A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian kernel of Euclidean space. As an important special case, kernels based on the geometry of multinomial families are derived, leading to kernel-based learning algorithms that apply naturally to discrete data. Bounds on covering numbers and Rademacher averages for the kernels are proved using bounds on the eigenvalues of the Laplacian on Riemannian manifolds. Experimental results are presented for document classification, for which the use of multinomial geometry is natural and well motivated, and improvements are obtained over the standard use of Gaussian or linear kernels, which have been the standard for text classification.

The path planning problem for arbitrary devices is first and foremost a geometrical problem. For the field of control theory, advanced mathematical techniques have been developed to describe and use geometry. In this paper, we use the notions of the flow of vector fields and geodesics in metric spaces to formalize and unify path planning problems. A path planning algorithm based on flow propagation is briefly discussed. Applications of the theory to motion planning for a robot arm, a maneuvering car, and Rubik's Cube are given. These very different problems (holonomic, non-holonomic and discrete, respectively) are all solved by the same unified procedure. 1 Introduction Path planning is the art of finding a path to move an arbitrary device from an initial state to a goal state. We believe it is possible and important to give one mathematical theory encompassing all path planning problems. Such a theory contributes towards a unified understanding of autonomous systems, and gives specif...

Abstract. For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein. 1.

We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the ψ-sum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids. 1

. In this paper the problem of path planning and path following for a car-like robot is considered. Some new algorithms are proposed, which, by experiments, have been proven to be quite robust. 1 Introduction From the point of view of systems and control, starting from the work of Brockett [3] several methods were developed for solving path planning and following problems for a car-like robot [4],[5],[9],[10],[13],[16]. However, most control algorithms presented in the literature either use a simple kinematic model of the following form x = v cos ` (1) y = v sin ` (2) ` = ! (3) and/or do not consider the actual constraints on the range of steering angles. Where in the equations x and y are cartesian coordinates of the middle point on the front axle , ` is orientation angle, v is longitudinal velocity, ! is angular velocity. The more realistic model can be derived based on the assumption that the vector of longitudinal velocity is parallel to the front wheels. This more sophisticat...

Artificial Intelligence focuses on the question of how to design systems to exhibit intelligent behaviour in complex environments. Complex global behaviours can emerge from simple systems acting in a complex environment; however, this emergence requires that the systems' internal structure reflect essential structures in the environment. This paper examines the algebraic structure of a system's actions. We find that these actions often possess a self-similar local neighborhood structure that permits analysis and synthesis to be performed locally yet produce global, intelligent behaviours. A procedure for finding this local structure is presented, and illustrated with examples. 1. Introduction Simon (1969) relates the parable of an ant traversing a beach. The path of the ant is quite complex, reflecting the need to deal with unforseen obstacles. He points out that when this path is copied onto a piece of paper, it could just as well be the path of an expert skier on a ski slope....

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Abstract. A Lie-Rinehart algebra (A, L) consists of a commutative algebra A and a Lie algebra L with additional structure which generalizes the mutual structure of interaction between the algebra of smooth functions and the Lie algebra of smooth vector fields on a smooth manifold. Lie-Rinehart algebras provide the correct categorical language to solve the problem whether Kähler quantization commutes with reduction which, in turn, may be seen as a descent problem.

A Lie-Rinehart algebra (A, L) consists of a commutative algebra A and a Lie algebra L with additional structure which generalizes the mutual structure of interaction between the algebra of smooth functions and the Lie algebra of smooth vector fields on a smooth manifold. Lie-Rinehart algebras provide the correct categorical language to solve the problem whether Kähler quantization commutes with reduction which, in turn, may be seen as a descent problem.

In this paper, we study Lie Rinehart bialgebras over a commutative algebra, the algebraic generalization of Lie algebroids. More precisely, we analyze the structure of action Lie Rinehart bialgebras over the polynomial ring K[t] induced by actions of Lie algebras on K[t]. 1