Langevin and hessian with fisher approximation stochastic sampling for parameter estimation of structured covariance

Abstract

We have studied two efficient sampling methods, Langevin and Hessian adapted Metropolis Hastings (MH), applied to a parameter estimation problem of the mathematical model (Lorentzian, Laplacian, Gaussian) that describes the Power Spectral Density (PSD) of a texture. The novelty brought by this paper consists in the exploration of textured images modeled by centered, stationary Gaussian fields using directional stochastic sampling methods. Our main contribution is the study of the behavior of the previously mentioned two samplers and the improvement of the Hessian MH method by using the Fisher information matrix instead of the Hessian to increase the stability of the algorithm and the computational speed. The directional methods yield superior performances as compared to the more popular Independent and standard Random Walk MH for the PSD described by the three models, but can easily be adapted to any target law respecting the differentiability constraint. The Fisher MH produces the best results as it combines the advantages of the Hessian, i.e., approaches the most probable regions of the target in a single iteration, and of the Langevin MH, as it requires only first order derivative computations.

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