Abstract
In this work we consider numerical efficiency and convergence rates for solvers of non-convex multi-penalty formulations when reconstructing sparse signals from noisy linear measurements. We extend an existing approach, based on reduction to an augmented single-penalty formulation, to the non-convex setting and discuss its computational intractability in large-scale applications. To circumvent this limitation, we propose an alternative single-penalty reduction based on infimal convolution that shares the benefits of the augmented approach but is computationally less dependent on the problem size. We provide linear convergence rates for both approaches, and their dependence on design parameters. Numerical experiments substantiate our theoretical findings.
Highlights
In many real-life applications one is interested in recovering a structured signal from few corrupted linear measurements
It commonly appears in signal processing and compressed sensing applications, where noise is added to the signal both before and after the measurement process occurs
Via the restricted isometry property (RIP)-constant δs theorem 2.7 gives a direct dependence of the convergence rate on the sparsity of the solution and the properties of the matrix, whereas theorem 2.11 is harder to interpret: it is straight-forward to deduce the existence of parameter regimes in which linear convergence occurs but hard to quantify the rate in terms of the parameters
Summary
In many real-life applications one is interested in recovering a structured signal from few corrupted linear measurements. One particular challenge lies in separating the ground-truth from pre-measurement noise since any such corruption is amplified during the measurement process, a phenomenon known as noise folding [2] or input noise model [1]. It commonly appears in signal processing and compressed sensing applications, where noise is added to the signal both before and after the measurement process occurs. Alternating minimization does not lend itself to an easy analysis of the convergence rate
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