Sparsity in max-plus algebra and systems

Abstract

We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the ℓ 1 residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations. * The paper was published in the Discrete Event (2008). A unification of max-type algebras and their duals using weighted lattices with applications to nonlinear dynamical systems was presented in Maragos (2017). Meanwhile, in the last decade, we have experienced an increase of interest in sparsity in linear equations and linear systems. A solution of a linear equation is sparse when it has many zero elements. The reason we are interested in such solutions, is that they need less elements to describe the same information. They provide us a way of compressing the available data, throwing away those that are unnecessary (Donoho, 2006). They also reveal the structure of partially known signals (Candès et al, 2006) or systems (Chen et al, 2009). In control systems, sparsity has been sought in the sense of minimizing the number of sensors or actuators, subject to energy (Tzoumas et al, 2016;Summers et al, 2016) or observability-controllability constraints (Pequito et al, 2016).Although sparsity has been extensively studied in the linear setting (Elad, 2010), it is still not much developed in more general nonlinear settings. In this work, we aim to define and study sparsity in the max-plus algebraic setting. A sparse solution of a max-plus equation is a solution with many non-informative elements, i.e. the infinite elements. As in the linear case, such solutions use the least number of elements to describe the same information, thus yielding compressed data. But there are many other applications where sparsity could be relevant. For example, in max-plus systems (either static or dynamical), finding sparse inputs implies that we are activating fewer actuators/machines, thus, saving resources. Similarly, the problem of selecting few sensors to observe a max-plus system could be expressed in terms of designing sparse output matrices. Another application co...

Tropical Geometry and Machine Learning

“…It can also be directly seen from the adjunction (δ, ε) where (35) The solutions of (32) and of (33) for the ∞ case have been further analyzed in [15] both algebraically and combinatorially. It is also possible to search and find sparse solutions of either the exact equation (32) or the approximate problem (33), as done in [109], where sparsity here means a large number of −∞ values in the solution vector.…”

Sparsity in Max-Plus Algebra and Applications in Multivariate Convex Regression

“…vectors which consist of as many uninformative (−∞) elements as possible. In particular, we focus on generalizing the problem of computing the sparsest solution of the max-plus equation, which was introduced in [15]. Such solutions describe the same information with the least number of elements.…”

“…Hence, they can lead to a significant reduction in memory and computational time-see, for example, the pruning problem in optimal control [16]. Sparse solutions have also been employed to recover underlying sparse systems in max-plus system identification [15]. In general, an exact solution to the max-plus equation might not exist due to data-corruption or model-mismatch [15].…”

“…Sparse solutions have also been employed to recover underlying sparse systems in max-plus system identification [15]. In general, an exact solution to the max-plus equation might not exist due to data-corruption or model-mismatch [15]. For this reason, we consider the problem of finding a sparse approximate solution, i.e.…”

“…Contributions: In summary, we pose a generalized inverse problem with sparsity and "lateness" constraints for matrix max-plus equations, where the approximation error is in terms of any p norm, for p < ∞. This formulation is more general than [15], where only the 1 norm was considered. We present the supermodular structure of the problem, which allows us to solve it approximately but efficiently via a greedy algorithm.…”