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

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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.