Regularization with Approximated $L^2$ Maximum Entropy Method

Jean-Michel Loubes (1) , Paul Rochet (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We tackle the inverse problem of reconstructing an unknown finite measure $\mu$ from a noisy observation of a generalized moment of $\mu$ defined as the integral of a continuous and bounded operator $\Phi$ with respect to $\mu$. When only a quadratic approximation $\Phi_m$ of the operator is known, we introduce the $L^2$ approximate maximum entropy solution as a minimizer of a convex functional subject to a sequence of convex constraints. Under several assumptions on the convex functional, the convergence of the approximate solution is established and rates of convergence are provided.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jean-Michel Loubes, Paul Rochet. Regularization with Approximated $L^2$ Maximum Entropy Method. 2009. ⟨hal-00389698⟩