A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous -time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture employs a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibited by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is then determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example syst...

We provide a classification of graphical models according to their representation as exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical (DAG) models and chain graphs with no hidden variables, including DAG models with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. The main results of this paper are that graphical models are SEFs and that many graphical models are not CEFs. That is, roughly speaking, graphical models when viewed as exponential families correspond to a set of smooth manifolds of various dimensions and usually not to a single smooth manifold. These results are discussed in the context of model selection. Keywords : Bayesian networks, graphical models, hidden variables, cur...

We derive the Varshamov--Gilbert and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over $\mathbb R$ and $\mathbb C$. The distance between two $k$-planes is defined as $\rho(p,q)=(\sin^2\theta_1 \dots \sin^2\theta_k)^{1/2}$, where $\theta_i, 1\le i\le k$, are the principal angles between $p$ and $q$.

Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt visually depict how quaternions interpolate rotation. This paper introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into a concise quaternion expression. This notation allows the paper to more cl...

Recently, the Isomap algorithm has been proposed for learning a nonlinear manifold from a set of unorganized high-dimensional data points. It is based on extending the classical multidimensional scaling method for dimension reduction. In this paper, we present a continuous version of Isomap which we call continuum isomap and show that manifold learning in the continuous framework is reduced to an eigenvalue problem of an integral operator. We also show that the continuum isomap can perfectly recover the underlying natural parametrization if the nonlinear manifold can be isometrically embedded onto an Euclidean space. Several numerical examples are given to illustrate the algorithm.

We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined. 1 Introduction A graphical model is a family of probability distributions. The set of distributions associated with a graphical model are usually define...

Graphics cards for personal computers have recently undergone a radical transformation from fixed-function graphics pipelines to multi-processor, programmable architectures. Multi-processor architectures are clearly advantageous for graphics for the simple reason that graphics computations are naturally concurrent, mapping well to stateless stream processing. They therefore parallelize easily and need no random access to memory with its problematic latencies. This paper presents Vertigo, a purely functional, Haskell-embedded language for 3D graphics and an optimizing compiler that generates graphics processor code. The language integrates procedural surface modeling, shading, and texture generation, and the compiler exploits the unusual processor architecture. The shading sublanguage is based on a simple and precise semantic model, in contrast to previous shading languages. Geometry and textures are also defined via a very simple denotational semantics. The formal semantics yields not only programs that are easy to understand and reason about, but also very efficient implementation, thanks to a compiler based on partial evaluation and symbolic optimization, much in the style of Pan [2]. Haskell’s overloading facility is extremely useful throughout Vertigo. For instance, math operators are used not just for floating point numbers, but also expressions (for differentiation and compilation), tuples, and functions. Typically, these overloadings cascade, as in the case of surfaces, which may be combined via math operators, though they are really functions over tuples of expressions on floating point numbers. Shaders may be composed with the same notational convenience. Functional dependencies are exploited for vector spaces, cross products, and derivatives.

LambdaMART is the boosted tree version of LambdaRank, which is based on RankNet. RankNet, LambdaRank, and LambdaMART have proven to be very successful algorithms for solving real world ranking problems: for example an ensemble of LambdaMART rankers won Track 1 of the 2010 Yahoo! Learning To Rank Challenge. The details of these algorithms are spread across several papers and reports, and so here we give a self-contained, detailed and complete description of them. 1

The motion of a biomolecule greatly depends on the engulfing solution, which is mostly water. Instead of representing individual water molecules, it is desirable to develop implicit solvent models that nevertheless accurately represent the contribution of the solvent interaction to the motion. In such models, hydrophobicity is expressed as a weighted sum of atomic surface areas. The derivatives of these weighted areas contribute to the force that drives the motion. In this paper we give formulas for the weighted and unweighted area derivatives of a molecule modeled as a space-filling diagram made up of balls in motion. Other than the radii and the centers of the balls, the formulas are given in terms of the sizes of circular arcs of the boundary and edges of the power diagram. We also give inclusion–exclusion formulas for these sizes.

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