Banach Spaces with Applications to Compressed Sensing and Greedy Convergence

这个项目将使用Banach空间理论来研究数据压缩应用中的一些问题。它将改进最近的具有限制等距性质(RIP)的矩阵的显式构造，从而更接近于概率构造的最优效率。在压缩感知中，RIP矩阵被用来通过比信号的测量向量小得多的测量向量来重建稀疏信号。包含在Banach空间中的向量可以用涉及基元素或冗余字典(如框架)的有限表达式来近似。通常，这些近似值由诸如X贪婪算法或阈值贪婪算法之类的贪婪算法来选择。这些算法的收敛结果取决于Banach空间的几何性质，如一致光滑性。它们还依赖于词典的无条件属性。Banach空间理论中需要解决的一个重要相关公开问题是证明与向量到大系数集的非线性投影相对应的常量族(Elton常数)是一致有界的。与此密切相关的是用有限的系数字母表替换任意实数系数的量化问题。具有良好量化性质的词典的存在将与Banach空间的几何性质有关。定量的结果将在有限维环境中得到。Banach空间是称为向量的对象的集合，这些对象可以相加或乘以数字来形成其他向量。有一个矢量之间的距离的概念，它类似于我们居住的三维世界中熟悉的点之间的距离的概念。数学家们已经发现，Banach空间提供了正确的框架，可以在其中建立泛函分析和偏微分方程等主要数学领域的公式。科学家和工程师还使用Banach空间对流体力学、信号处理和金融等应用领域的问题进行建模。有许多不同的Banach空间，它们可以通过诸如光滑性和凸性等几何性质来区分。属于Banach空间的单个向量通常由称为系数的无限数字串来标识。数据压缩中的一个重要问题是找到一种方法来选择最重要的系数，以便得到的有限串向量与原始向量的距离较短。这个项目将集中在这个问题和其他涉及Banach空间几何性质的问题上。

This project will use Banach space theory to investigate some problems that are motivated by applications to data compression. It will improve the recent explicit constructions of matrices with the Restricted Isometry Property (RIP) so as to match more closely the optimal efficiency of probabilistic constructions. RIP matrices are used in compressed sensing to reconstruct a sparse signal by a measurements vector of much smaller dimension than that of the signal. A vector contained in a Banach space can be approximated by finite expressions involving the elements of a basis or a redundant dictionary such as a frame. Typically these approximants are selected by a greedy algorithm such as the X greedy Algorithm or the Thresholding Greedy Algorithm. Convergence results for these algorithms depend on geometrical properties of the Banach space such as uniform smoothness. They also depend on unconditionality properties of the dictionary. An important related open problem in Banach space theory to be resolved is to show that the family of constants (Elton constants) corresponding to the nonlinear projection of a vector onto sets of large coefficients is uniformly bounded. Closely connected to this is the quantization problem of replacing arbitrary real coefficients by a finite alphabet of coefficients. The existence of dictionaries with good quantization properties will be related to the geometrical properties of the Banach space. Quantitative results will be obtained in the finite-dimensional setting. A Banach space is a collection of objects called vectors which can be added together or multiplied by numbers to form other vectors. There is a concept of distance between vectors which is analogous to the familiar notion of distance between the points in the three-dimensional world which we inhabit. Mathematicians have found that Banach spaces provide the correct framework in which to formulate major areas of mathematics such as Functional Analysis and Partial Differential Equations. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, and finance. There are many different Banach spaces which can be distinguished from each other by geometrical properties such as smoothness and convexity. An individual vector belonging to a Banach space is usually identified by an infinite string of numbers called coefficients. An important problem in data compression is to find a method to select the most significant coefficients so that the resulting finite string vector is a short distance from the original vector. The project will concentrate on this and other problems involving geometrical properties of Banach spaces.