A Geometric Analysis Of Convex Demixing (English Edition)

A Geometric Analysis Of Convex Demixing (English Edition)

Demixing is the problem of identifying multiple informative signals from a single observation of their superposition. Demixing problems arise frequently in scientific applications, including fields as diverse as geophysics [TBM79], astrophysics [SDC03], image segmentation [ESQD05], machine learning [CSPW11], robust statistics [CLMW11], and audio processing [AEJ+12]. The underlying observation in each of these examples is typically high-dimensional—often with millions or even billions of variables—so that effective demixing procedures require efficient computational methods. This thesis proposes a generic computational framework for demixing based on convex optimization. By applying methods from integral geometry, we develop a general theory that characterizes the performance of our demixing approach under a probabilistic model. These results apply to a number of particular cases where the general theory predicts the outcome of numerical experiments with a high level of precision. We begin with a concrete example that illustrates the theoretical guarantees developed in this thesis.

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