On the complexity of piecewise affine system identification

TL;DR

This paper investigates the computational complexity of identifying piecewise affine systems, showing polynomial complexity in data size but exponential in data dimension, and establishes NP-hardness of the problem.

Contribution

It demonstrates that global optimality for PWA system identification can be achieved with polynomial complexity in data size but is NP-hard overall.

Findings

Global minimization is polynomial in data size for certain PWA maps.

The problem is NP-hard, indicating inherent computational difficulty.

Exponential complexity in data dimension is unavoidable for exact solutions.

Abstract

The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a global minimizer of the error. Previous work showed that a global solution could be obtained for continuous PWA maps with a worst-case complexity exponential in the number of data. In this paper, we show how global optimality can be reached for a slightly more general class of possibly discontinuous PWA maps with a complexity only polynomial in the number of data, however with an exponential complexity with respect to the data dimension. This result is obtained via an analysis of the intrinsic classification subproblem of associating the data points to the different modes. In addition, we prove that the problem is NP-hard, and thus that the exponential complexity in the dimension is a natural expectation for any exact algorithm.