A geometry in the set of solutions to ill-posed linear problems with box constraints: Applications to probabilities on discrete sets

TL;DR

This paper characterizes the geometric structure of solutions to ill-posed linear problems with box constraints, revealing how solutions depend on data via Lagrange multipliers within a Riemannian framework derived from a Fermi-Dirac entropy.

Contribution

It provides a novel geometric description of constrained solutions using a Riemannian metric from a Fermi-Dirac entropy, linking solution behavior to data changes.

Findings

Solutions form a surface parameterized by Lagrange multipliers.

The solution set lies in the orthogonal complement of the kernel of A.

The geometry is induced by a Hessian of a Fermi-Dirac type entropy.

Abstract

When there are no constraints upon the solutions of the equation $\mathbf{A}\mathbf{\xi}= \mathbf{y},$ where $\mathbf{A}$ is a $K\times N-$matrix, $\mathbf{\xi}\in\mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^K$ a given vector, the description of the set of solutions as $\mathbf{y}$ varies in $\mathbb{R}^K$ is well known. But this is not so when the solutions are required to satisfy $\mathbf{\xi} \in \mathcal{K}\prod_{i\leq j\leq N} [a_j,b_j],$ for finite $a_j\leq b_j: 1\leq j\leq N.$ Here we provide a description of the set of solutions as a surface in the constraint set, parameterized by the Lagrange multipliers that come up in a related optimization problem in which $\mathbf{A}\mathbf{\xi} = \mathbf{y}$ appears as a constraint. It is the dependence of the Lagrange multipliers on the data vector $\mathbf{y}$ that determines how the solution changes as the datum changes. The geometry on the solutions is inherited from a Riemannian geometry on the set of constraints induced by the Hessian of an entropy of the Fermi-Dirac type which is the objective in the restatement of the optimization problem mentioned above. We prove that the set of solutions is contained in $\ker(\mathbf{A})^\perp$ in the metric defined as the Hessian of the entropy.