Through the haze: a non-convex approach to blind gain calibration for linear random sensing models

Abstract

Computational sensing strategies often suffer from calibration errors in the physical implementation of their ideal sensing models. Such uncertainties are typically addressed by using multiple, accurately chosen training signals to recover the missing information on the sensing model, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not employ any training signal, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each subject to an unknown positive multiplicative factor (or gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable, carefully chosen initialisation point. An analysis of this algorithm allows us to show that it converges to the exact solution provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Interestingly, we show that this requirement grows linearly (up to log factors) in the number of unknowns of the problem. This sample complexity is found both in absence of prior information, as well as when subspace priors are available for both the signal and gains, allowing a further reduction of the number of observations required for our recovery guarantees to hold. Moreover, in the presence of noise we show how our descent algorithm yields a solution whose accuracy degrades gracefully with the amount of noise affecting the measurements. Finally, we present some numerical experiments in an imaging context, where our algorithm allows for a simple solution to blind calibration of the gains in a sensor array.

The basins of attraction of the global minimizers of the non-convex sparse spike estimation problem

“…It is not obvious that the unconstrained gradient descent defined in iterations (8) and the corresponding notion of basin of attraction is suitable to perform constrained minimization (7). In fact, we show in this paper (essentially through Lemma 3.1) that the global minimum of constrained minimization (7) has a basin of attraction.…”

“…where E = R k(d+1) or E = Θ k,ǫ and g(θ) = f (φ(θ)). Note that when E = Θ k,ǫ , performing minimization (7) allows to recover the minima of the ideal minimization (2), yielding stable recovery guarantees under a RIP assumption. Hence we are particularly interested in this case.…”

“…Hence we are particularly interested in this case. When E = R k(d+1) , we speak about unconstrained minimization for minimization (7).…”

“…In this work, we are interested in minimizing g defined in (7). Since g is a smooth function, a classical method to minimize g is to consider a fixed step gradient descent.…”

“…Conditions on the number of measurements guarantee the success of the technique. In the area of blind deconvolution and bi-convex programming, recent works have exploited similar ideas [22,7].…”