Stability analysis of switched ARX models and application to learning with guarantees

The main subject of this work is the stability analysis of Switched Auto-Regressive models with eXogenous inputs (SARX), which constitute a reference class for switched and hybrid system identification. The work introduces novel conditions for the arbitrary switching stability of multiple-input multiple-output SARX models which exploit the peculiar structure of their state-space realization. The analysis relies on the properties of block companion matrices, and partly leverages results from the theory of non-negative matrices, without nevertheless asking for an input–output positive behavior of the model. The novel stability conditions have a simple formulation in terms of linear co-positive common Lyapunov functions, and come at a remarkably low computational cost, being solvable by Linear Programming. The low computational burden is particularly attractive in an identification context, as it allows to efficiently constrain learning procedures in order to obtain SARX models with stability guarantees. The latter is itself a contribution of the work, as it fills a gap in the literature on the estimation of SARX models. The results are validated on a particular learning technique based on Regression Trees – a well known machine learning algorithm – which has shown remarkable accuracy in experimental environments.     

完美匹配树的次大和次小的最大特征值

本文讨论完美匹配树的次大和次小的最大特征值问题,得到了次大的最大特征值的上界的明确表达式并确定了达到此上界的极树,同时也得到了次小的最大特征值的下界并确定了相应的极树。     

Complexes of Graphs with Bounded Matching Size

For positive integers k,n, we investigate the simplicial complex of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex of all bipartite graphs G on bipartition such that there is no matching of size k in G, and obtain results similar to those obtained for . S. Linusson and V. Welker supported by EC’s IHRP program through grant HPRN-CT-2001-00272. J. Shareshian partially supported by National Science Foundation grants DMS-0070757 and DMS-0030483.     

树的联的end-正则性

明确给出了具有正则自同态幺半群的两个树的联．     

TNT: An Interpretable Tree-Network-Tree Learning Framework using Knowledge Distillation

Deep Neural Networks (DNNs) usually work in an end-to-end manner. This makes the trained DNNs easy to use, but they remain an ambiguous decision process for every test case. Unfortunately, the interpretability of decisions is crucial in some scenarios, such as medical or financial data mining and decision-making. In this paper, we propose a Tree-Network-Tree (TNT) learning framework for explainable decision-making, where the knowledge is alternately transferred between the tree model and DNNs. Specifically, the proposed TNT learning framework exerts the advantages of different models at different stages: (1) a novel James–Stein Decision Tree (JSDT) is proposed to generate better knowledge representations for DNNs, especially when the input data are in low-frequency or low-quality; (2) the DNNs output high-performing prediction result from the knowledge embedding inputs and behave as a teacher model for the following tree model; and (3) a novel distillable Gradient Boosted Decision Tree (dGBDT) is proposed to learn interpretable trees from the soft labels and make a comparable prediction as DNNs do. Extensive experiments on various machine learning tasks demonstrated the effectiveness of the proposed method.     

Hierarchical differential evolution for minimal cut sets identification: Application to nuclear safety systems

In this paper, we present a Hierarchical Differential Evolution (HDE) algorithm for minimal cut set (mcs) identification of coherent and non-coherent Fault Trees (FTs). In realistic application of large-size systems, problems may be encountered in handling a large number of gates and events. In this work, to avoid any approximation, mcs identification is originally transformed into a hierarchical optimization problem, stated as the search for the minimum combination of cut sets that can guarantee the best coverage of all the minterms that make the system fail: during the first step of the iterative search, a multiple-population, parallel search policy is used to expedite the convergence of the second step of the exploration algorithm. The proposed hierarchical method is applied to the Reactor Protection System (RPS) of a Pressurized Water Reactor (PWR) and to the the Airlock System (AS) of a CANadian Deuterium Uranium (CANDU) reactor. Results are evaluated with respect to the accuracy and computational demand of the solution found.     

Combinatorics of non-ambiguous trees

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.     

Dynamical percolation on general trees

Häggström et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time ${t\in D}H?ggstr?m et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time , in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation. Research supported in part by a grant from the National Science Foundation.