Abstract
Compressed sensing is the art of reconstructing structured $n$-dimensional vectors from substantially fewer measurements than naively anticipated. A plethora of analytic reconstruction guarantees support this credo. The strongest among them are based on deep results from large-dimensional probability theory that require a considerable amount of randomness in the measurement design. Here, we demonstrate that derandomization techniques allow for considerably reducing the amount of randomness that is required for such proof strategies. More, precisely we establish uniform s-sparse reconstruction guarantees for $C s \log (n)$ measurements that are chosen independently from strength-four orthogonal arrays and maximal sets of mutually unbiased bases, respectively. These are highly structured families of $\tilde{C} n^2$ vectors that imitate signed Bernoulli and standard Gaussian vectors in a (partially) derandomized fashion.
Highlights
One can show that at most n+1 different orthonormal bases may exist that have this property in a pairwise fashion [21, Theorem 3.5]. Such a set of n + 1 bases is called a maximal set of mutually unbiased bases (MMUB)
With probability at least 1 − 2e−cm, any s-sparse x ∈ Rn can be recovered from y = Ax by solving (1). This result readily generalizes to measurements that are sampled from a maximal set of mutually unbiased bases
The nullspace property, as well as its connection to uniform s-sparse recovery readily generalizes to complex-valued s-sparse vectors
Summary
Compressed sensing is the art of effectively reconstructing structured signals from substantially fewer measurements than would naively be required for standard techniques like least squares. Our results highlight that this technique almost allows bridging the gap between existing proof techniques for generic and structured measurements: the results are still strong but require slightly more randomness than choosing vectors uniformly from a bounded orthogonal system, such as Fourier or Hadamard vectors. While we are not able to fully overcome this drawback here, the methods described in this work do limit the amount of randomness required to generate individual structured measurements. We believe that this may help to reduce the discrepancy between “what can be proved” and “what can be done” in a variety of concrete applications
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