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 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 global optimum 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 is actually linear (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 their provably exact recovery. Moreover, in the presence of noise we show how our 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, for which our algorithm allows for a simple solution to blind calibration of the gains in an imaging sensor array.
['Valerio Cambareri', 'Laurent Jacques']
Through the Haze: A Non-Convex Approach to Blind Calibration for Linear Random Sensing Models