Regularization with Approximated $L^2$ Maximum Entropy Method

TL;DR

This paper introduces an $L^2$ approximate maximum entropy method for reconstructing unknown measures from noisy generalized moment observations, providing convergence guarantees and rates when only approximate operators are known.

Contribution

It proposes a novel $L^2$ approximate maximum entropy approach for inverse measure reconstruction with theoretical convergence analysis.

Findings

Convergence of the approximate solution is proven under certain assumptions.

Rates of convergence are established for the proposed method.

The method effectively reconstructs measures from noisy, approximate data.

Abstract

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.