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

压缩传感理论有望彻底改变遥感，如雷达，生物医学成像，甚至可能是数字摄影。从这个理论的主要见解是，一个可压缩的信号可以获得比一个信号具有高的信息含量少得多的努力。然而，这些结果通常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些脱节。这项工作探讨了缩小理论与实践之间差距的最新趋势。代替数字信号，模拟信号的模型用于定义压缩性。信号空间包括连续变化的可能性，而不会在信号恢复过程中产生伪影。这个想法将应用于雷达，X射线晶体学和其他传感系统。这项工作也有望应用于神经网络，构成现代机器学习算法的基础。虽然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编码信息可以被准确地检索。带框架的冗余、稳定扩展已经成为数据科学、信号采集和通信中许多数学应用的核心，从遥感到基于分组的通信、无线通信、光纤通信或量子通信，以及最近的压缩传感和超分辨率。尽管基于帧的扩展和信号采集取得了成功，但在假定的程式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，信号采集通常由特定的线性泛函来描述，而不是随机选择的非结构化泛函。该研究项目解决了在基础层面上提高连续和离散表示空间之间兼容性的需求。模拟信号的一个典型模型由无限维希尔伯特空间给出，该空间具有再生核和相对于连续的、高度相干的向量族的相关扩展。信号稀疏性的一个自然度量是在这种情况下，其扩展所需的核函数的最小数量。信号采集通常基于从群不变的离散泛函族中采样。该项目的预期成果包括：（1）使用相对于连续变形稳定的稀疏诱导范数，精确恢复在有限维或无限维再现核空间中稀疏合成并用物理相关传感模型测量的信号;（2）相位恢复，基于帧系数的幅度的信号恢复，在诸如多变量Paley-Wiener空间的再生核希尔伯特空间中，这将使用稀疏性来证明测量的注入性、稳定性以及在一般类的核空间中恢复算法的可行性;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中的Mallat散射变换的版本。散射变换从数据中提取非线性特征，这些特征是分类问题中的强大描述符。它是从模拟域中理想特性的角度设计的，但它的应用主要是有限大小的数字化数据，需要适当调整权利要求。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.