# Statistical approach to linear inverse problems

## TL;DR

This paper discusses a classical statistical approach to linear inverse problems with additive noise, emphasizing stability, feasible solution regions, and a near-optimal nonlinear filtering method without Bayesian assumptions.

## Contribution

It introduces a non-Bayesian statistical framework for stable inverse solutions, characterizes the feasible solution region, and presents a nonlinear filter close to the optimal Kolmogorov-Wiener filter.

## Key findings

- Feasible solution region is an elongated ellipsoid shaped by Fisher information.
- Spectrum of Fisher matrix determines inverse problem stability.
- Proposes a nonlinear filter near the Kolmogorov-Wiener optimal filter.

## Abstract

The main features of the statistical approach to inverse problems are described on the example of a linear model with additive noise. The approach does not use any Bayesian hypothesis regarding an unknown object; instead, the standard statistical requirements for the procedure for finding a desired object estimate are presented. In this way, it is possible to obtain stable and efficient inverse solutions in the framework of classical statistical theory. The exact representation is given for the feasible region of inverse solutions, i.e., the set of inverse estimates that are in agreement, in the statistical sense, with the data and available a priory information. The typical feasible region has the form of an extremely elongated hole ellipsoid, the orientation and shape of which are determined by the Fisher information matrix. It is the spectrum of the Fisher matrix that provides an exhaustive description of the stability of the inverse problem under consideration. The method of constructing a nonlinear filter close to the optimal Kolmogorov-Wiener filter is presented.