Frame Compatibility: Discrete Versus Continuous Redundant Expansions, Strategies for Narrowing the Digital-Analog Gap

压缩传感理论有望给雷达、生物医学成像、甚至数字摄影等遥感技术带来革命性的变化。这一理论的主要观点是，可压缩信号的获取比信息含量高的信号要少得多。然而，这些结果通常是基于已经数字化的信号和随机测量的传感器的数学模型，这使得它们与现实的物理信号和设备有点脱节。这项工作探索了缩小理论与实践之间差距的最新趋势。使用模拟信号的模型来定义可压缩性，而不是数字信号。信号空间包括在信号恢复过程中连续改变而不产生伪影的可能性。这一想法将应用于雷达、X射线结晶学和其他传感系统。这项工作还有望应用于构成现代机器学习算法基础的神经网络。尽管机器学习的应用完全与数字数据有关，但连续模型的使用确保了可以准确地检索编码信息。在数据科学、信号采集和通信中，从遥感到基于分组的无线、光纤或量子通信，以及最近在压缩传感和超分辨率中的许多数学应用，带帧的冗余、稳定的展开已经成为核心。尽管基于帧的信号扩展和获取取得了成功，但在假定的风格化数学信号和测量模型与模拟域中的相应物理模型之间经常存在不匹配。例如，信号捕获通常由特定的线性泛函描述，而不是随机选择的非结构化泛函。这项研究项目解决了在基本水平上改善连续和离散表示空间之间的兼容性的需要。模拟信号的典型模型是由无限维希尔伯特空间给出的，该空间具有再生核和关于连续的、高度相干的矢量族的相关展开。在这种情况下，信号稀疏性的自然度量是其展开所需的最小数量的核函数。信号采集通常基于从群不变的离散泛函家族中进行采样。该项目的预期结果包括：(1)使用稀疏诱导范数对在有限或无限维再生核空间中稀疏合成的信号进行精确恢复，并使用物理相关的感测模型测量信号，该范数对于连续变形是稳定的；(2)相位恢复，基于帧系数的幅度，在诸如多元Paley-Wiener空间的再生核Hilbert空间中，这将利用稀疏性来证明测量的可注入性、稳定性和恢复算法在一般核空间中的可行性；以及(3)基于相位恢复和稀疏性的具有近似可逆性的冗余表示形式的Mallat散射变换。散射变换从数据中提取非线性特征，这些特征在分类问题中是强大的描述符。它是从模拟域中所需特性的观点来设计的，但其应用主要是有限大小的数字化数据，因此需要适当地修改权利要求。研究人员将使用相位恢复和稀疏性来证明变换的近似可逆性，这是验证数据的忠实编码所必需的。

The theory of compressed sensing promises to revolutionize remote sensing such as radar, biomedical imaging, and perhaps even digital photography. The main insight from this theory is that a compressible signal can be acquired with much less effort than a signal with a high information content. However, these results are commonly based on mathematical models for signals that are already digitized and for sensors that measure randomly, which makes them somewhat disconnected from realistic physical signals and apparatuses. This work explores recent trends in narrowing the gap between theory and practice. Instead of digital signals, models for analog signals are used to define compressibility. The signal space includes the possibility of continuous changes without producing artifacts in the signal recovery procedure. This idea will be applied to radar, X-ray crystallography, and other sensing systems. The work is also anticipated to have application to neural networks that form the basis for modern machine learning algorithms. Although the application to machine learning is entirely concerned with digital data, the use of continuous models ensures that encoded information can be retrieved accurately. Redundant, stable expansions with frames have become central to many applications of mathematics in data science, signal acquisition, and communications, from remote sensing to packet-based, wireless, fiber optical, or quantum communications and recently in compressed sensing and super-resolution. Despite the successes of the frame-based expansion and acquisition of signals, there is often a mismatch between the stylized mathematical signal and measurement models that are assumed and the corresponding physical models in the analog domain. For example, signal acquisition is typically described by specific linear functionals, not randomly chosen, unstructured ones. This research project addresses the need to improve compatibility between continuous and discrete representation spaces on a fundamental level. A typical model for analog signals is given by infinite-dimensional Hilbert spaces with a reproducing kernel and an associated expansion with respect to a continuous, highly coherent family of vectors. A natural measure of sparsity of a signal is in this setting the minimal number of kernel functions needed in its expansion. Signal acquisition is usually based on sampling from a group-invariant, discrete family of functionals. The expected outcomes of the project include: (1) accurate recovery for signals that are sparsely synthesized in a finite- or infinite-dimensional reproducing kernel space and measured with physically relevant sensing models, using a sparsity-inducing norm that is stable with respect to continuous deformations; (2) phase retrieval, signal recovery based on magnitudes of frame coefficients, in reproducing kernel Hilbert spaces such as multivariate Paley-Wiener spaces, which will be done using sparsity to demonstrate injectivity of measurements, stability, and feasibility of recovery algorithms in a general class of kernel spaces; and (3) a version of Mallat's scattering transform in a redundant representation with approximate invertibility based on phase retrieval and sparsity. The scattering transform extracts nonlinear features from data that are powerful descriptors in classification problems. It is designed from a viewpoint of desirable properties in the analog domain, but its application is mostly to digitized data of limited size, for which the claims need to be properly adapted. The investigators will use phase retrieval and sparsity to demonstrate the approximate invertibility of the transform, which is needed to verify the faithful encoding of data.