CIF: Medium: Collaborative Research: Subspace Matching and Approximation on the Continuum

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In a diverse range of application areas--including astronomicalimaging, microscopy, wireless communications, radar, sonar, andmany others--one must confront the problem of estimating orextracting information from a signal that is governed by a smallnumber of unknown (continuous-valued) parameters. Customized toeach of these applications, unique methods for handling theseunknown parameters have been proposed in the past. This researchinvolves the development of a unified framework for addressingproblems in all of these areas systematically, efficiently, withawareness of the fundamental limits, and with provable accuracy.There are two main components of this framework. The firstcomponent involves the development of models where the signals ofinterest lie within or close to a subspace or a union of subspaces.In the case where the underlying parameters for the signal arecontinuous-valued, this is a challenging task. Rather than attemptto simply discredit these problems (a program potentially fraughtwith difficulty), this research involves the development of finiteapproximations for a locally infinite range of subspaces usingefficient local subspace fits whose dimensions match the effectivenumber of local degrees of freedom. The second component of thisframework consists of techniques for identifying the range ofsubspaces responsible for generating a signal from a set ofdiscrete observations, confronting difficulties posed by thecoherence, by the continuous nature of the parameter space, and bythe fact that some signals may be synthesized from a narrow (butcontinuously-indexed and therefore infinite) range of subspaces.These components combine to enable the efficient acquisition,processing, and estimation of signals arising from theparameterized subspaces model.