The MIN PFS problem and piecewise linear model estimation

Abstract

We consider a new combinatorial optimization problem related to linear systems (MIN PFS) that consists, given an infeasible system, in finding a partition into a minimum number of feasible subsystems. MIN PFS allows formalization of the fundamental problem of piecewise linear model estimation, which is an attractive alternative when modeling a wide range of nonlinear phenomena. Since MIN PFS turns out to be NP-hard to approximate within every factor strictly smaller than 3/2 and we are mainly interested in real-time applications, we propose a greedy strategy based on randomized and thermal variants of the classical Agmon–Motzkin–Schoenberg relaxation method for solving systems of linear inequalities. Our method provides good approximate solutions in a short amount of time. The potential of our approach and the performance of our algorithm are demonstrated on two challenging problems from image and signal processing. The first one is that of detecting line segments in digital images and the second one that of modeling time-series using piecewise linear autoregressive models. In both cases the MIN PFS-based approach presents various advantages with respect to conventional alternatives, including wider range of applicability, lower computational requirements and no need for a priori assumptions regarding the underlying structure of the data.