This paper surveys locally weighted learning, a form of lazy learning and memorybased learning, and focuses on locally weighted linear regression. The survey discusses distance functions, smoothing parameters, weighting functions, local model structures, regularization of the estimates and bias, assessing predictions, handling noisy data and outliers, improving the quality of predictions by tuning t parameters, interference between old and new data, implementing locally weighted learning e ciently, and applications of locally weighted learning. A companion paper surveys how locally weighted learning can be used in robot learning and control.

We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.

This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel B-splines are introduced to compute a C²-continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarse-tofine hierarchy of control lattices to generate a sequence of bicubic B-spline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using B-spline refinement to reduce the sum of these functions into one equivalent B-spline function. Experimental results demonstrate that high-fidelity reconstruction is possible from a selected set of sparse and irregular samples.

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems. List of Contents 1. Introduction 2. Rendering of Trivariate Functions 3. Tensor Product Schemes 4. Point Schemes 4.1 Shepard's Methods 4.2 Radial Interpolants 4.2.1 Hardy Multiquadrics 4.2.2 Duchon Thin Plate Splines 5. Natural Neighbor Interpolation 6. k-dimensional Triangulations 7. Tetrahedral Schemes 7.1 Polynomial Schemes 7.2 Rational Schemes 8. Simplicial Schemes 8.1 Polynomial Schemes 8.2 Rational Schemes 8.3 A Transfinite Scheme 9. Multivariate Splines 10. Transfinite Hypercubal Methods 11. Derivative Generation 12. Interpolation on the sphere and other surfa...

Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most important methods of scattered data interpolation in two-dimensional and in three-dimensional spaces. We review both single-valued cases, where the underlying function has the form f:R #R, and multivalued cases, where the underlying function is f:R . The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (Clough--Tocher) interpolation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neighbor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scattered data interpolation and scattered data fitting.

The edge-insertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.

This paper explores a memory-based approach to robot learning, using memorybased neural networks to learn models of the task to be performed. Steinbuch and Taylor presented neural network designs to explicitly store training data and do nearest neighbor lookup in the early 1960s. In this paper their nearest neighbor network is augmented with a local model network, which fits a local model to a set of nearest neighbors. This network design is equivalent to a statistical approach known as locally weighted regression, in which a local model is formed to answer each query, using a weighted regression in which nearby points (similar experiences) are weighted more than distant points (less relevant experiences). We illustrate this approach by describing how it has been used to enable a robot to learn a difficult juggling task. Keywords: memory-based, robot learning, locally weighted regression, nearest neighbor, local models. 1 Introduction An important problem in motor learning is approxim...

The classical approaches in fuzzy control (Zadeh and Mamdani) deal with dense rule bases. When this is not the case, i.e. in sparse rule bases, one has to choose another method. Fuzzy rule interpolation (proposed first by Koczy and Hirota [15]) offers a possibility to construct fuzzy controllers (KH controllers) under such conditions. The main result of this paper shows that the KH interpolation method is stable. It also contributes to the application oriented use of Balazs-Shepard interpolation operators investigated extensively by researchers in approximation theory. The numerical analysis aspect of the result contributes to the well-known problem of finding a stable interpolation method in the following sense.

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I prove that amongst all triangulations of a planar point set the Delaunay triangulation maximizes the arithmetic mean of the inradii of the triangles. 1 Introduction A triangulation of a set of points is a partition of the convex hull into triangles. The Delaunay triangulation is a well known triangulation, being the planar dual of the famous Voronoi diagram. Most applications of triangulations require that the triangulation should avoid `skinny' triangles. Many different measures of the skinniness of a triangle have been proposed. One of these is the inradius (radius of the inscribed circle) [14, 19]. In this paper I prove that the Delaunay triangulation is the triangulation that maximizes the arithmetic mean of the inradius. 1.1 Background Triangulating sets of points is a very important problem in computational geometry; there are far too many applications in computational geometry and other fields to mention here. (See the surveys [4, 6, 1]) There are many different possible ...