Interpolation | Artificial neural networks | Numerical analysis
A radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977.RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications. (Wikipedia).
A short refresher on vectors. Before I introduce vector-based functions, it's important to look at vectors themselves and how they are represented in python™ and the IPython Notebook using SymPy.
From playlist Life Science Math: Vectors
What is the formula for component form of a vector
http://www.freemathvideos.com in this video series I will show you how to find the angle of a vector when given in component form or as a linear combination. To understand the direction of a vector it is important to go back to the unit circle and determine how we can find the angle when
From playlist Vectors
Linear Algebra 4.7 Change of Basis
My notes are available at http://asherbroberts.com/ (so you can write along with me). Elementary Linear Algebra: Applications Version 12th Edition by Howard Anton, Chris Rorres, and Anton Kaul
From playlist Linear Algebra
Math 060 Fall 2017 111317C Orthonormal Bases
Motivation: how to obtain the coordinate vector with respect to a given basis? Definition: orthogonal set. Example. Orthogonal implies linearly independent. Orthonormal sets. Example of an orthonormal set. Definition: orthonormal basis. Properties of orthonormal bases. Example: Fou
From playlist Course 4: Linear Algebra (Fall 2017)
Introduction to Change of Basis
This video introduces a change of basis and show how to convert between the standard basis and a nonstandard basis coordinates.
From playlist Vectors: Change of Basis
Calculating the matrix of a linear transformation with respect to a basis B. Here is the case where the input basis is the same as the output basis. Check out my Vector Space playlist: https://www.youtube.com/watch?v=mU7DHh6KNzI&list=PLJb1qAQIrmmClZt_Jr192Dc_5I2J3vtYB Subscribe to my ch
From playlist Linear Transformations
The MLE for the weight vector in a Gaussian linear regression model when using basis functions (assuming a known variance). A playlist of these Machine Learning videos is available here: http://www.youtube.com/view_play_list?p=D0F06AA0D2E8FFBA
From playlist Machine Learning
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Representing multivariate random signals using principal components. Principal component analysis identifies the basis vectors that describe the la
From playlist Random Signal Characterization
From playlist Linear Algebra Ch 8 (updated Jan2021)
Lecture 04-Jack Simons Electronic Structure Theory- Linear combinations of atomic orbitals
The Hartree-Fock molecular orbitals; LCAO-MO expansion; Hartree-Fock equations in matrix form; one- and two-electron integrals; the iterative SCF process; scaling with basis set size; how virtual orbitals change with basis set; core, valence, polarization, and diffuse basis functions; Slat
From playlist U of Utah: Jack Simons' Electronic Structure Theory course
Lecture 05-Jack Simons Electronic Structure Theory- Basis sets
Basis set notations; complete-basis extrapolation of the Hartree-Fock and correlation energies. (1)Jack Simons Electronic Structure Theory- Session 1- Born-Oppenheimer approximation http://www.youtube.com/watch?v=Z5cq7JpsG8I (2)Jack Simons Electronic Structure Theory- Session 2- Hartr
From playlist U of Utah: Jack Simons' Electronic Structure Theory course
Virginie Ehrlacher - Multi-center decomposition of molecular densities: a mathematical perspective
Recorded 04 May 2022. Virginie Ehrlacher of the École Nationale des Ponts-et-Chaussées presents "Multi-center decomposition of molecular densities: a mathematical perspective" at IPAM's Large-Scale Certified Numerical Methods in Quantum Mechanics Workshop. Abstract: The aim of this talk is
From playlist 2022 Large-Scale Certified Numerical Methods in Quantum Mechanics
Radial Basis Function Networks are not talked about a lot these days, but they are very interesting and useful. Handwriting demo: http://macheads101.com/demos/handwriting/?c=rbf Resizing images with RBF networks: https://github.com/unixpickle/rbfscale#results Distance formula in kNN vid
From playlist Machine Learning
Interpolations and Mappings with Applications in Image Processing
In this talk, Markus van Almsick reviews the most popular and most advanced interpolation methods and discusses their merits and shortcomings. The Wolfram Language provides many interpolation methods to construct continuous functions from discrete data points. Furthermore, interpolations a
From playlist Wolfram Technology Conference 2020
Lecture 16 - Radial Basis Functions
Radial Basis Functions - An important learning model that connects several machine learning models and techniques. Lecture 16 of 18 of Caltech's Machine Learning Course - CS 156 by Professor Yaser Abu-Mostafa. View course materials in iTunes U Course App - https://itunes.apple.com/us/cours
From playlist Machine Learning Course - CS 156
Jacek Dziubański: Selected results in real harmonic analysis in the rational Dunkl setting
HYBRID EVENT Recorded during the meeting "Modern Analysis Related to Root Systems with Applications" the October 19, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathe
From playlist Virtual Conference
Greg Fasshauer: Some recent insights into computing with positive definite kernels
Abstract: In this talk I will discuss recent joint work with Mike McCourt (SigOpt, San Francisco) that has led to progress on the numerically stable computation of certain quantities of interest when working with positive definite kernels to solve scattered data interpolation (or kriging)
From playlist Numerical Analysis and Scientific Computing
Dual basis definition and proof that it's a basis In this video, given a basis beta of a vector space V, I define the dual basis beta* of V*, and show that it's indeed a basis. We'll see many more applications of this concept later on, but this video already shows that it's straightforwar
From playlist Dual Spaces
Maryna Viazovska - 2/6 Automorphic Forms and Optimization in Euclidean Space
Hadamard Lectures 2019 The goal of this lecture course, “Automorphic Forms and Optimization in Euclidean Space”, is to prove the universal optimality of the E8 and Leech lattices. This theorem is the main result of a recent preprint “Universal Optimality of the E8 and Leech Lattices and I
From playlist Hadamard Lectures 2019 - Maryna Viazovska - Automorphic Forms and Optimization in Euclidean Space