Banach Spaces with Applications to Compressed Sensing and Greedy Convergence

这个项目将使用Banach空间理论来研究一些问题，这些问题是由数据压缩的应用所激发的。它将改进最近的显式构造矩阵的限制等距性质（RIP），以便更接近的概率构造的最佳效率。 RIP矩阵用于压缩感知，通过比信号的维度小得多的测量向量来重建稀疏信号。一个包含在Banach空间中的向量可以用包含基元素或冗余字典（如框架）的有限表达式来近似。典型地，这些近似值通过贪婪算法（诸如X贪婪算法或保持贪婪算法）来选择。 这些算法的收敛结果依赖于Banach空间的几何性质，如一致光滑性。它们还依赖于字典的无条件性属性。一个重要的相关的公开问题在Banach空间理论要解决的是，家庭的常数（埃尔顿常数）对应的非线性投影的一个向量集的大系数是一致有界的。 与此密切相关的是用有限的系数字母表替换任意真实的系数的量化问题。具有良好量子化性质的字典的存在性与Banach空间的几何性质有关。定量结果将在有限维环境中获得。一个Banach空间是一个被称为向量的对象的集合，这些对象可以被加在一起或乘以数字以形成其他向量。向量之间有一个距离的概念，它类似于我们所熟悉的三维世界中点之间距离的概念。 数学家们发现，Banach空间提供了正确的框架，在其中制定数学的主要领域，如功能分析和偏微分方程。Banach空间也被科学家和工程师用于模拟应用领域的问题，如流体力学，信号处理和金融。有许多不同的Banach空间，它们可以通过几何性质（如光滑性和凸性）相互区分。属于Banach空间的单个向量通常由称为系数的无限长的数字串来标识。 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，使得得到的有限字符串向量与原始向量的距离很短。该项目将集中在这个和其他问题涉及几何性质的Banach空间。

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.