This talk will focus on the relatively new field of compressed sensing, which is based on the observation that sparse signals can be recovered from what was previously thought to be highly incomplete information. We first give a brief overview of the subject, followed by a discussion of two underlying concepts in this field, namely the Restricted Isometry Property and the Johnson-Lindenstrauss Lemma. The latter part of the talk will focus on a "near equivalence" of these concepts recently discovered by the speaker and F. Krahmer. As a consequence of this result, we can improve on the best-known bounds for dimensionality reduction using structured, or "fast" linear embeddings. We also highlight a connection to cross-validation of the error incurred by compressed sensing algorithms. Finally, we discuss the Restricted Isometry Property for structured measurement matrices formed by subsampling orthonormal polynomial and spherical harmonic systems, along with several engineering applications.
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