Annals of Mathematics and Artificial Intelligence最新文献

Systems of decision rules and decision trees are widely used as a means for knowledge representation, as classifiers, and as algorithms. They are among the most interpretable models for classifying and representing knowledge. The study of relationships between these two models is an important task of computer science. It is easy to transform a decision tree into a decision rule system. The inverse transformation is a more difficult task. In this paper, we study unimprovable upper and lower bounds on the minimum depth of decision trees derived from decision rule systems with discrete attributes depending on the various parameters of these systems. To illustrate the process of transformation of decision rule systems into decision trees, we generalize well known result for Boolean functions to the case of functions of k-valued logic.

There are many real life applications where data can not be effectively represented in Hilbert spaces and/or where the data points are uncertain. In this context, we address the issue of binary classification in Banach spaces in presence of uncertainty. We show that a number of results from classical support vector machines theory can be appropriately generalized to their robust counterpart in Banach spaces. These include the representer theorem, strong duality for the associated optimization problem as well as their geometrical interpretation. Furthermore, we propose a game theoretical interpretation of the class separation problem when the underlying space is reflexive and smooth. The proposed Nash equilibrium formulation draws connections and emphasizes the interplay between class separation in machine learning and game theory in the general setting of Banach spaces.

This paper concerns preference elicitation and learning of decision models in the context of multicriteria decision making. We propose an approach to learn a representation of preferences by a non-additive multiattribute utility function, namely a Choquet or bi-Choquet integral. This preference model is parameterized by one-dimensional utility functions measuring the attractiveness of consequences w.r.t. various point of views and one or two set functions (capacities) used to weight the coalitions and control the intensity of interactions among criteria, on the positive and possibly the negative sides of the utility scale. Our aim is to show how we can successively learn marginal utilities from properly chosen preference examples and then learn where the interactions matter in the overall model. We first present a preference elicitation method to learn spline representations of marginal utilities on every component of the model. Then we propose a sparse learning approach based on adaptive (L_1)-regularization for determining a compact Möbius representation fitted to the observed preferences. We present numerical tests to compare different regularization methods. We also show the advantages of our approach compared to basic methods that do not seek sparsity or that force sparsity a priori by requiring k-additivity.

In the context of Multiple Criteria Decision Aiding, decision makers often face problems with multiple conflicting criteria that justify the use of preference models to help advancing towards a decision. In order to determine the parameters of these preference models, preference elicitation makes use of preference learning algorithms, usually taking as input holistic judgments, i.e., overall preferences on some of the alternatives, expressed by the decision maker. Tools to achieve this goal in the context of a ranking model based on multiple reference profiles are usually based on mixed-integer linear programming, Boolean satisfiability formulation or metaheuristics. However, they are usually unable to handle realistic problems involving many criteria and a large amount of input information. We propose here an evolutionary metaheuristic in order to address this issue. Extensive experiments illustrate its ability to handle problem instances that previous proposals cannot.

Nondeterministic planning is the process of computing plans or policies of actions achieving given goals, when there is nondeterministic uncertainty about the initial state and/or the outcomes of actions. This process encompasses many precise computational problems, from classical planning, where there is no uncertainty, to contingent planning, where the agent has access to observations about the current state. Fundamental to these problems is belief tracking, that is, obtaining information about the current state after a history of actions and observations. At an abstract level, belief tracking can be seen as maintaining and querying the current belief state, that is, the set of states consistent with the history. We take a knowledge compilation perspective on these processes, by defining the queries and transformations which pertain to belief tracking. We study them for propositional domains, considering a number of representations for belief states, actions, observations, and goals. In particular, for belief states, we consider explicit propositional representations with and without auxiliary variables, as well as implicit representations by the history itself; and for actions, we consider propositional action theories as well as ground PDDL and conditional STRIPS. For all combinations, we investigate the complexity of relevant queries (for instance, whether an action is applicable at a belief state) and transformations (for instance, revising a belief state by an observation); we also discuss the relative succinctness of representations. Though many results show an expected tradeoff between succinctness and tractability, we identify some interesting combinations. We also discuss the choice of representations by existing planners in light of our study.

While most models of human choice are linear to ease interpretation, it is not clear whether linear models are good models of human decision making. And while prior studies have investigated how task conditions and group characteristics, such as personality or socio-demographic background, influence human decisions, no prior works have investigated how to use less personal information for choice prediction. We propose a deep learning model based on self-attention and cross-attention to model human decision making which takes into account both subject-specific information and task conditions. We show that our model can consistently predict human decisions more accurately than linear models and other baseline models while remaining interpretable. In addition, although a larger amount of subject specific information will generally lead to more accurate choice prediction, collecting more surveys to gather subject background information is a burden to subjects, as well as costly and time-consuming. To address this, we introduce a training scheme that reduces the number of surveys that must be collected in order to achieve more accurate predictions.

In this paper we propose a new notion of a clique reliability. The clique reliability is understood as the ratio of the number of statistically significant links in a clique to the number of edges of the clique. This notion relies on a recently proposed original technique for separating inferences about pairwise connections between vertices of a network into significant and admissible ones. In this paper, we propose an extension of this technique to the problem of clique detection. We propose a method of step-by-step construction of a clique with a given reliability. The results of constructing cliques with a given reliability using data on the returns of stocks included in the Dow Jones index are presented.

Topological representations of binary digital images usually take into consideration different adjacency types between colors. Within the cubical-voxel 3D binary image context, we design an algorithm for computing the isotopic model of an image, called (6, 26)-Homological Region Adjacency Tree ((6, 26)-Hom-Tree). This algorithm is based on a flexible graph scaffolding at the inter-voxel level called Homological Spanning Forest model (HSF). Hom-Trees are edge-weighted trees in which each node is a maximally connected set of constant-value voxels, which is interpreted as a subtree of the HSF. This representation integrates and relates the homological information (connected components, tunnels and cavities) of the maximally connected regions of constant color using 6-adjacency and 26-adjacency for black and white voxels, respectively (the criteria most commonly used for 3D images). The Euler-Poincaré numbers (which may as well be computed by counting the number of cells of each dimension on a cubical complex) and the connected component labeling of the foreground and background of a given image can also be straightforwardly computed from its Hom-Trees. Being (I_D) a 3D binary well-composed image (where D is the set of black voxels), an almost fully parallel algorithm for constructing the Hom-Tree via HSF computation is implemented and tested here. If (I_D) has (m_1{times } m_2{times } m_3) voxels, the time complexity order of the reproducible algorithm is near (O(log (m_1{+}m_2{+}m_3))), under the assumption that a processing element is available for each cubical voxel. Strategies for using the compressed information of the Hom-Tree representation to distinguish two topologically different images having the same homological information (Betti numbers) are discussed here. The topological discriminatory power of the Hom-Tree and the low time complexity order of the proposed implementation guarantee its usability within machine learning methods for the classification and comparison of natural 3D images.

We consider comparative dissimilarity relations on pairs on fuzzy description profiles, the latter providing a fuzzy set-based representation of pairs of objects. Such a relation expresses the idea of “no more dissimilar than” and is used by a decision maker when performing a case-based decision task under vague information. We first limit ourselves to those relations admitting a weighted (varvec{L}^p) distance representation, for which we provide an axiomatic characterization in case the relation is complete, transitive and defined on the entire space of pairs of fuzzy description profiles. Next, we switch to the more general class of comparative dissimilarity relations representable by a Choquet (varvec{L}^p) distance, parameterized by a completely alternating normalized capacity.