Fast inference of ill-posed problems within a convex space

TL;DR

This paper introduces a fast, scalable algorithm for exploring high-dimensional convex polytopes to determine the probability distribution of parameters consistent with low-dimensional data, applicable in scientific and technological contexts.

Contribution

The paper presents a novel algorithm that efficiently explores convex polytopes with linear time scaling, improving upon existing numerical techniques for ill-posed problems.

Findings

Algorithm matches the accuracy of existing methods

Runs linearly with system size, enhancing scalability

Successfully applied to biological and communication network models

Abstract

In multiple scientific and technological applications we face the problem of having low dimensional data to be justified by a linear model defined in a high dimensional parameter space. The difference in dimensionality makes the problem ill-defined: the model is consistent with the data for many values of its parameters. The objective is to find the probability distribution of parameter values consistent with the data, a problem that can be cast as the exploration of a high dimensional convex polytope. In this work we introduce a novel algorithm to solve this problem efficiently. It provides results that are statistically indistinguishable from currently used numerical techniques while its running time scales linearly with the system size. We show that the algorithm performs robustly in many abstract and practical applications. As working examples we simulate the effects of restricting reaction fluxes on the space of feasible phenotypes of a {\em genome} scale E. Coli metabolic network and infer the traffic flow between origin and destination nodes in a real communication network.