Gelfand Widths of ℓ1-Balls
This chapter makes a detour to the geometry of Banach spaces. First, it highlights a strong connection between compressive sensing and Gelfand widths, with implications on the minimal number of measurements needed for stable sparse recovery with an arbitrary measurement matrix. Then two-sided estimates for the Gelfand widths of ℓ 1-balls are established, as well as two-sided estimates for Kolmogorov widths. Finally, compressive sensing techniques are applied to give a proof of Kashin’s decomposition theorem.
Keywordsadaptive measurement scheme nonadaptive measurement scheme stability ℓ1-minimization Gelfand width Kolmogorov width Kashin’s decomposition theorem
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