Introduction to Machine Learning. Recitation 11
|
|
- Phebe Parrish
- 5 years ago
- Views:
Transcription
1 Introduction to Machine Learning Lecturer: Regev Schweiger Recitation Fall Seester Scribe: Regev Schweiger. Kernel Ridge Regression We now take on the task of kernel-izing ridge regression. Let x,..., x R d, and y R. Recall that ridge regression solves the following proble: arg in a R d y Xa 2 + λ a 2 where λ is penalty coefficient. Equating the gradient to 0 result in the solution we have seen in class: â = (X T X + λi d ) X T y Note that (X T X + λi d )X T = X T XX T + λx T = X T (XX T + λi n ) Multiplying (X T X + λi d ) at the left and (XX T + λi n ) at the right, we get X T (XX T + λi n ) = (X T X + λi d ) X T Therefore, the optial solution is equivalently, â = X T (XX T + λi n ) y Given a new point x, our regression estiate will be x T â = x T X T (XX T + λi n ) y We would now like to ebed our points to a space H, with x i φ(x i ), and perfor ridge regression after the transforation. It is easy to see that, given the forulation above, we can replace all the expressions involving X with kernel expressions. First define K as the atrix for which K i,j = K(x i, x j ). Siilarly define k as the vector for which k i = φ(x) T φ(x i ) = K(x, x i ). Thus, given a new point x, our regression estiate will be φ(x) T â = k T ( K + λi n ) y Note that, as usual, we cannot write down â explicitly, but we can apply it to the transforation of new points.
2 2 Lecture.2 PCA as axiizing variance We have seen how the PCA algorith can be derived in the context of iniizing the reconstruction error. More forally, assue we have a set of input vectors x,..., x, where x i R d. Denote the principal coponents by the coluns of V, as v,..., v r ; the orthonorality constraints iply that V T V = I. The PCA proble was: V = arg in V R d r x i V V T x i 2. (.) i= We now consider another possible criterion. Let s assue r =, that is, we would like to find the best line in soe sense. One intuitive criterion is the line, which if we project all points on, will give axial epirical variance. The epirical variance of a set of easureents, a,..., a, is (a i i= a j ) 2 j= Assue without loss of generality that the data points are centered at zero, that is x i = 0 i= If that is not the case, we ean-center the data. Therefore, it is easy to say that i= vt x i = 0 for each v. Therefore, the epirical variance of the set of projection is siply the ean of squares. Therefore, the criterion we like for the first direction is: v = arg ax v = (v T x i ) 2 For the next direction, we would like to capture the variance on directions we have not yet seen. Forally, we would like directions orthogonal to previous directions. Assue we found already v,..., v r. Then, the r-th direction is: i= v r = argax v =,v v,...,v r (v T x i ) 2 We can instead forulate that to find all r directions together, to get: i= argax V R d r,v T V =I r j= (vj T x i ) 2 i=
3 .3. PCA EXAMPLE 3 It is easy to see that the optiization function is: r j= (vj T x i ) 2 = i= r (vj T x i ) 2 = i= j= V T x i 2 i= To suarize, a sensible criterion for diensionality reduction would be to choose V so that the variance of projections is axiized, i.e., intuitively the structure of the data is preserved as uch as possible: argax V R d r,v T V =I We note, however, the following equality, based on Pythagoras: V T x i 2. (.2) i= x i 2 = V V T x i 2 + x i V V T x i 2. And it is easy to see that V V T x i 2 = V T x i 2 due to the orthonorality of V. Since x i does not depend on V, we see that iniizing the reconstruction error is equivalent to axiizing the variance. The goal in principal coponent analysis (PCA) is therefore to iniize the reconstruction error (see Equation.), and to axiize the projected variance (Equation.2). Eigenvalues. An iportant observation is the following. We know that the solution of PCA is the eigenvectors of the epirical covariance atrix. What are the eigenvalues? The variance axiization criterion gives an intuitive interpretation. Let λ,..., λ n be the eigenvalues of C = i= x ix T i ; i.e., Cv j = λ i v j. We seeked to axiize i= (vt x i ) 2. Plugging in v = v j, we get ( (vj T x i ) 2 = ) vt j x i x T i v j = vj T Cv j = λ j i= i= That is, λ j, the j-th eigenvalue, is the epirical variance of the projection on the j-th principal axis..3 PCA exaple.3. Background The DNA in our cells contains long chains of four cheical building blocks adenine, thyine, cytosine, and guanine, abbreviated A, T, C, and G. More than 6 billion of these
4 4 Lecture cheical bases, strung together in 23 pairs of chroosoes, exist in a huan cell. These genetic sequences contain inforation that influences our physical traits, our likelihood of suffering fro disease, and the responses of our bodies to substances that we encounter in the environent. The genetic sequences of different people are rearkably siilar. When the chroosoes of two huans are copared, their DNA sequences can be identical for hundreds of bases. But at about one in every,200 bases, on average, the sequences will differ. Differences in individual bases are by far the ost coon type of genetic variation. One person ight have an A at that location, while another person has a G. These genetic differences are known as single nucleotide polyorphiss, or SNPs (pronounced snips ). There are approxiately 0 illion SNPs estiated to occur coonly in the huan genoe. Each distinct spelling of a chroosoal region is called an allele, and a collection of alleles in a person s chroosoes is known as a genotype. In the ost coon case, there are only two alleles for all population at each SNP position. Data describing the genotype data for individuals, often does not specify the bases explicitly. Instead, one allele (per position) is selected as a reference allele. Then, at that position, the nuber of non-reference alleles is presented: 0 if both alleles in that position, in the chroosoe pair, were identical to the reference allele for that position; if only one of the was the reference allele; and 2 if neither were the reference alleles..3.2 Novebre et al., 2008 In the work of Novebre et al. 2008, Nature, 3,92 European individuals were genotyped at 500,568 positions (soe details are oitted for siplicity). They applied PCA with r = 2 and presented the projections of all genoes on these two principal axes:
5 .3. PCA EXAMPLE 5 Each individuals is denoted by colored two-letters, denoting their country of origin. It can be seen that the projections reflect the geography of Europe well.
3.3 Variational Characterization of Singular Values
3.3. Variational Characterization of Singular Values 61 3.3 Variational Characterization of Singular Values Since the singular values are square roots of the eigenvalues of the Heritian atrices A A and
More informationMachine Learning Basics: Estimators, Bias and Variance
Machine Learning Basics: Estiators, Bias and Variance Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics in Basics
More informationPrincipal Components Analysis
Principal Coponents Analysis Cheng Li, Bingyu Wang Noveber 3, 204 What s PCA Principal coponent analysis (PCA) is a statistical procedure that uses an orthogonal transforation to convert a set of observations
More informationCOS 424: Interacting with Data. Written Exercises
COS 424: Interacting with Data Hoework #4 Spring 2007 Regression Due: Wednesday, April 18 Written Exercises See the course website for iportant inforation about collaboration and late policies, as well
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationModel Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationBootstrapping Dependent Data
Bootstrapping Dependent Data One of the key issues confronting bootstrap resapling approxiations is how to deal with dependent data. Consider a sequence fx t g n t= of dependent rando variables. Clearly
More informationKernel Methods and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley ENSIAG 2 / osig 1 Second Seester 2012/2013 Lesson 20 2 ay 2013 Kernel ethods and Support Vector achines Contents Kernel Functions...2 Quadratic
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE7C (Spring 018: Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee7c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee7c@berkeley.edu October 15,
More informationAn l 1 Regularized Method for Numerical Differentiation Using Empirical Eigenfunctions
Journal of Matheatical Research with Applications Jul., 207, Vol. 37, No. 4, pp. 496 504 DOI:0.3770/j.issn:2095-265.207.04.0 Http://jre.dlut.edu.cn An l Regularized Method for Nuerical Differentiation
More informationCS Lecture 13. More Maximum Likelihood
CS 6347 Lecture 13 More Maxiu Likelihood Recap Last tie: Introduction to axiu likelihood estiation MLE for Bayesian networks Optial CPTs correspond to epirical counts Today: MLE for CRFs 2 Maxiu Likelihood
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More informationSupport Vector Machines. Maximizing the Margin
Support Vector Machines Support vector achines (SVMs) learn a hypothesis: h(x) = b + Σ i= y i α i k(x, x i ) (x, y ),..., (x, y ) are the training exs., y i {, } b is the bias weight. α,..., α are the
More information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More informationare equal to zero, where, q = p 1. For each gene j, the pairwise null and alternative hypotheses are,
Page of 8 Suppleentary Materials: A ultiple testing procedure for ulti-diensional pairwise coparisons with application to gene expression studies Anjana Grandhi, Wenge Guo, Shyaal D. Peddada S Notations
More informationMachine Learning: Fisher s Linear Discriminant. Lecture 05
Machine Learning: Fisher s Linear Discriinant Lecture 05 Razvan C. Bunescu chool of Electrical Engineering and Coputer cience bunescu@ohio.edu Lecture 05 upervised Learning ask learn an (unkon) function
More informationUnsupervised Learning: Dimension Reduction
Unsupervised Learning: Diension Reduction by Prof. Seungchul Lee isystes Design Lab http://isystes.unist.ac.kr/ UNIST Table of Contents I.. Principal Coponent Analysis (PCA) II. 2. PCA Algorith I. 2..
More informationPattern Recognition and Machine Learning. Learning and Evaluation for Pattern Recognition
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lesson 1 4 October 2017 Outline Learning and Evaluation for Pattern Recognition Notation...2 1. The Pattern Recognition
More informationBoosting with log-loss
Boosting with log-loss Marco Cusuano-Towner Septeber 2, 202 The proble Suppose we have data exaples {x i, y i ) i =... } for a two-class proble with y i {, }. Let F x) be the predictor function with the
More informationOptimal Jamming Over Additive Noise: Vector Source-Channel Case
Fifty-first Annual Allerton Conference Allerton House, UIUC, Illinois, USA October 2-3, 2013 Optial Jaing Over Additive Noise: Vector Source-Channel Case Erah Akyol and Kenneth Rose Abstract This paper
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2016/2017 Lessons 9 11 Jan 2017 Outline Artificial Neural networks Notation...2 Convolutional Neural Networks...3
More informationBest Arm Identification: A Unified Approach to Fixed Budget and Fixed Confidence
Best Ar Identification: A Unified Approach to Fixed Budget and Fixed Confidence Victor Gabillon Mohaad Ghavazadeh Alessandro Lazaric INRIA Lille - Nord Europe, Tea SequeL {victor.gabillon,ohaad.ghavazadeh,alessandro.lazaric}@inria.fr
More informationMultivariate Methods. Matlab Example. Principal Components Analysis -- PCA
Multivariate Methos Xiaoun Qi Principal Coponents Analysis -- PCA he PCA etho generates a new set of variables, calle principal coponents Each principal coponent is a linear cobination of the original
More informationOn the Impact of Kernel Approximation on Learning Accuracy
On the Ipact of Kernel Approxiation on Learning Accuracy Corinna Cortes Mehryar Mohri Aeet Talwalkar Google Research New York, NY corinna@google.co Courant Institute and Google Research New York, NY ohri@cs.nyu.edu
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee227c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee227c@berkeley.edu October
More informationNon-Parametric Non-Line-of-Sight Identification 1
Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
More informationTopic 5a Introduction to Curve Fitting & Linear Regression
/7/08 Course Instructor Dr. Rayond C. Rup Oice: A 337 Phone: (95) 747 6958 E ail: rcrup@utep.edu opic 5a Introduction to Curve Fitting & Linear Regression EE 4386/530 Coputational ethods in EE Outline
More informationSlide10. Haykin Chapter 8: Principal Components Analysis. Motivation. Principal Component Analysis: Variance Probe
Slide10 Motivation Haykin Chapter 8: Principal Coponents Analysis 1.6 1.4 1.2 1 0.8 cloud.dat 0.6 CPSC 636-600 Instructor: Yoonsuck Choe Spring 2015 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 How can we
More informationExperimental Design For Model Discrimination And Precise Parameter Estimation In WDS Analysis
City University of New York (CUNY) CUNY Acadeic Works International Conference on Hydroinforatics 8-1-2014 Experiental Design For Model Discriination And Precise Paraeter Estiation In WDS Analysis Giovanna
More informationSupport Vector Machines. Goals for the lecture
Support Vector Machines Mark Craven and David Page Coputer Sciences 760 Spring 2018 www.biostat.wisc.edu/~craven/cs760/ Soe of the slides in these lectures have been adapted/borrowed fro aterials developed
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.it.edu 16.323 Principles of Optial Control Spring 2008 For inforation about citing these aterials or our Ters of Use, visit: http://ocw.it.edu/ters. 16.323 Lecture 10 Singular
More informationBayes Decision Rule and Naïve Bayes Classifier
Bayes Decision Rule and Naïve Bayes Classifier Le Song Machine Learning I CSE 6740, Fall 2013 Gaussian Mixture odel A density odel p(x) ay be ulti-odal: odel it as a ixture of uni-odal distributions (e.g.
More informationSupport Vector Machines. Machine Learning Series Jerry Jeychandra Blohm Lab
Support Vector Machines Machine Learning Series Jerry Jeychandra Bloh Lab Outline Main goal: To understand how support vector achines (SVMs) perfor optial classification for labelled data sets, also a
More informationOBJECTIVES INTRODUCTION
M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and
More informationLeast Squares Fitting of Data
Least Squares Fitting of Data David Eberly, Geoetric Tools, Redond WA 98052 https://www.geoetrictools.co/ This work is licensed under the Creative Coons Attribution 4.0 International License. To view a
More informationDEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS
ISSN 1440-771X AUSTRALIA DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS An Iproved Method for Bandwidth Selection When Estiating ROC Curves Peter G Hall and Rob J Hyndan Working Paper 11/00 An iproved
More informationMulti-Scale/Multi-Resolution: Wavelet Transform
Multi-Scale/Multi-Resolution: Wavelet Transfor Proble with Fourier Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. A serious drawback in transforing to the
More informationRecovering Data from Underdetermined Quadratic Measurements (CS 229a Project: Final Writeup)
Recovering Data fro Underdeterined Quadratic Measureents (CS 229a Project: Final Writeup) Mahdi Soltanolkotabi Deceber 16, 2011 1 Introduction Data that arises fro engineering applications often contains
More informationPrincipal Component Analysis
CSci 5525: Machine Learning Dec 3, 2008 The Main Idea Given a dataset X = {x 1,..., x N } The Main Idea Given a dataset X = {x 1,..., x N } Find a low-dimensional linear projection The Main Idea Given
More informationInspection; structural health monitoring; reliability; Bayesian analysis; updating; decision analysis; value of information
Cite as: Straub D. (2014). Value of inforation analysis with structural reliability ethods. Structural Safety, 49: 75-86. Value of Inforation Analysis with Structural Reliability Methods Daniel Straub
More informationMultiple Testing Issues & K-Means Clustering. Definitions related to the significance level (or type I error) of multiple tests
StatsM254 Statistical Methods in Coputational Biology Lecture 3-04/08/204 Multiple Testing Issues & K-Means Clustering Lecturer: Jingyi Jessica Li Scribe: Arturo Rairez Multiple Testing Issues When trying
More informationBipartite subgraphs and the smallest eigenvalue
Bipartite subgraphs and the sallest eigenvalue Noga Alon Benny Sudaov Abstract Two results dealing with the relation between the sallest eigenvalue of a graph and its bipartite subgraphs are obtained.
More informationDistributed Subgradient Methods for Multi-agent Optimization
1 Distributed Subgradient Methods for Multi-agent Optiization Angelia Nedić and Asuan Ozdaglar October 29, 2007 Abstract We study a distributed coputation odel for optiizing a su of convex objective functions
More informationInteractive Markov Models of Evolutionary Algorithms
Cleveland State University EngagedScholarship@CSU Electrical Engineering & Coputer Science Faculty Publications Electrical Engineering & Coputer Science Departent 2015 Interactive Markov Models of Evolutionary
More informationIntroduction to Machine Learning. PCA and Spectral Clustering. Introduction to Machine Learning, Slides: Eran Halperin
1 Introduction to Machine Learning PCA and Spectral Clustering Introduction to Machine Learning, 2013-14 Slides: Eran Halperin Singular Value Decomposition (SVD) The singular value decomposition (SVD)
More informationComputational and Statistical Learning Theory
Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher
More informationA New Class of APEX-Like PCA Algorithms
Reprinted fro Proceedings of ISCAS-98, IEEE Int. Syposiu on Circuit and Systes, Monterey (USA), June 1998 A New Class of APEX-Like PCA Algoriths Sione Fiori, Aurelio Uncini, Francesco Piazza Dipartiento
More informationUsing EM To Estimate A Probablity Density With A Mixture Of Gaussians
Using EM To Estiate A Probablity Density With A Mixture Of Gaussians Aaron A. D Souza adsouza@usc.edu Introduction The proble we are trying to address in this note is siple. Given a set of data points
More informationEstimating Parameters for a Gaussian pdf
Pattern Recognition and achine Learning Jaes L. Crowley ENSIAG 3 IS First Seester 00/0 Lesson 5 7 Noveber 00 Contents Estiating Paraeters for a Gaussian pdf Notation... The Pattern Recognition Proble...3
More informationIntelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes
More informationOn the Use of A Priori Information for Sparse Signal Approximations
ITS TECHNICAL REPORT NO. 3/4 On the Use of A Priori Inforation for Sparse Signal Approxiations Oscar Divorra Escoda, Lorenzo Granai and Pierre Vandergheynst Signal Processing Institute ITS) Ecole Polytechnique
More informationSupport Vector Machines MIT Course Notes Cynthia Rudin
Support Vector Machines MIT 5.097 Course Notes Cynthia Rudin Credit: Ng, Hastie, Tibshirani, Friedan Thanks: Şeyda Ertekin Let s start with soe intuition about argins. The argin of an exaple x i = distance
More informationCh 12: Variations on Backpropagation
Ch 2: Variations on Backpropagation The basic backpropagation algorith is too slow for ost practical applications. It ay take days or weeks of coputer tie. We deonstrate why the backpropagation algorith
More informationLinear Transformations
Linear Transforations Hopfield Network Questions Initial Condition Recurrent Layer p S x W S x S b n(t + ) a(t + ) S x S x D a(t) S x S S x S a(0) p a(t + ) satlins (Wa(t) + b) The network output is repeatedly
More informationA Simplified Analytical Approach for Efficiency Evaluation of the Weaving Machines with Automatic Filling Repair
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving
More informationNyström Method vs Random Fourier Features: A Theoretical and Empirical Comparison
yströ Method vs : A Theoretical and Epirical Coparison Tianbao Yang, Yu-Feng Li, Mehrdad Mahdavi, Rong Jin, Zhi-Hua Zhou Machine Learning Lab, GE Global Research, San Raon, CA 94583 Michigan State University,
More informationLower Bounds for Quantized Matrix Completion
Lower Bounds for Quantized Matrix Copletion Mary Wootters and Yaniv Plan Departent of Matheatics University of Michigan Ann Arbor, MI Eail: wootters, yplan}@uich.edu Mark A. Davenport School of Elec. &
More informationOn Hyper-Parameter Estimation in Empirical Bayes: A Revisit of the MacKay Algorithm
On Hyper-Paraeter Estiation in Epirical Bayes: A Revisit of the MacKay Algorith Chune Li 1, Yongyi Mao, Richong Zhang 1 and Jinpeng Huai 1 1 School of Coputer Science and Engineering, Beihang University,
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationTowards Gauge Invariant Bundle Adjustment: A Solution Based on Gauge Dependent Damping
EXTENDED VERSION SHORT VERSION APPEARED IN THE 9TH ICCV, NICE, FRANCE, OCTOBER 003. Towards Gauge Invariant Bundle Adjustent: A Solution Based on Gauge Dependent Daping Adrien Bartoli INRIA Rhône-Alpes,
More informationConsistent Multiclass Algorithms for Complex Performance Measures. Supplementary Material
Consistent Multiclass Algoriths for Coplex Perforance Measures Suppleentary Material Notations. Let λ be the base easure over n given by the unifor rando variable (say U over n. Hence, for all easurable
More informationSoft-margin SVM can address linearly separable problems with outliers
Non-linear Support Vector Machines Non-linearly separable probles Hard-argin SVM can address linearly separable probles Soft-argin SVM can address linearly separable probles with outliers Non-linearly
More informationA remark on a success rate model for DPA and CPA
A reark on a success rate odel for DPA and CPA A. Wieers, BSI Version 0.5 andreas.wieers@bsi.bund.de Septeber 5, 2018 Abstract The success rate is the ost coon evaluation etric for easuring the perforance
More informationEMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS
EMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS Jochen Till, Sebastian Engell, Sebastian Panek, and Olaf Stursberg Process Control Lab (CT-AST), University of Dortund,
More informationSupplementary to Learning Discriminative Bayesian Networks from High-dimensional Continuous Neuroimaging Data
Suppleentary to Learning Discriinative Bayesian Networks fro High-diensional Continuous Neuroiaging Data Luping Zhou, Lei Wang, Lingqiao Liu, Philip Ogunbona, and Dinggang Shen Proposition. Given a sparse
More informationpaper prepared for the 1996 PTRC Conference, September 2-6, Brunel University, UK ON THE CALIBRATION OF THE GRAVITY MODEL
paper prepared for the 1996 PTRC Conference, Septeber 2-6, Brunel University, UK ON THE CALIBRATION OF THE GRAVITY MODEL Nanne J. van der Zijpp 1 Transportation and Traffic Engineering Section Delft University
More informationResearch Article Robust ε-support Vector Regression
Matheatical Probles in Engineering, Article ID 373571, 5 pages http://dx.doi.org/10.1155/2014/373571 Research Article Robust ε-support Vector Regression Yuan Lv and Zhong Gan School of Mechanical Engineering,
More informationLecture 13 Eigenvalue Problems
Lecture 13 Eigenvalue Probles MIT 18.335J / 6.337J Introduction to Nuerical Methods Per-Olof Persson October 24, 2006 1 The Eigenvalue Decoposition Eigenvalue proble for atrix A: Ax = λx with eigenvalues
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationSupport Vector Machine Classification of Uncertain and Imbalanced data using Robust Optimization
Recent Researches in Coputer Science Support Vector Machine Classification of Uncertain and Ibalanced data using Robust Optiization RAGHAV PAT, THEODORE B. TRAFALIS, KASH BARKER School of Industrial Engineering
More informationThe Methods of Solution for Constrained Nonlinear Programming
Research Inventy: International Journal Of Engineering And Science Vol.4, Issue 3(March 2014), PP 01-06 Issn (e): 2278-4721, Issn (p):2319-6483, www.researchinventy.co The Methods of Solution for Constrained
More informationA Self-Organizing Model for Logical Regression Jerry Farlow 1 University of Maine. (1900 words)
1 A Self-Organizing Model for Logical Regression Jerry Farlow 1 University of Maine (1900 words) Contact: Jerry Farlow Dept of Matheatics Univeristy of Maine Orono, ME 04469 Tel (07) 866-3540 Eail: farlow@ath.uaine.edu
More informationPAC-Bayes Analysis Of Maximum Entropy Learning
PAC-Bayes Analysis Of Maxiu Entropy Learning John Shawe-Taylor and David R. Hardoon Centre for Coputational Statistics and Machine Learning Departent of Coputer Science University College London, UK, WC1E
More informationThis model assumes that the probability of a gap has size i is proportional to 1/i. i.e., i log m e. j=1. E[gap size] = i P r(i) = N f t.
CS 493: Algoriths for Massive Data Sets Feb 2, 2002 Local Models, Bloo Filter Scribe: Qin Lv Local Models In global odels, every inverted file entry is copressed with the sae odel. This work wells when
More informationLeonardo R. Bachega*, Student Member, IEEE, Srikanth Hariharan, Student Member, IEEE Charles A. Bouman, Fellow, IEEE, and Ness Shroff, Fellow, IEEE
Distributed Signal Decorrelation and Detection in Sensor Networks Using the Vector Sparse Matrix Transfor Leonardo R Bachega*, Student Meber, I, Srikanth Hariharan, Student Meber, I Charles A Bouan, Fellow,
More informationIntroduction to Kernel methods
Introduction to Kernel ethods ML Workshop, ISI Kolkata Chiranjib Bhattacharyya Machine Learning lab Dept of CSA, IISc chiru@csa.iisc.ernet.in http://drona.csa.iisc.ernet.in/~chiru 19th Oct, 2012 Introduction
More informationHighly Robust Error Correction by Convex Programming
Highly Robust Error Correction by Convex Prograing Eanuel J. Candès and Paige A. Randall Applied and Coputational Matheatics, Caltech, Pasadena, CA 9115 Noveber 6; Revised Noveber 7 Abstract This paper
More informationIntelligent Systems: Reasoning and Recognition. Artificial Neural Networks
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley MOSIG M1 Winter Seester 2018 Lesson 7 1 March 2018 Outline Artificial Neural Networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationPrediction by random-walk perturbation
Prediction by rando-walk perturbation Luc Devroye School of Coputer Science McGill University Gábor Lugosi ICREA and Departent of Econoics Universitat Popeu Fabra lucdevroye@gail.co gabor.lugosi@gail.co
More informationLecture 21. Interior Point Methods Setup and Algorithm
Lecture 21 Interior Point Methods In 1984, Kararkar introduced a new weakly polynoial tie algorith for solving LPs [Kar84a], [Kar84b]. His algorith was theoretically faster than the ellipsoid ethod and
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2016 Lessons 7 14 Dec 2016 Outline Artificial Neural networks Notation...2 1. Introduction...3... 3 The Artificial
More informationW-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS
W-BASED VS LATENT VARIABLES SPATIAL AUTOREGRESSIVE MODELS: EVIDENCE FROM MONTE CARLO SIMULATIONS. Introduction When it coes to applying econoetric odels to analyze georeferenced data, researchers are well
More informationarxiv: v1 [stat.ot] 7 Jul 2010
Hotelling s test for highly correlated data P. Bubeliny e-ail: bubeliny@karlin.ff.cuni.cz Charles University, Faculty of Matheatics and Physics, KPMS, Sokolovska 83, Prague, Czech Republic, 8675. arxiv:007.094v
More informationStochastic Subgradient Methods
Stochastic Subgradient Methods Lingjie Weng Yutian Chen Bren School of Inforation and Coputer Science University of California, Irvine {wengl, yutianc}@ics.uci.edu Abstract Stochastic subgradient ethods
More informationLecture October 23. Scribes: Ruixin Qiang and Alana Shine
CSCI699: Topics in Learning and Gae Theory Lecture October 23 Lecturer: Ilias Scribes: Ruixin Qiang and Alana Shine Today s topic is auction with saples. 1 Introduction to auctions Definition 1. In a single
More informationA note on the multiplication of sparse matrices
Cent. Eur. J. Cop. Sci. 41) 2014 1-11 DOI: 10.2478/s13537-014-0201-x Central European Journal of Coputer Science A note on the ultiplication of sparse atrices Research Article Keivan Borna 12, Sohrab Aboozarkhani
More informationESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS. A Thesis. Presented to. The Faculty of the Department of Mathematics
ESTIMATING AND FORMING CONFIDENCE INTERVALS FOR EXTREMA OF RANDOM POLYNOMIALS A Thesis Presented to The Faculty of the Departent of Matheatics San Jose State University In Partial Fulfillent of the Requireents
More informationSharp Time Data Tradeoffs for Linear Inverse Problems
Sharp Tie Data Tradeoffs for Linear Inverse Probles Saet Oyak Benjain Recht Mahdi Soltanolkotabi January 016 Abstract In this paper we characterize sharp tie-data tradeoffs for optiization probles used
More informationA Low-Complexity Congestion Control and Scheduling Algorithm for Multihop Wireless Networks with Order-Optimal Per-Flow Delay
A Low-Coplexity Congestion Control and Scheduling Algorith for Multihop Wireless Networks with Order-Optial Per-Flow Delay Po-Kai Huang, Xiaojun Lin, and Chih-Chun Wang School of Electrical and Coputer
More informationASSUME a source over an alphabet size m, from which a sequence of n independent samples are drawn. The classical
IEEE TRANSACTIONS ON INFORMATION THEORY Large Alphabet Source Coding using Independent Coponent Analysis Aichai Painsky, Meber, IEEE, Saharon Rosset and Meir Feder, Fellow, IEEE arxiv:67.7v [cs.it] Jul
More informationWhen Short Runs Beat Long Runs
When Short Runs Beat Long Runs Sean Luke George Mason University http://www.cs.gu.edu/ sean/ Abstract What will yield the best results: doing one run n generations long or doing runs n/ generations long
More informationProbability Distributions
Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples
More informationOptimal quantum detectors for unambiguous detection of mixed states
PHYSICAL REVIEW A 69, 06318 (004) Optial quantu detectors for unabiguous detection of ixed states Yonina C. Eldar* Departent of Electrical Engineering, Technion Israel Institute of Technology, Haifa 3000,
More informationC na (1) a=l. c = CO + Clm + CZ TWO-STAGE SAMPLE DESIGN WITH SMALL CLUSTERS. 1. Introduction
TWO-STGE SMPLE DESIGN WITH SMLL CLUSTERS Robert G. Clark and David G. Steel School of Matheatics and pplied Statistics, University of Wollongong, NSW 5 ustralia. (robert.clark@abs.gov.au) Key Words: saple
More informationBest Arm Identification: A Unified Approach to Fixed Budget and Fixed Confidence
Best Ar Identification: A Unified Approach to Fixed Budget and Fixed Confidence Victor Gabillon, Mohaad Ghavazadeh, Alessandro Lazaric To cite this version: Victor Gabillon, Mohaad Ghavazadeh, Alessandro
More informationCSE525: Randomized Algorithms and Probabilistic Analysis May 16, Lecture 13
CSE55: Randoied Algoriths and obabilistic Analysis May 6, Lecture Lecturer: Anna Karlin Scribe: Noah Siegel, Jonathan Shi Rando walks and Markov chains This lecture discusses Markov chains, which capture
More informationAlgorithms for parallel processor scheduling with distinct due windows and unit-time jobs
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 57, No. 3, 2009 Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs A. JANIAK 1, W.A. JANIAK 2, and
More information