Sparse regularized total least squares for sensing applications

This paper focuses on solving sparse reconstruction problems where we have noise in both the observations and the dictionary. Such problems appear for instance in compressive sampling applications where the compression matrix is not exactly known due to hardware non-idealities. But it also has merits in sensing applications, where the atoms of the dictionary are used to describe a continuous field (frequency, space, angle, …). Since there are only a finite number of atoms, they can only approximately represent the field, unless we allow the atoms to move, which can be done by modeling them as noisy. In most works on sparse reconstruction, only the observations are considered noisy, leading to problems of the least squares (LS) type with some kind of sparse regularization. In this paper, we also assume a noisy dictionary and we try to combat both noise terms by casting the problem into a sparse regularized total least squares (SRTLS) framework. To solve it, we derive an alternating descent algorithm that converges to a stationary point at least. Our algorithm is tested on some illustrative sensing problems.