Abstract
This chapter develops a theoretical analysis of the convex programming method for recovering a structured signal from independent random linear measurements. This technique delivers bounds for the sampling complexity that are similar to recent results for standard Gaussian measurements, but the argument applies to a much wider class of measurement ensembles. To demonstrate the power of this approach, the chapter presents a short analysis of phase retrieval by trace-norm minimization. The key technical tool is a framework, due to Mendelson and coauthors, for bounding a nonnegative empirical process.
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Notes
- 1.
The extended real numbers \(\overline{\mathbb{R}}:= \mathbb{R} \cup \{\pm \infty \}\). A proper convex function takes at least one finite value but never the value \(-\infty \).
- 2.
A Rademacher random variable takes the two values ± 1 with equal probability.
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Acknowledgements
JAT gratefully acknowledges support from ONR award N00014-11-1002, AFOSR award FA9550-09-1-0643, and a Sloan Research Fellowship. Thanks are also due to the Moore Foundation.
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Tropp, J.A. (2015). Convex Recovery of a Structured Signal from Independent Random Linear Measurements. In: Pfander, G. (eds) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19749-4_2
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