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Protovalue functions: A laplacian framework for learning representation and control in markov decision processes
 Journal of Machine Learning Research
, 2006
"... This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by d ..."
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Cited by 66 (10 self)
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This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators (ii) A specific instantiation of this approach where global basis functions called protovalue functions (PVFs) are formed using the eigenvectors of the graph Laplacian on an undirected graph formed from state transitions induced by the MDP (iii) A threephased procedure called representation policy iteration comprising of a sample collection phase, a representation learning phase that constructs basis functions from samples, and a final parameter estimation phase that determines an (approximately) optimal policy within the (linear) subspace spanned by the (current) basis functions. (iv) A specific instantiation of the RPI framework using leastsquares policy iteration (LSPI) as the parameter estimation method (v) Several strategies for scaling the proposed approach to large discrete and continuous state spaces, including the Nyström extension for outofsample interpolation of eigenfunctions, and the use of Kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored MDPs (vi) Finally, a series of illustrative discrete and continuous control tasks, which both illustrate the concepts and provide a benchmark for evaluating the proposed approach. Many challenges remain to be addressed in scaling the proposed framework to large MDPs, and several elaboration of the proposed framework are briefly summarized at the end.
GradientBased Adaptation of General Gaussian Kernels
, 2005
"... Gradientbased optimizing of gaussian kernel functions is considered. The gradient for the adaptation of scaling and rotation of the input space is computed to achieve invariance against linear transformations. This is done by using the exponential map as a parameterization of the kernel parameter m ..."
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Cited by 15 (7 self)
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Gradientbased optimizing of gaussian kernel functions is considered. The gradient for the adaptation of scaling and rotation of the input space is computed to achieve invariance against linear transformations. This is done by using the exponential map as a parameterization of the kernel parameter manifold. By restricting the optimization to a constant trace subspace, the kernel size can be controlled. This is, for example, useful to prevent overfitting when minimizing radiusmargin generalization performance measures. The concepts are demonstrated by training hard margin support vector machines on toy data.
A formal model for semantic web service composition
 IN ISWC’06
, 2006
"... Automated composition of Web services or the process of forming new value added Web services is one of the most promising challenges in the semantic Web service research area. Semantics is one of the key elements for the automated composition of Web services because such a process requires rich mach ..."
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Cited by 10 (2 self)
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Automated composition of Web services or the process of forming new value added Web services is one of the most promising challenges in the semantic Web service research area. Semantics is one of the key elements for the automated composition of Web services because such a process requires rich machineunderstandable descriptions of services that can be shared. Semantics enables Web service to describe their capabilities and processes, nevertheless there is still some work to be done. Indeed Web services described at functional level need a formal context to perform the automated composition of Web services. The suggested model (i.e., Causal link matrix) is a necessary starting point to apply problemsolving techniques such as regressionbased search for Web service composition. The model supports a semantic context in order to find a correct, complete, consistent and optimal plan as a solution. In this paper an innovative and formal model for an AI planningoriented composition is presented.
NumberLike Objects and the Extended Lie Correspondence, arXiv
 math.RA/0604179 v1 8
, 2006
"... We classify finite dimensional division real associativeZ2algebras, introduce compositionZ2algebras, and extend the CampbellBakerHausdorff series and Lie correspondence in the context of linear HuLiu Leibniz algebras. The impetus to my research is the curiosity to generalize the beautiful theor ..."
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Cited by 3 (3 self)
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We classify finite dimensional division real associativeZ2algebras, introduce compositionZ2algebras, and extend the CampbellBakerHausdorff series and Lie correspondence in the context of linear HuLiu Leibniz algebras. The impetus to my research is the curiosity to generalize the beautiful theory of Lie algebras. A natural criterion of judging whether a new algebraic object is a good generalization of a Lie algebra is whether there is a fair counterpart of the Lie correspondence between connected linear groups and linear Lie algebras. One way of introducing new algebraic objects which satisfy the criterion is to add some new algebraic structures to Lie algebras. In particular, if a Leibniz algebra structure is added to a nonsemisimple Lie algebra, then the resulting algebraic object can be used to establish the extended Lie correspondence. This is one of the results of this paper. Another result of this paper is the extension of Frobeius ’ theorem, which is an unexpected result appearing in my search for the generalization of Lie algebras. It is well known that Frobeius ’ theorem asserts that only finite dimensional division real associative algebras are the field R of the real numbers, the field C of the complex numbers, and Hamilton’s quaternion algebra H. In the context of finite dimensional division real associative Z2algebras, the list in Frobeius ’ theorem is extended to a new list consisting of eight objects, where the first three objects are the ordinary finite dimensional division real associative algebras (without divisors of zero), and the last five objects are finite dimensional real associative Z2algebras with divisors of zero. Hence, it is reasonable to regard finite dimensional division real associative Z2algebras as numberlike objects. Throughout this paper the associative algebras considered are assumed to have an identity. 1 1 Basic Definisions In this section we recall some basic definitions introduced in [6]. Definition 1.1 A Lie algebra (L, +, [ ,]) is called a HuLiu Leibniz algebra if there exists a binary operation 〈, 〉 : L × L → L such that the following two properties hold: (i) 〈, 〉 satisfies the Leibniz identity:
HUNITARY AND LORENTZ MATRICES: A REVIEW
"... Abstract. Many properties of Hunitary and Lorentz matrices are derived using elementary methods. Complex matrices which are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H, are called Hunitary, and real matrices that are orthogonal with respect to t ..."
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Abstract. Many properties of Hunitary and Lorentz matrices are derived using elementary methods. Complex matrices which are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H, are called Hunitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix, are called Lorentz. The focus is on the analogues of singular value and CS decompositions for general Hunitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable Hunitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and Hunitary, products of reflections, stability and robust stability. Key words. Lorentz matrices, indefinite inner product. AMS subject classifications. 15A63 1. Introduction. Let Mn = Mn(IF) be the algebra of n×n matrices with entries in the field IF = C, the complex numbers, or IF = IR, the real numbers. If H ∈ Mn is an invertible Hermitian (symmetric in the real case) matrix, a matrix A ∈ Mn is called Hunitary if A ∗ HA = H.
Exponentiations over the universal enveloping algebra of sl2(C), submitted
"... Abstract. We construct, by modeltheoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra sl2(C). MSC: 16S30, 17B10, 03C60. Key words: universal enveloping algebra, exponential map, asymptotic cone ..."
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Cited by 1 (0 self)
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Abstract. We construct, by modeltheoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra sl2(C). MSC: 16S30, 17B10, 03C60. Key words: universal enveloping algebra, exponential map, asymptotic cone
THE nBODY PROBLEM IN SPACES OF CONSTANT CURVATURE
, 2008
"... Abstract. We generalize the Newtonian nbody problem to spaces of curvature κ = constant, and study the motion in the 2dimensional case. For κ> 0, the equations of motion encounter noncollision singularities, which occur when two bodies are antipodal. These singularities of the equations are respo ..."
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Cited by 1 (1 self)
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Abstract. We generalize the Newtonian nbody problem to spaces of curvature κ = constant, and study the motion in the 2dimensional case. For κ> 0, the equations of motion encounter noncollision singularities, which occur when two bodies are antipodal. These singularities of the equations are responsible for the existence of some hybrid solution singularities that end up in finite time in a collisionantipodal configuration. We also point out the existence of several classes of relative equilibria, including those generated by hyperbolic rotations for κ < 0. In the end, we prove Saari’s conjecture when the bodies are on a geodesic that rotates circularly or hyperbolically. Our approach also shows that each of the spaces κ < 0, κ = 0, and κ> 0 is characterized by certain orbits, which don’t occur in the other cases, a fact that might us help determine the nature of the physical space. 1 2 F. Diacu, E. PérezChavela, and M. Santoprete Contents
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
Designing Smooth Motions of Rigid Objects Computing Curves in Lie Groups
, 2003
"... Consider the problem of designing the path of a camera in 3D. As we may identify each camera position with a member of the Euclidean motions, SE(3), the problem may be recast mathematically as constructing interpolating curves on the (nonEuclidean) space SE(3). There exist many ways to formulate th ..."
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Consider the problem of designing the path of a camera in 3D. As we may identify each camera position with a member of the Euclidean motions, SE(3), the problem may be recast mathematically as constructing interpolating curves on the (nonEuclidean) space SE(3). There exist many ways to formulate this problem, and indeed many solutions. In this thesis we shall examine solutions based on simple geometric constructions, with the goal of discovering well behaved and computable solutions. In affine spaces there exist elegant solutions to the problem of curve design, which are collectively known as the techniques of Computer Aided Geometric Design (CAGD). The approach of this thesis will be the generalization of these methods and an examination of computation on matrix Lie groups. In particular, the Lie groups SO(3) and SE(3) will be examined in some detail. Table of Contents