On non-negative solutions to large systems of random linear equations

Abstract

Systems of random linear equations may or may not have solutions with all components being non-negative. The question is, e.g., of relevance when the unknowns are concentrations or population sizes. In the present paper we show that if such systems are large the transition between these two possibilities occurs at a sharp value of the ratio between the number of unknowns and the number of equations. We analytically determine this threshold as a function of the statistical properties of the random parameters and show its agreement with numerical simulations. We also make contact with two special cases that have been studied before: the storage problem of a perceptron and the resource competition model of MacArthur.

Aspects of a Phase Transition in High-Dimensional Random Geometry

“…Similar to optimal mixed strategies considered in the previous subsection, the solution typically has a number of entries that are strictly zero (species that died out), the remaining ones being positive (surviving species). Again for and , a sharp transition at a critical value separates situations with typically no non-negative solution from those in which typically such a solution can be found [ 4 ].…”

“…Denoting the normal of this hyperplane by , we hence have the following duality: either the system ( 31 ) has a non-negative solution , or there exists a vector with If the is drawn independently from a distribution with finite first and second cumulant R and , respectively, and the components are independent random numbers with average B and variance , the dual problem ( 33 ) may be analyzed along the lines of ( 14 )–( 16 ). The result for the typical entropy of solution vectors reads [ 4 ] where the parameter characterizes the distributions of and . The main difference to ( 16 ) is the additional extremum over regularized by the penalty term proportional to .…”

“…The problem is closely related to a phase transition found recently in MacArthur’s resource competition model [ 4 , 6 , 7 ], in which a community of purely competing species builds up a collective cooperative phase above a critical threshold of the biodiversity.…”

“…A large class of problems in random geometry is concerned with the collocation of points in high-dimensional space. Applications range from optimization of financial portfolios [ 1 ], binary classifications of data strings [ 2 ] and optimal stategies in game theory [ 3 ] to the existence of non-negative solutions to systems of linear equations [ 4 , 5 ], the emergence of cooperation in competitive ecosystems [ 6 , 7 ], and linear programming with random parameters [ 8 ]. It is frequently relevant to consider the case where both the number of points T and the dimension of space N tend to infinity.…”

Aspects of a phase transition in high-dimensional random geometry

“…A large class of problems in random geometry is concerned with the collocation of points in high-dimensional space. Applications range from optimization of financial portfolios [1], binary classifications of data strings [2] and optimal stategies in game theory [3] to the existence of non-negative solutions to systems of linear equations [4,5], the emergence of cooperation in competitive ecosystems [6,7], and linear programming with random parameters [8]. It is frequently relevant to consider the case where both the number of points T and the dimension of space N tend to infinity.…”