During the past decade, much progress has made in refining the principle that high-dimensional statistical problems are tractable when they exhibit some form of low-dimensional structure. However, in practice, it is often unclear whether or not structural assumptions are justified by data, and the problem of validating such assumptions is unresolved in many contexts. In this talk, I will focus on the context of compressed sensing (CS) --- a signal processing framework that is built on the structural assumption of sparsity. Although the theory of CS offers strong guarantees for recovering sparse signals, many aspects of the recovery process depend on prior knowledge of the signal's sparsity level --- a parameter which is rarely known in practice. Towards a resolution of this issue, I will introduce a generalized family of sparsity parameters that can be estimated in a way that is free of structural assumptions. In connection with signal recovery, I will show that the error rate of the Basis Pursuit Denoising algorithm can be bounded tightly in terms of these parameters. Lastly, I will present consistency results for the proposed sparsity estimation procedure, including a CLT, which allows for the hypothesis of sparsity to be tested in a precise sense.
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