Using automatic differentiation for the solution of the minimum p-norm estimation problem in magnetoencephalography

The minimum-norm estimate is a popular reconstruction technique to localize the electrical activity on the human cortex for given measurements of a magnetic field outside the head. The standard approach minimizes the Euclidean norm of the current density distribution of the underlying dipole moments. However, for certain magnetic fields whose current density distribution is known to be focal, the traditional approach based on the Euclidean norm tends to over-smooth the reconstructions. To overcome these difficulties, a minimum p-norm approach with 1<p<2 is taken to increase the focality when p approaches unity. A Newton-type optimization algorithm is investigated in order to avoid potential numerical instabilities caused by reweighted least-squares algorithms. The reverse mode of automatic differentiation is used to efficiently evaluate the underlying gradient of the cost function.