Nature. This book was released on 2021-09-07 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a unified exposition of some fundamental theoretical problems in high-dimensional statistics. It specifically considers the canonical problems of detection and support estimation for sparse signals observed with noise. Novel phase-transition results are obtained for the signal support estimation problem under a variety of statistical risks. Based on a surprising connection to a concentration of maxima probabilistic phenomenon, the authors obtain a complete characterization of the exact support recovery problem for thresholding estimators under dependent errors.

Download or read book High Dimensional Optimization and Probability written by Ashkan Nikeghbali and published by Springer Nature. This book was released on 2022-08-04 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents extensive research devoted to a broad spectrum of mathematics with emphasis on interdisciplinary aspects of Optimization and Probability. Chapters also emphasize applications to Data Science, a timely field with a high impact in our modern society. The discussion presents modern, state-of-the-art, research results and advances in areas including non-convex optimization, decentralized distributed convex optimization, topics on surrogate-based reduced dimension global optimization in process systems engineering, the projection of a point onto a convex set, optimal sampling for learning sparse approximations in high dimensions, the split feasibility problem, higher order embeddings, codifferentials and quasidifferentials of the expectation of nonsmooth random integrands, adjoint circuit chains associated with a random walk, analysis of the trade-off between sample size and precision in truncated ordinary least squares, spatial deep learning, efficient location-based tracking for IoT devices using compressive sensing and machine learning techniques, and nonsmooth mathematical programs with vanishing constraints in Banach spaces. The book is a valuable source for graduate students as well as researchers working on Optimization, Probability and their various interconnections with a variety of other areas. Chapter 12 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Download or read book High Dimensional Data Analysis with Low Dimensional Models written by John Wright and published by Cambridge University Press. This book was released on 2022-01-13 with total page 717 pages. Available in PDF, EPUB and Kindle. Book excerpt: Connects fundamental mathematical theory with real-world problems, through efficient and scalable optimization algorithms.

Download or read book Computational Advantages in High Dimensions written by Stephen White and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although computational tasks in high dimensions are often characterized as suffering from a ``curse of dimensionality,'' many random objects in high dimensions also exhibit significant concentration of measure, a phenomenon which has been characterized as a countervailing ``blessing of dimensionality.'' With high-dimensional data more prevalent than ever, understanding both how to avoid the pitfalls and how to leverage the advantages of high-dimensional randomness are now essential skills for any practitioner of data science, statistics, or applied probability. In this dissertation, we present three examples whereby the dimensional structure of a problem can be exploited to improve upon existing results or solve otherwise intractable problems. In Chapter 2, we study the sparse phase retrieval problem of recovering a sparse signal from only the magnitude of its Fourier transform. By exploiting the additional freedom provided in multiple dimensions, we show that in dimension greater than one the sparse phase retrieval problem admits solutions which are several orders of magnitude faster than comparable one-dimensional approaches. In Chapter 3, we consider the problem of sparse dictionary learning, a problem in machine learning in which we seek to recover a sparsely-used dictionary matrix $\bfD$ from many samples of the form $\bfy_i = \bfD\bfx_i$. We present a polynomial-time algorithm which recovers overcomplete matrices satisfying the restricted isometry property even when sparsity is nearly linear in the dimension of the samples $\bfy_i$. We prove recovery guarantees which are significantly stronger than previously known alternatives for the overcomplete, linear-sparsity regime. Finally, in Chapter 4 we present a solution to the problem of organism detection in metagenomics. By leveraging the high dimensional structure of genomic data and its interaction with existing dimension reduction strategies, we are able to develop a surprisingly simple algorithm that significantly outperforms existing approaches. Numerical simulations confirm all theoretical results.

Download or read book Phase Retrieval from Locally Supported Measurements written by Brian Patrick Preskitt and published by . This book was released on 2018 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we study a new approach to the problem of phase retrieval, which is the task of reconstructing a complex-valued signal from magnitude-only measurements. This problem occurs naturally in several specialized imaging applications such as electron microscopy and X-ray crystallography. Although solutions were first proposed for this problem as early as the 1970s, these algorithms have lacked theoretical guarantees of success, and phase retrieval has suffered from a considerable gap between practice and theory for almost the entire history of its study. A common technique in fields that use phase retrieval is that of ptychography, where measurements are collected by only illuminating small sections of the sample at any time. We refer to measurements designed in this way as local measurements, and in this dissertation, we develop and expand the theory for solving phase retrieval in measurement regimes of this kind. Our first contribution is a basic model for this setup in the case of a one-dimensional signal, along with an algorithm that robustly solves phase retrieval under this model. This work is unique in many ways that represent substantial improvements over previously existing solutions: perhaps most significantly, many of the recovery guarantees in recent work rely on the measurements being generated by a random process, while we devise a class of measurements for which the conditioning of the system is known and quickly checkable. These advantages constitute major progress towards producing theoretical results for phase retrieval that are directly usable in laboratory settings. Chapter 1 conducts a survey of the history of phase retrieval and its applications, as well as the recent literature on the subject. Chapter 2 presents co-authored results defining and establishing the setting and solution of the base model explored in this dissertation. Chapter 3 expands the theory on what measurement schemes are admissible in our model, including an analysis of conditioning and runtime. Chapter 4 introduces an alternate solution for angular synchronization that yields helpful theoretical results. Chapter 5 brings our model nearer to the actual practice of ptychography. Chapter 6 extends the base model to two dimensions.

Download or read book Parallel Computing is Everywhere written by S. Bassini and published by IOS Press. This book was released on 2018-03-07 with total page 852 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most powerful computers work by harnessing the combined computational power of millions of processors, and exploiting the full potential of such large-scale systems is something which becomes more difficult with each succeeding generation of parallel computers. Alternative architectures and computer paradigms are increasingly being investigated in an attempt to address these difficulties. Added to this, the pervasive presence of heterogeneous and parallel devices in consumer products such as mobile phones, tablets, personal computers and servers also demands efficient programming environments and applications aimed at small-scale parallel systems as opposed to large-scale supercomputers. This book presents a selection of papers presented at the conference: Parallel Computing (ParCo2017), held in Bologna, Italy, on 12 to 15 September 2017. The conference included contributions about alternative approaches to achieving High Performance Computing (HPC) to potentially surpass exa- and zetascale performances, as well as papers on the application of quantum computers and FPGA processors. These developments are aimed at making available systems better capable of solving intensive computational scientific/engineering problems such as climate models, security applications and classic NP-problems, some of which cannot currently be managed by even the most powerful supercomputers available. New areas of application, such as robotics, AI and learning systems, data science, the Internet of Things (IoT), and in-car systems and autonomous vehicles were also covered. As always, ParCo2017 attracted a large number of notable contributions covering present and future developments in parallel computing, and the book will be of interest to all those working in the field.

Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1995 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Phase Retrieval in Two Dimensions written by Garry Neil Newsam and published by . This book was released on 1986 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Fourier Ptychographic Imaging written by Guoan Zheng and published by Morgan & Claypool Publishers. This book was released on 2016-06-30 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates the concept of Fourier ptychography, a new imaging technique that bypasses the resolution limit of the employed optics. In particular, it transforms the general challenge of high-throughput, high-resolution imaging from one that is coupled to the physical limitations of the optics to one that is solvable through computation. Demonstrated in a tutorial form and providing many MATLAB® simulation examples for the reader, it also discusses the experimental implementation and recent developments of Fourier ptychography. This book will be of interest to researchers and engineers learning simulation techniques for Fourier optics and the Fourier ptychography concept.

Download or read book High Dimensional Chaotic and Attractor Systems written by Vladimir G. Ivancevic and published by Springer Science & Business Media. This book was released on 2007-02-06 with total page 711 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate–level textbook is devoted to understanding, prediction and control of high–dimensional chaotic and attractor systems of real life. The objective is to provide the serious reader with a serious scientific tool that will enable the actual performance of competitive research in high–dimensional chaotic and attractor dynamics. From introductory material on low-dimensional attractors and chaos, the text explores concepts including Poincaré’s 3-body problem, high-tech Josephson junctions, and more.

Download or read book The Phase Retrieval Problem written by David Aaron Barmherzig and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The phase retrieval problem is an inverse problem which consists of recovering a signal from a set of squared magnitude measurements. One version of this problem, often known as Fourier phase retrieval, arises ubiquitously in scientific imaging fields (such as diffraction imaging, crystallography, and optics, etc.) where one seeks to recover an image or signal from squared magnitude measurements of its Fourier transform. Another version, known as Gaussian phase retrieval, is manifested as the study of solving random systems of quadratic equations, and constitutes an important problem in the field of nonconvex optimization. The first part of this thesis introduces a general mathematical framework for the holographic phase retrieval problem. In this problem, which arises in holographic coherent diffraction imaging, a "reference" portion of the signal to be recovered via (Fourier) phase retrieval is a priori known from experimental design. A general formula is also derived for the expected recovery error when the measurement data is corrupted by Poisson shot noise. This facilitates an optimization perspective towards reference design and analysis, which is then employed towards quantifying the performance of various known reference choices. Based on insights gained from these results, a new "dual-reference" design is proposed which consists of two reference portions - being "block" and "pinhole" shaped regions - adjacent to the imaging specimen. Expected error analysis on data following a Poisson shot noise model shows that the dual-reference scheme produces uniformly superior performance over the leading single-reference schemes. Numerical experiments on simulated data corroborate these theoretical results, and demonstrate the advantage of the dual-reference design. Based on this work, a prototype experiment for holographic coherent diffraction imaging using a dual-reference has been designed at the SLAC National Accelerator Laboratory. The second part studies the one-dimensional Fourier phase retrieval problem, as well as the closely related spectral factorization problem. In its first chapter, a comprehensive exposition of the problem theory is provided. This includes a full characterization of its general nonuniqueness, as well as the special cases for which unique solutions exists. In the second chapter, a semidefinite programming formulation is derived for the Fourier phase retrieval problem. It is shown that this approach provides guaranteed recovery whenever there exists a unique phase retrieval solution. A correspondence is also established between solutions of the phase retrieval SDP, and sum-of-squares decompositions of Laurent and trigonometric polynomials. In the third chapter, a least-squares formulation is presented for the one-dimensional Fourier phase retrieval and spectral factorization problems. This formulation allows for the successful implementation of numerous first- and second-order optimization methods. In the third part, a biconvex formulation of the Gaussian phase retrieval problem is introduced. This allows for alternating-projection algorithms, such as ADMM and block coordinate descent, to be successfully applied to Gaussian phase retrieval. Both theoretical guarantees and numerical simulations demonstrate the success of these methods.

Download or read book High Rate High Dimensional Quantum Key Distribution Systems written by Nurul T. Islam and published by Springer. This book was released on 2018-10-01 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes a broad research program on quantum communication. Here, a cryptographic key is exchanged by two parties using quantum states of light and the security of the system arises from the fundamental properties of quantum mechanics. The author developed new communication protocols using high-dimensional quantum states so that more than one classical bit is transferred by each photon. This approach helps circumvent some of the non-ideal properties of the experimental system, enabling record key rates on metropolitan distance scales. Another important aspect of the work is the encoding of the key on high-dimensional phase-randomized weak coherent states, combined with so-called decoy states to thwart a class of possible attacks on the system. The experiments are backed up by a rigorous security analysis of the system, which accounts for all known device non-idealities. The author goes on to demonstrate a scalable approach for increasing the dimension of the quantum states, and considers attacks on the system that use optimal quantum cloning techniques. This thesis captures the current state-of-the-art of the field of quantum communication in laboratory systems, and demonstrates that phase-randomized weak coherent states have application beyond quantum communication.

Download or read book Advanced High Resolution Tomography in Regenerative Medicine written by Alessandra Giuliani and published by Springer. This book was released on 2018-11-19 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers the state-of-the-art research on advanced high-resolution tomography, exploring its role in regenerative medicine. and also explores the 3D interactions between tissues, cells, and biomaterials. Various multidisciplinary paths in regenerative medicine are covered, including X-ray microtomography and its role in regenerative medicine, synchrotron radiation-based microtomography and phase contrast tomography, the challenge of the vascularization of regenerated tissues, lung and cartilage imaging, and more. This is an ideal book for biomedical engineers, biologists, physicists, clinicians, and students who want to pursue their studies in the field of regenerative medicine. This book also: Reviews in detail the algorithms and software used for the 3D exploration of regenerated tissue Covers the latest research on the use of X-ray microtomography for muscle diseases Details applications of synchrotron radiation tomography in orthopedics and dentistry

Download or read book A First Course in Random Matrix Theory written by Marc Potters and published by Cambridge University Press. This book was released on 2020-12-03 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: An intuitive, up-to-date introduction to random matrix theory and free calculus, with real world illustrations and Big Data applications.