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53
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 92 (11 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.
Mirrors in motion: Epipolar geometry and motion estimation
 In International Journal on Computer Vision
, 2003
"... In this paper we consider the images taken from pairs of parabolic catadioptric cameras separated by discrete motions. Despite the nonlinearity of the projection model, the epipolar geometry arising from such a system, like the perspective case, can be encoded in a bilinear form, the catadioptric fu ..."
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Cited by 35 (1 self)
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In this paper we consider the images taken from pairs of parabolic catadioptric cameras separated by discrete motions. Despite the nonlinearity of the projection model, the epipolar geometry arising from such a system, like the perspective case, can be encoded in a bilinear form, the catadioptric fundamental matrix. We show that all such matrices have equal Lorentzian singular values, and they define a ninedimensional manifold in the space of 4 × 4 matrices. Furthermore, this manifold can be identified with a quotient of two Lie groups. We present a method to estimate a matrix in this space, so as to obtain an estimate of the motion. We show that the estimation procedures are robust to modest deviations from the ideal assumptions. 1.
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 re ..."
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Cited by 28 (14 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
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 21 (5 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.
Relative equilibria in the 3dimensional curved nbody problem
 Memoirs Amer. Math. Soc
"... Abstract. We consider the 3dimensional gravitational nbody problem, n ≥ 2, in spaces of constant Gaussian curvature κ 6 = 0, i.e. on spheres S3κ, for κ> 0, and on hyperbolic manifolds H3κ, for κ < 0. Our goal is to define and study relative equilibria, which are orbits whose mutual distances ..."
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Cited by 20 (13 self)
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Abstract. We consider the 3dimensional gravitational nbody problem, n ≥ 2, in spaces of constant Gaussian curvature κ 6 = 0, i.e. on spheres S3κ, for κ> 0, and on hyperbolic manifolds H3κ, for κ < 0. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of S3κ and H3κ. Then we prove several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and we describe their qualitative behaviour. In particular, we show that in S3κ the bodies move either on circles or on Clifford tori, whereas in H3κ they move either on circles or on hyperbolic cylinders. Then we construct
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 20 (9 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.
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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Cited by 6 (0 self)
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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.
Markov invariants, plethysms, and phylogenetics
, 2007
"... We explore model based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that t ..."
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Cited by 4 (3 self)
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We explore model based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the LogDet distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analysing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.