Regularization in Reproducing Kernel Banach Spaces
|
|
- Ruth Burns
- 5 years ago
- Views:
Transcription
1 .... Regularization in Reproducing Kernel Banach Spaces Guohui Song School of Mathematical and Statistical Sciences Arizona State University Comp Math Seminar, September 16, 2010 Joint work with Dr. Fred Hickernell (Illinois Institute of Technology) Dr. Haizhang Zhang (Sun Yat-Sen University) Guohui Song Comp Math 1 of 21
2 Outline 1 Scattered Data Approximation 2 Reproducing Kernel Hilbert Spaces 3 Reproducing Kernel Banach Spaces Guohui Song Comp Math 2 of 21
3 Scattered Data Approximation Setting: Given data {(xj, y j ) : j = 1, 2,..., n} in R d R. Find a function Pf which is a good fit to the given data.. Question 1... What is a good fit?.. Guohui Song Comp Math 4 of 21
4 A 1-D Example scattered data points Guohui Song Comp Math 5 of 21
5 A Regularization Approach We want to control the closeness to the given data the complexity of the function A regularization approach: Target function: L(f, y, H) := n j=1 (f (x j) y j ) 2 + λ f 2 H Pf := arg min L(f, y, H). f H Some fancy names: penalized least square, ridge regression, smoothing spline. Question 2... What is the hypothesis space H?.. Guohui Song Comp Math 6 of 21
6 Reproducing Kernel Hilbert Spaces (RKHS) We need H is a Hilbert space f n f H 0 = f n (x) f (x) 0 for all x X RKHS: a Hilbert space H on which the point evaluation functional is continuous. f (x) M x f H for all x X. Question 3....Ẉhere is the kernel? Guohui Song Comp Math 8 of 21
7 Kernel Suppose X is a subset of R d. K is a real-valued function on X X : K : X X R K is a kernel if for any positive integer m and X := {x1,..., x m } X, the kernel gram matrix K m := [K(x j, x l ) : 1 j, l m] is symmetric and positive semi-definite. Guohui Song Comp Math 9 of 21
8 Connections Between RKHS and Kernels [Aronszajn, 1950] There is a bijective mapping from the RKHS to the set of kernels such that K(, x) H for any x X, f (x) = (f ( ), K(, x)) H for any f H. Some properties of RKHS HK and the kernel K H 0 := span{k(, x) : x X } is dense in H K. For any f = m j=1 c jk(, x j ) H 0, f HK = K 1/2 m c 2. Guohui Song Comp Math 10 of 21
9 Some Examples of RKHS and Kernels Sobolev space H 2 (R): K(s, t) = 3 3 ( ) 3 e 2 s t sin s t 2 + π 6. C 0 Matérn kernel: K(s, t) = e s t Gaussian kernel: K(s, t) = e w(s t) 2, w > 0. Sinc kernel: K(s, t) = sinc(s t). Polynomial kernels: K(s, t) = (st) d, d = 1, 2,.... L 2 (R) is NOT a RKHS. Guohui Song Comp Math 11 of 21
10 Regularization in the RKHS H K [Kimeldorf and Wahba, 1971] Representer Theorem : Target function: L(f, y, H K ) = n j=1 (f (x j) y j ) 2 + λ f 2 H K. Let S n := span{k(, x j) : j = 1, 2,..., n}. The optimization problem reduces to finite-dimensional: min L(f, y, H K) = min L(f, y, H K) f H K f S n The minimizer is explicitly given: Pf = n α jk(, x j), where α = (K n + λi n) 1 y. j=1 Guohui Song Comp Math 12 of 21
11 Reproducing Kernel Banach Spaces We try to construct a Banach space B point evaluation functional δ x is continuous on B A specific construction Let B 0 := span{k(, x) : x X }. For any f = m j=1 c jk(, x j ) B 0, define f B := c 1. δ x is continuous on B 0 if K(, ) is uniformly bounded. Let B be the Banach completion of B 0 with the norm B. Guohui Song Comp Math 14 of 21
12 Some Properties of RKBS [Song2010+] Point evaluation functional is continuous on B if and only if α j K(, x j ) = 0 = α = 0. j=1 [Song2010+] Reproducing property still holds. Define a bilinear form <, > on B 0 B 0 such that < m m α jk(, x j), β jk(, x j) >= α T K mβ j=1 j=1 The bilinear form <, > can be extended to B B such that < f, K(, x) >= f (x), x X, f B. Guohui Song Comp Math 15 of 21
13 Regularization in RKBS Target function: L(f, y, B) = n j=1 (f (x j) y j ) 2 + λ f B. Recall Sn = span{k(, x j ) : j = 1, 2,..., n}. Does the optimization problem reduce to finite-dimensional??? min L(f, y, B) = min L(f, y, B) f B f S n If it can reduce to finite-dimensional, how to find the minimizer Pf = n j=1 α jk(, x j )? Guohui Song Comp Math 16 of 21
14 Regularization and Interpolation Define the interpolation space I n (y) = {f B : f (x j ) = y j, j = 1, 2,..., n}. [Song2010+] The following two statements are equivalent. min L(f, y, B) = min L(f, y, B), for all y R n. f B f S n min f B = min f B, for all y R n. f I n(y) f I n(y) S n Note that In (y) S n has only one element when K n is invertible. We only need to show that the minimal norm interpolation problem admits a minimizer in the finite-dimensional space S n. Guohui Song Comp Math 17 of 21
15 Representer Theorem in RKBS Let k(x) := (K(x, x1 ),..., K(x, x n )) T. [Song2010+] Minimal norm interpolation min f B = min f B, for all y R n f I n (y) f I n (y) S n K n 1 k(x) 1 1, for all x X. [Song2010+] Regularization min L(f, y, B) = min L(f, y, B), for all y R n f B f S n K n 1 k(x) 1 1, for all x X. Guohui Song Comp Math 18 of 21
16 Some Examples The condition Kn 1 k(x) 1 1 is not easy to check. We have only been able to find two kernels satisfying it so far. K(s, t) = min{s, t} st, s, t [0, 1] K(s, t) = e s t, s, t R Counter examples that does not satisfy this condition Gaussian kernels: K(s, t) = e (s t)2, Sinc Kernel: K(s, t) = sinc(s t), s, t R s, t R Guohui Song Comp Math 19 of 21
17 How to find the minimizer? { } n min L(f, y, B) = min (f (x j ) y j ) 2 + λ c 1 : f = n c j K(, x j ) f S n j=1 We do not have a closed form of the minimizer. Standard optimization methods may do, but we still need efficient methods especially for large size of data. j=1 Guohui Song Comp Math 20 of 21
18 Thank you! Guohui Song Comp Math 21 of 21
MATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 2 Part 3: Native Space for Positive Definite Kernels Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2014 fasshauer@iit.edu MATH
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 12, 2007 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationOnline Gradient Descent Learning Algorithms
DISI, Genova, December 2006 Online Gradient Descent Learning Algorithms Yiming Ying (joint work with Massimiliano Pontil) Department of Computer Science, University College London Introduction Outline
More informationA graph based approach to semi-supervised learning
A graph based approach to semi-supervised learning 1 Feb 2011 Two papers M. Belkin, P. Niyogi, and V Sindhwani. Manifold regularization: a geometric framework for learning from labeled and unlabeled examples.
More informationReproducing Kernel Hilbert Spaces
9.520: Statistical Learning Theory and Applications February 10th, 2010 Reproducing Kernel Hilbert Spaces Lecturer: Lorenzo Rosasco Scribe: Greg Durrett 1 Introduction In the previous two lectures, we
More informationReproducing Kernel Hilbert Spaces Class 03, 15 February 2006 Andrea Caponnetto
Reproducing Kernel Hilbert Spaces 9.520 Class 03, 15 February 2006 Andrea Caponnetto About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationThe Learning Problem and Regularization Class 03, 11 February 2004 Tomaso Poggio and Sayan Mukherjee
The Learning Problem and Regularization 9.520 Class 03, 11 February 2004 Tomaso Poggio and Sayan Mukherjee About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 11, 2009 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More information10-701/ Recitation : Kernels
10-701/15-781 Recitation : Kernels Manojit Nandi February 27, 2014 Outline Mathematical Theory Banach Space and Hilbert Spaces Kernels Commonly Used Kernels Kernel Theory One Weird Kernel Trick Representer
More informationKernels MIT Course Notes
Kernels MIT 15.097 Course Notes Cynthia Rudin Credits: Bartlett, Schölkopf and Smola, Cristianini and Shawe-Taylor The kernel trick that I m going to show you applies much more broadly than SVM, but we
More informationCIS 520: Machine Learning Oct 09, Kernel Methods
CIS 520: Machine Learning Oct 09, 207 Kernel Methods Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture They may or may not cover all the material discussed
More informationMIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications. Class 04: Features and Kernels. Lorenzo Rosasco
MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications Class 04: Features and Kernels Lorenzo Rosasco Linear functions Let H lin be the space of linear functions f(x) = w x. f w is one
More informationStat542 (F11) Statistical Learning. First consider the scenario where the two classes of points are separable.
Linear SVM (separable case) First consider the scenario where the two classes of points are separable. It s desirable to have the width (called margin) between the two dashed lines to be large, i.e., have
More informationSemi-Nonparametric Inferences for Massive Data
Semi-Nonparametric Inferences for Massive Data Guang Cheng 1 Department of Statistics Purdue University Statistics Seminar at NCSU October, 2015 1 Acknowledge NSF, Simons Foundation and ONR. A Joint Work
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationStatistical learning on graphs
Statistical learning on graphs Jean-Philippe Vert Jean-Philippe.Vert@ensmp.fr ParisTech, Ecole des Mines de Paris Institut Curie INSERM U900 Seminar of probabilities, Institut Joseph Fourier, Grenoble,
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 33: Adaptive Iteration Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter 33 1 Outline 1 A
More informationReproducing Kernel Hilbert Spaces
Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 9, 2011 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 33: Adaptive Iteration Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter 33 1 Outline 1 A
More informationMATH 829: Introduction to Data Mining and Analysis Support vector machines and kernels
1/12 MATH 829: Introduction to Data Mining and Analysis Support vector machines and kernels Dominique Guillot Departments of Mathematical Sciences University of Delaware March 14, 2016 Separating sets:
More informationKarhunen-Loève decomposition of Gaussian measures on Banach spaces
Karhunen-Loève decomposition of Gaussian measures on Banach spaces Jean-Charles Croix GT APSSE - April 2017, the 13th joint work with Xavier Bay. 1 / 29 Sommaire 1 Preliminaries on Gaussian processes 2
More informationOutline. Motivation. Mapping the input space to the feature space Calculating the dot product in the feature space
to The The A s s in to Fabio A. González Ph.D. Depto. de Ing. de Sistemas e Industrial Universidad Nacional de Colombia, Bogotá April 2, 2009 to The The A s s in 1 Motivation Outline 2 The Mapping the
More informationData fitting by vector (V,f)-reproducing kernels
Data fitting by vector (V,f-reproducing kernels M-N. Benbourhim to appear in ESAIM.Proc 2007 Abstract In this paper we propose a constructive method to build vector reproducing kernels. We define the notion
More informationCOS 424: Interacting with Data
COS 424: Interacting with Data Lecturer: Rob Schapire Lecture #14 Scribe: Zia Khan April 3, 2007 Recall from previous lecture that in regression we are trying to predict a real value given our data. Specically,
More informationBernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION
Malaysian Journal of Mathematical Sciences 6(2): 25-36 (202) Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces Noli N. Reyes and Rosalio G. Artes Institute of Mathematics, University of
More information3. Some tools for the analysis of sequential strategies based on a Gaussian process prior
3. Some tools for the analysis of sequential strategies based on a Gaussian process prior E. Vazquez Computer experiments June 21-22, 2010, Paris 21 / 34 Function approximation with a Gaussian prior Aim:
More informationReproducing Kernel Banach Spaces for Machine Learning
Proceedings of International Joint Conference on Neural Networks, Atlanta, Georgia, USA, June 14-19, 2009 Reproducing Kernel Banach Spaces for Machine Learning Haizhang Zhang, Yuesheng Xu and Jun Zhang
More informationCompressive Inference
Compressive Inference Weihong Guo and Dan Yang Case Western Reserve University and SAMSI SAMSI transition workshop Project of Compressive Inference subgroup of Imaging WG Active members: Garvesh Raskutti,
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationIntroduction to Machine Learning
Introduction to Machine Learning Kernel Methods Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574 1 / 21
More informationCausal Inference by Minimizing the Dual Norm of Bias. Nathan Kallus. Cornell University and Cornell Tech
Causal Inference by Minimizing the Dual Norm of Bias Nathan Kallus Cornell University and Cornell Tech www.nathankallus.com Matching Zoo It s a zoo of matching estimators for causal effects: PSM, NN, CM,
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 14: The Power Function and Native Space Error Estimates Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu
More information5.6 Nonparametric Logistic Regression
5.6 onparametric Logistic Regression Dmitri Dranishnikov University of Florida Statistical Learning onparametric Logistic Regression onparametric? Doesnt mean that there are no parameters. Just means that
More informationDirect Learning: Linear Classification. Donglin Zeng, Department of Biostatistics, University of North Carolina
Direct Learning: Linear Classification Logistic regression models for classification problem We consider two class problem: Y {0, 1}. The Bayes rule for the classification is I(P(Y = 1 X = x) > 1/2) so
More informationBayesian Aggregation for Extraordinarily Large Dataset
Bayesian Aggregation for Extraordinarily Large Dataset Guang Cheng 1 Department of Statistics Purdue University www.science.purdue.edu/bigdata Department Seminar Statistics@LSE May 19, 2017 1 A Joint Work
More informationDiffeomorphic Warping. Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel)
Diffeomorphic Warping Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel) What Manifold Learning Isn t Common features of Manifold Learning Algorithms: 1-1 charting Dense sampling Geometric Assumptions
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationLecture 10: Support Vector Machine and Large Margin Classifier
Lecture 10: Support Vector Machine and Large Margin Classifier Applied Multivariate Analysis Math 570, Fall 2014 Xingye Qiao Department of Mathematical Sciences Binghamton University E-mail: qiao@math.binghamton.edu
More informationApproximate Kernel PCA with Random Features
Approximate Kernel PCA with Random Features (Computational vs. Statistical Tradeoff) Bharath K. Sriperumbudur Department of Statistics, Pennsylvania State University Journées de Statistique Paris May 28,
More informationCan we do statistical inference in a non-asymptotic way? 1
Can we do statistical inference in a non-asymptotic way? 1 Guang Cheng 2 Statistics@Purdue www.science.purdue.edu/bigdata/ ONR Review Meeting@Duke Oct 11, 2017 1 Acknowledge NSF, ONR and Simons Foundation.
More informationMercer s Theorem, Feature Maps, and Smoothing
Mercer s Theorem, Feature Maps, and Smoothing Ha Quang Minh, Partha Niyogi, and Yuan Yao Department of Computer Science, University of Chicago 00 East 58th St, Chicago, IL 60637, USA Department of Mathematics,
More informationRKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets Class 22, 2004 Tomaso Poggio and Sayan Mukherjee
RKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets 9.520 Class 22, 2004 Tomaso Poggio and Sayan Mukherjee About this class Goal To introduce an alternate perspective of RKHS via integral operators
More informationKernel Learning via Random Fourier Representations
Kernel Learning via Random Fourier Representations L. Law, M. Mider, X. Miscouridou, S. Ip, A. Wang Module 5: Machine Learning L. Law, M. Mider, X. Miscouridou, S. Ip, A. Wang Kernel Learning via Random
More informationMultivariate Interpolation with Increasingly Flat Radial Basis Functions of Finite Smoothness
Multivariate Interpolation with Increasingly Flat Radial Basis Functions of Finite Smoothness Guohui Song John Riddle Gregory E. Fasshauer Fred J. Hickernell Abstract In this paper, we consider multivariate
More informationKernels A Machine Learning Overview
Kernels A Machine Learning Overview S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola, Stéphane Canu, Mike Jordan and Peter
More informationSolving the 3D Laplace Equation by Meshless Collocation via Harmonic Kernels
Solving the 3D Laplace Equation by Meshless Collocation via Harmonic Kernels Y.C. Hon and R. Schaback April 9, Abstract This paper solves the Laplace equation u = on domains Ω R 3 by meshless collocation
More informationSupport Vector Machines
Wien, June, 2010 Paul Hofmarcher, Stefan Theussl, WU Wien Hofmarcher/Theussl SVM 1/21 Linear Separable Separating Hyperplanes Non-Linear Separable Soft-Margin Hyperplanes Hofmarcher/Theussl SVM 2/21 (SVM)
More informationStructured Prediction
Structured Prediction Ningshan Zhang Advanced Machine Learning, Spring 2016 Outline Ensemble Methods for Structured Prediction[1] On-line learning Boosting AGeneralizedKernelApproachtoStructuredOutputLearning[2]
More informationApproximation Theory on Manifolds
ATHEATICAL and COPUTATIONAL ETHODS Approximation Theory on anifolds JOSE ARTINEZ-ORALES Universidad Nacional Autónoma de éxico Instituto de atemáticas A.P. 273, Admon. de correos #3C.P. 62251 Cuernavaca,
More informationElements of Positive Definite Kernel and Reproducing Kernel Hilbert Space
Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department
More informationOnline gradient descent learning algorithm
Online gradient descent learning algorithm Yiming Ying and Massimiliano Pontil Department of Computer Science, University College London Gower Street, London, WCE 6BT, England, UK {y.ying, m.pontil}@cs.ucl.ac.uk
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationKernel Ridge Regression. Mohammad Emtiyaz Khan EPFL Oct 27, 2015
Kernel Ridge Regression Mohammad Emtiyaz Khan EPFL Oct 27, 2015 Mohammad Emtiyaz Khan 2015 Motivation The ridge solution β R D has a counterpart α R N. Using duality, we will establish a relationship between
More informationDivide and Conquer Kernel Ridge Regression. A Distributed Algorithm with Minimax Optimal Rates
: A Distributed Algorithm with Minimax Optimal Rates Yuchen Zhang, John C. Duchi, Martin Wainwright (UC Berkeley;http://arxiv.org/pdf/1305.509; Apr 9, 014) Gatsby Unit, Tea Talk June 10, 014 Outline Motivation.
More informationHilbert Space Methods in Learning
Hilbert Space Methods in Learning guest lecturer: Risi Kondor 6772 Advanced Machine Learning and Perception (Jebara), Columbia University, October 15, 2003. 1 1. A general formulation of the learning problem
More informationFunction Spaces. 1 Hilbert Spaces
Function Spaces A function space is a set of functions F that has some structure. Often a nonparametric regression function or classifier is chosen to lie in some function space, where the assume structure
More informationReduction of Model Complexity and the Treatment of Discrete Inputs in Computer Model Emulation
Reduction of Model Complexity and the Treatment of Discrete Inputs in Computer Model Emulation Curtis B. Storlie a a Los Alamos National Laboratory E-mail:storlie@lanl.gov Outline Reduction of Emulator
More informationAnalysis of Five Diagonal Reproducing Kernels
Bucknell University Bucknell Digital Commons Honors Theses Student Theses 2015 Analysis of Five Diagonal Reproducing Kernels Cody Bullett Stockdale cbs017@bucknelledu Follow this and additional works at:
More informationSupport Vector Machines for Classification: A Statistical Portrait
Support Vector Machines for Classification: A Statistical Portrait Yoonkyung Lee Department of Statistics The Ohio State University May 27, 2011 The Spring Conference of Korean Statistical Society KAIST,
More informationKernel Methods. Machine Learning A W VO
Kernel Methods Machine Learning A 708.063 07W VO Outline 1. Dual representation 2. The kernel concept 3. Properties of kernels 4. Examples of kernel machines Kernel PCA Support vector regression (Relevance
More informationAN INTRODUCTION TO THE THEORY OF REPRODUCING KERNEL HILBERT SPACES
AN INTRODUCTION TO THE THEORY OF REPRODUCING KERNEL HILBERT SPACES VERN I PAULSEN Abstract These notes give an introduction to the theory of reproducing kernel Hilbert spaces and their multipliers We begin
More informationNotes on Regularized Least Squares Ryan M. Rifkin and Ross A. Lippert
Computer Science and Artificial Intelligence Laboratory Technical Report MIT-CSAIL-TR-2007-025 CBCL-268 May 1, 2007 Notes on Regularized Least Squares Ryan M. Rifkin and Ross A. Lippert massachusetts institute
More informationKernel Methods. Jean-Philippe Vert Last update: Jan Jean-Philippe Vert (Mines ParisTech) 1 / 444
Kernel Methods Jean-Philippe Vert Jean-Philippe.Vert@mines.org Last update: Jan 2015 Jean-Philippe Vert (Mines ParisTech) 1 / 444 What we know how to solve Jean-Philippe Vert (Mines ParisTech) 2 / 444
More informationMath 240 (Driver) Qual Exam (9/12/2017)
1 Name: I.D. #: Math 240 (Driver) Qual Exam (9/12/2017) Instructions: Clearly explain and justify your answers. You may cite theorems from the text, notes, or class as long as they are not what the problem
More informationSupport Vector Machine
Support Vector Machine Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Linear Support Vector Machine Kernelized SVM Kernels 2 From ERM to RLM Empirical Risk Minimization in the binary
More informationSpatial Process Estimates as Smoothers: A Review
Spatial Process Estimates as Smoothers: A Review Soutir Bandyopadhyay 1 Basic Model The observational model considered here has the form Y i = f(x i ) + ɛ i, for 1 i n. (1.1) where Y i is the observed
More informationAn Introduction to Kernel Methods 1
An Introduction to Kernel Methods 1 Yuri Kalnishkan Technical Report CLRC TR 09 01 May 2009 Department of Computer Science Egham, Surrey TW20 0EX, England 1 This paper has been written for wiki project
More informationBeyond the Point Cloud: From Transductive to Semi-Supervised Learning
Beyond the Point Cloud: From Transductive to Semi-Supervised Learning Vikas Sindhwani, Partha Niyogi, Mikhail Belkin Andrew B. Goldberg goldberg@cs.wisc.edu Department of Computer Sciences University of
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationDIRECT ERROR BOUNDS FOR SYMMETRIC RBF COLLOCATION
Meshless Methods in Science and Engineering - An International Conference Porto, 22 DIRECT ERROR BOUNDS FOR SYMMETRIC RBF COLLOCATION Robert Schaback Institut für Numerische und Angewandte Mathematik (NAM)
More informationRepresenter theorem and kernel examples
CS81B/Stat41B Spring 008) Statistical Learning Theory Lecture: 8 Representer theorem and kernel examples Lecturer: Peter Bartlett Scribe: Howard Lei 1 Representer Theorem Recall that the SVM optimization
More informationUNBOUNDED OPERATORS ON HILBERT SPACES. Let X and Y be normed linear spaces, and suppose A : X Y is a linear map.
UNBOUNDED OPERATORS ON HILBERT SPACES EFTON PARK Let X and Y be normed linear spaces, and suppose A : X Y is a linear map. Define { } Ax A op = sup x : x 0 = { Ax : x 1} = { Ax : x = 1} If A
More informationTHE GAUSSIAN RADON TRANSFORM AND MACHINE LEARNING
THE GAUSSIAN RADON TRANSFORM AND MACHINE LEARNING IRINA HOLMES AND AMBAR N. SENGUPTA Abstract. There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as
More informationReproducing Kernel Banach Spaces for Machine Learning
Journal of Machine Learning Research 10 (2009) 2741-2775 Submitted 2/09; Revised 7/09; Published 12/09 Reproducing Kernel Banach Spaces for Machine Learning Haizhang Zhang Yuesheng Xu Department of Mathematics
More informationScattered Data Approximation of Noisy Data via Iterated Moving Least Squares
Scattered Data Approximation o Noisy Data via Iterated Moving Least Squares Gregory E. Fasshauer and Jack G. Zhang Abstract. In this paper we ocus on two methods or multivariate approximation problems
More informationOutline of Fourier Series: Math 201B
Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C
More informationStability of Kernel Based Interpolation
Stability of Kernel Based Interpolation Stefano De Marchi Department of Computer Science, University of Verona (Italy) Robert Schaback Institut für Numerische und Angewandte Mathematik, University of Göttingen
More informationKernels for Multi task Learning
Kernels for Multi task Learning Charles A Micchelli Department of Mathematics and Statistics State University of New York, The University at Albany 1400 Washington Avenue, Albany, NY, 12222, USA Massimiliano
More informationKernel methods and the exponential family
Kernel methods and the exponential family Stéphane Canu 1 and Alex J. Smola 2 1- PSI - FRE CNRS 2645 INSA de Rouen, France St Etienne du Rouvray, France Stephane.Canu@insa-rouen.fr 2- Statistical Machine
More informationNonparametric Regression. Badr Missaoui
Badr Missaoui Outline Kernel and local polynomial regression. Penalized regression. We are given n pairs of observations (X 1, Y 1 ),...,(X n, Y n ) where Y i = r(x i ) + ε i, i = 1,..., n and r(x) = E(Y
More informationKernel-based Approximation. Methods using MATLAB. Gregory Fasshauer. Interdisciplinary Mathematical Sciences. Michael McCourt.
SINGAPORE SHANGHAI Vol TAIPEI - Interdisciplinary Mathematical Sciences 19 Kernel-based Approximation Methods using MATLAB Gregory Fasshauer Illinois Institute of Technology, USA Michael McCourt University
More informationCS 7140: Advanced Machine Learning
Instructor CS 714: Advanced Machine Learning Lecture 3: Gaussian Processes (17 Jan, 218) Jan-Willem van de Meent (j.vandemeent@northeastern.edu) Scribes Mo Han (han.m@husky.neu.edu) Guillem Reus Muns (reusmuns.g@husky.neu.edu)
More informationChapter 9. Support Vector Machine. Yongdai Kim Seoul National University
Chapter 9. Support Vector Machine Yongdai Kim Seoul National University 1. Introduction Support Vector Machine (SVM) is a classification method developed by Vapnik (1996). It is thought that SVM improved
More informationStatistical Convergence of Kernel CCA
Statistical Convergence of Kernel CCA Kenji Fukumizu Institute of Statistical Mathematics Tokyo 106-8569 Japan fukumizu@ism.ac.jp Francis R. Bach Centre de Morphologie Mathematique Ecole des Mines de Paris,
More informationTRANSLATION INVARIANCE OF FOCK SPACES
TRANSLATION INVARIANCE OF FOCK SPACES KEHE ZHU ABSTRACT. We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.
More informationReproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
Noname manuscript No. (will be inserted by the editor) Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators Gregory E. Fasshauer Qi Ye Abstract
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 6: Scattered Data Interpolation with Polynomial Precision Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu
More information1. SS-ANOVA Spaces on General Domains. 2. Averaging Operators and ANOVA Decompositions. 3. Reproducing Kernel Spaces for ANOVA Decompositions
Part III 1. SS-ANOVA Spaces on General Domains 2. Averaging Operators and ANOVA Decompositions 3. Reproducing Kernel Spaces for ANOVA Decompositions 4. Building Blocks for SS-ANOVA Spaces, General and
More informationComplexity and regularization issues in kernel-based learning
Complexity and regularization issues in kernel-based learning Marcello Sanguineti Department of Communications, Computer, and System Sciences (DIST) University of Genoa - Via Opera Pia 13, 16145 Genova,
More informationKernel Methods. Outline
Kernel Methods Quang Nguyen University of Pittsburgh CS 3750, Fall 2011 Outline Motivation Examples Kernels Definitions Kernel trick Basic properties Mercer condition Constructing feature space Hilbert
More informationAre Loss Functions All the Same?
Are Loss Functions All the Same? L. Rosasco E. De Vito A. Caponnetto M. Piana A. Verri November 11, 2003 Abstract In this paper we investigate the impact of choosing different loss functions from the viewpoint
More informationLecture 4 Colorization and Segmentation
Lecture 4 Colorization and Segmentation Summer School Mathematics in Imaging Science University of Bologna, Itay June 1st 2018 Friday 11:15-13:15 Sung Ha Kang School of Mathematics Georgia Institute of
More informationIntroduction to Bases in Banach Spaces
Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers
More informationMath Solutions to homework 5
Math 75 - Solutions to homework 5 Cédric De Groote November 9, 207 Problem (7. in the book): Let {e n } be a complete orthonormal sequence in a Hilbert space H and let λ n C for n N. Show that there is
More informationNormality of adjointable module maps
MATHEMATICAL COMMUNICATIONS 187 Math. Commun. 17(2012), 187 193 Normality of adjointable module maps Kamran Sharifi 1, 1 Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316,
More informationThe Subspace Information Criterion for Infinite Dimensional Hypothesis Spaces
Journal of Machine Learning Research 3 (22) 323-359 Submitted 12/1; Revised 8/2; Published 11/2 The Subspace Information Criterion for Infinite Dimensional Hypothesis Spaces Masashi Sugiyama Department
More informationKernel Bayes Rule: Nonparametric Bayesian inference with kernels
Kernel Bayes Rule: Nonparametric Bayesian inference with kernels Kenji Fukumizu The Institute of Statistical Mathematics NIPS 2012 Workshop Confluence between Kernel Methods and Graphical Models December
More informationBasis Expansion and Nonlinear SVM. Kai Yu
Basis Expansion and Nonlinear SVM Kai Yu Linear Classifiers f(x) =w > x + b z(x) = sign(f(x)) Help to learn more general cases, e.g., nonlinear models 8/7/12 2 Nonlinear Classifiers via Basis Expansion
More informationComputation Of Asymptotic Distribution. For Semiparametric GMM Estimators. Hidehiko Ichimura. Graduate School of Public Policy
Computation Of Asymptotic Distribution For Semiparametric GMM Estimators Hidehiko Ichimura Graduate School of Public Policy and Graduate School of Economics University of Tokyo A Conference in honor of
More information