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A globally convergent algorithm for MAP estimation in the linear model with non-Gaussian priors

We develop a framework for analyzing non-Gaussian densities in terms of the curvature of the density function itself rather than moments of the random variable. The framework suggests a new criterion for sub- and super-gaussianity of densities that is seen to be of a wider range of application than the commonly used kurtosis criterion. We show that the notion of relative curvature introduced can be seen as a generalization of the notion of convexity, where classical convexity of a function is seen as a relationship between the function and a linear model. We use the curvature framework to derive an inequality that holds for all functions that are super-Gaussian in the sense of the proposed criterion. This inequality allows proof of global convergence of a certain re-weighted minimum norm algorithm by providing a weighting matrix that yields descent without line search. The algorithm is equivalent to the FOCUSS algorithm (Rao, B.D. and Gorodnitsky, I.F., IEEE Trans. Sig. Processing, vol.45, p.600-6, 1997; Rao and Kreutz-Delgado, K., IEEE Trans. Sig. Processing, vol.47, p.187-200, 1999) in the case of independent generalized Gaussian densities in the linear model.