Oversampled Adaptive Sensing with Random Projections: Analysis and Algorithmic Approaches

Abstract

Oversampled adaptive sensing (OAS) is a recently proposed Bayesian framework which sequentially adapts the sensing basis. In OAS, estimation quality is, in each step, measured by conditional mean squared errors (MSEs), and the basis for the next sensing step is adapted accordingly. For given average sensing time, OAS reduces the MSE compared to nonadaptive schemes, when the signal is sparse. This paper studies the asymptotic performance of Bayesian OAS, for unitarily invariant random projections. For sparse signals, it is shown that OAS with Bayesian recovery and hard adaptation significantly outperforms the minimum MSE bound for non-adaptive sensing. To address implementational aspects, two computationally tractable algorithms are proposed, and their performances are compared against the state-of-the-art non-adaptive algorithms via numerical simulations. Investigations depict that these lowcomplexity OAS algorithms, despite their suboptimality, outperform well-known non-adaptive schemes for sparse recovery, such as LASSO, with rather small oversampling factors. This gain grows, as the compression rate increases.

An Adaptive Bayesian Framework for Recovery of Sources with Structured Sparsity

“…The Bayesian OAS framework, introduced and analyzed in [1], [2], refers to the following sequential sensing procedure:…”

“…Note that in our simplified framework, A m+1 is restricted to be chosen from O F . We hence employ the worstcase adaptation strategy proposed in [1] and utilized in [2]: In subframe m, the adaptation function finds the permutation…”

“…On the other hand, one may restrict the ensemble of sensing matrices and/or postulate a different prior distribution, such that the estimation and sensing matrix construction is addressed in each subframe with low complexity. The investigations in [1] and [2] show that even by following the latter suboptimal approach, the OAS framework outperforms the benchmark.…”

“…The recently proposed oversampled adaptive sensing (OAS) framework has shown privileged performance for time-limited sensing in noisy environments [1], [2]. Unlike earlier adaptive approaches, e.g., [3]- [5], this scheme allows for oversampling.…”

“…In [1], it has been demonstrated that OAS achieves a considerable performance gain, when some prior information on the signal is available. The most well-known form of such prior information is sparsity which was explicitly studied in [1], [2]. Investigations have depicted that even suboptimal low-complexity OAS algorithms outperform well-known non-adaptive compressive sensing techniques in time-limited scenarios.…”