PCA & ICA. CE-717: Machine Learning Sharif University of Technology Spring Soleymani
|
|
- Chad Blair
- 5 years ago
- Views:
Transcription
1 PCA & ICA CE-717: Machine Learning Sharif University of Technology Spring 2015 Soleymani
2 Dimensionality Reduction: Feature Selection vs. Feature Extraction Feature selection Select a subset of a given feature set Feature extraction A linear or non-linear transform on the original feature space x 1 x d x i1 x id x 1 x d y 1 y d = f x 1 x d Feature Selection (d < d) Feature Extraction 2
3 Feature Extraction Mapping of the original data to another space Criterion for feature extraction can be different based on problem settings Unsupervised task: minimize the information loss (reconstruction error) Supervised task: maximize the class discrimination on the projected space Feature extraction algorithms Linear Methods Unsupervised: e.g., Principal Component Analysis (PCA) Supervised: e.g., Linear Discriminant Analysis (LDA) Also known as Fisher s Discriminant Analysis (FDA) 3
4 Feature Extraction Unsupervised feature extraction: X = x 1 (1) (1) x d (N) x d x 1 (N) Feature Extraction Supervised feature extraction: A mapping f: R d R d Or only the transformed data X = x 1 (1) (1) x d x d x 1 (N) (N) X = x 1 (1) (1) x d (N) x d x 1 (N) Y = y (1) y (N) Feature Extraction A mapping f: R d R d Or only the transformed data X = x 1 (1) (1) x d x d x 1 (N) (N) 4
5 Unsupervised Feature Reduction Visualization: projection of high-dimensional data onto 2D or 3D. Data compression: efficient storage, communication, or and retrieval. Pre-process: to improve accuracy by reducing features As a preprocessing step to reduce dimensions for supervised learning tasks Helps avoiding overfitting Noise removal E.g, noise in the images introduced by minor lighting variations, slightly different imaging conditions, 5
6 Linear Transformation For linear transformation, we find an explicit mapping f x = A T x that can transform also new data vectors. Original data Type equation here. = reduced data A T R d d x R d x R x = A T x d < d 6
7 Linear Transformation Linear transformation are simple mappings T xa x ( x a x) j j T j = 1,, d A = a 11 a 1d a d1 a dd a1 ad 7
8 Linear Dimensionality Reduction Unsupervised Principal Component Analysis (PCA) [we will discuss] Independent Component Analysis (ICA) [we will discuss] SingularValue Decomposition (SVD) Multi Dimensional Scaling (MDS) Canonical Correlation Analysis (CCA) 8
9 Principal Component Analysis (PCA) Also known as Karhonen-Loeve (KL) transform Principal Components (PCs): orthogonal vectors that are ordered by the fraction of the total information (variation) in the corresponding directions Find the directions at which data approximately lie When the data is projected onto first PC, the variance of the projected data is maximized 9
10 Principal Component Analysis (PCA) The best linear subspace (i.e. providing least reconstruction error of data): 10 Find mean reduced data The axes have been rotated to new (principal) axes such that: Principal axis 1 has the highest variance... Principal axis i has the i-th highest variance. The principal axes are uncorrelated Covariance among each pair of the principal axes is zero. Goal: reducing the dimensionality of the data while preserving the variation present in the dataset as much as possible. PCs can be found as the best eigenvectors of the covariance matrix of the data points.
11 Principal components If data has a Gaussian distribution N(μ, Σ), the direction of the largest variance can be found by the eigenvector of Σ that corresponds to the largest eigenvalue of Σ 5 4 v 2 v
12 Covariance Matrix μ x = μ 1 μ d = E(x 1 ) E(x d ) Σ = E x μ x x μ x T ML estimate of covariance matrix from data points x (i) N i=1 : N Σ = 1 N i=1 x (i) μ x (i) μ T = 1 N XT X X = x (1) x (N) = x (1) μ x (N) μ N μ = 1 N i=1 x (i) 12 Mean-centered data
13 PCA: Steps Input: N d data matrix X (each row contain a d dimensional data point) μ = 1 N i=1 N x (i) X Mean value of data points is subtracted from rows of X Σ = 1 N XT X (Covariance matrix) Calculate eigenvalue and eigenvectors of Σ Pick d eigenvectors corresponding to the largest eigenvalues and put them in the columns of A = [v 1,, v d ] X = XA First PC d -th PC 13
14 Another Interpretation: Least Squares Error PCs are linear least squares fits to samples, each orthogonal to the previous PCs: First PC is a minimum distance fit to a vector in the original feature space Second PC is a minimum distance fit to a vector in the plane perpendicular to the first PC 14
15 Another Interpretation: example 15
16 Another Interpretation: example 16
17 Least Squares Error and Maximum Variance Views Are Equivalent (1-dim Interpretation) Minimizing sum of square distances to the line is equivalent to maximizing the sum of squares of the projections on that line (Pythagoras). origin 17
18 PCA: Uncorrelated Features x = A T x R x = E x x T = E A T xx T A = A T E xx T A = A T R x A If A = [a 1,, a d ] where a 1,, a d are orthonormal eighenvectors of R x : R x = A T R x A = A T AΛA T A = Λ i j i, j = 1,, d E x i x j = 0 then mutually uncorrelated features are obtained Completely uncorrelated features avoid information redundancies 18
19 PCA Derivation (Correlation Version): Mean Square Error Approximation Incorporating all eigenvectors in A = [a 1,, a d ]: x = A T x Ax = AA T x = x x = Ax If d = d then x can be reconstructed exactly from x 19
20 PCA Derivation (Correlation Version): Relation between Eigenvalues and Variances The j-th largest eigenvalue of R x is the variance on the j-th PC: var x j = a j T R x a j = λ j 20
21 PCA Derivation (Correlation Version): Mean Square Error Approximation Incorporating only d eigenvectors corresponding to the largest eigenvalues A = [a 1,, a d ] (d < d) It minimizes MSE between x and x = Ax : 21 J A = E x x 2 = E x Ax 2 = E = E = d j=d +1 k=d +1 d j=d +1 d a j T E xx T a j = x j a j T a k x k = E d j=d +1 d j=d +1 a j T R x a j = d j=d +1 x j 2 = d j=d +1 x j a j d j=d +1 λ j 2 E x j 2 Sum of the d d smallest eigenvalues
22 PCA Derivation (Correlation Version): Mean Square Error Approximation In general, it can also be shown MSE is minimized compared to any other approximation of x by any d -dimensional orthonormal basis without first assuming that the axes are eigenvectors of the correlation matrix, this result can also be obtained. If the data is mean-centered in advance, R x and C x (covariance matrix) will be the same. However, in the correlation version when C x R x the approximation is not, in general, a good one (although it is a minimum MSE solution) 22
23 PCA on Faces: Eigenfaces ORL Database 23 Some Images
24 PCA on Faces: Eigenfaces Average face 1 st PC 6 th PC For eigen faces gray = 0, white > 0, black < 0 24
25 PCA on Faces: x is a = dimensional vector containing intensity of the pixels of this image Feature vector=[x 1,x 2,,x d ] x i = PC i T x The projection of x on the i-th PC = +x 1 +x 2 + +x 256 Average Face 25
26 PCA on Faces: Reconstructed Face d'=1 d'=2 d'=4 d'=8 d'=16 d'=32 d'=64 d'=128 d'=256 Original Image 26
27 Kernel PCA Kernel extension of PCA data (approximately) lies on a lower dimensional non-linear space 27
28 PCA and LDA: Drawbacks PCA drawback: An excellent information packing transform does not necessarily lead to a good class separability. The directions of the maximum variance may be useless for classification purpose LDA PCA LDA drawback 28 Singularity or under-sampled problem (when N < d) Example: gene expression data, images, text documents Can reduces dimension only to d C 1 (unlike PCA)
29 PCA vs. LDA Although LDA often provide more suitable features for classification tasks, PCA might outperform LDA in some situations: when the number of samples per class is small (overfitting problem of LDA) when the training data non-uniformly sample the underlying distribution when the number of the desired features is more than C 1 Advances in the last decade: Semi-supervised feature extraction E.g., PCA+LDA, Regularized LDA, Locally FDA (LFDA) 29
30 Independent Component Analysis (ICA) PCA: The transformed dimensions will be uncorrelated from each other Orthogonal linear transform Only uses second order statistics (i.e., covariance matrix) ICA: The transformed dimensions will be as independent as possible. Non-orthogonal linear transform High-order statistics can also used 30
31 Uncorrelated and Independent Gaussian Independent Uncorrelated Uncorrelated: cov X 1,X 2 = 0 Independent: P X 1,X 2 = P(X 1 )P(X 2 ) Non-Gaussian Independent Uncorrelated Uncorrelated Independent 31
32 ICA: Cocktail party problem Cocktail party problem d speakers are speaking simultaneously and any microphone records only an overlapping combination of these voices. Each microphone records a different combination of the speakers voices. Using these d microphone recordings, can we separate out the original d speakers speech signals? Mixing matrix A: Unmixing matrix A 1 : x = As s = A 1 x s j (i) : sound that speaker j was uttering at time i. 32 x j (i) : acoustic reading recorded by microphone j at time i.
33 ICA: Ambiguities We cannot determine the order of the independent components. There will be no way to distinguish between As and AP 1 Ps (P is a permutation matrix) There is no way to recover the correct scaling of sources There will be no way to distinguish between As and A/α αs Gaussian distribution N(0, I) of sources The distribution of any orthonormal transformation of the Gaussian has exactly the same distribution as that Gaussian 33
34 Gaussian distribution N(0, I) of sources x = As x = A s A = AR RR T = R T R = I E x x T = E ARR T A T = AA T = E xx T There is no way to tell if sources were mixed using A or A an arbitrary rotational component that cannot be determined from the data, and thus we cannot recover the original sources. So long as the data is not Gaussian, it is possible, given enough data, to recover the d independent sources. 34
35 ICA: Assumptions p x x = p s (A 1 x) A 1 Consider the assumption that the sources are independent: p s = d i=1 p(s i ) W = A 1 s i = w i T x What we ll choose for the cdf, sigmoid function σ(s) = 1/(1 + e s ) as a reasonable default function Hence, p(s) = σ (s). 35
36 ICA: Log likelihood l W = n i=1 d j=1 log σ w j T x (i) + log W Stochastic gradient descent: 1 2σ w T 1 x (i) W = W + α x i 1 2σ w T d x (i) T + W 1 T W W = W W 1 T 36
37 Summary PCA is a linear dimensionality reduction method that finds an orthonormal basis (minimizing MSE) ICA finds a linear transformation to make components as independent as possible 37
Independent Components Analysis
CS229 Lecture notes Andrew Ng Part XII Independent Components Analysis Our next topic is Independent Components Analysis (ICA). Similar to PCA, this will find a new basis in which to represent our data.
More informationPrincipal Component Analysis
Principal Component Analysis Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [based on slides from Nina Balcan] slide 1 Goals for the lecture you should understand
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Feature Extraction Hamid R. Rabiee Jafar Muhammadi, Alireza Ghasemi, Payam Siyari Spring 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2/ Agenda Dimensionality Reduction
More informationDimension Reduction (PCA, ICA, CCA, FLD,
Dimension Reduction (PCA, ICA, CCA, FLD, Topic Models) Yi Zhang 10-701, Machine Learning, Spring 2011 April 6 th, 2011 Parts of the PCA slides are from previous 10-701 lectures 1 Outline Dimension reduction
More informationPCA, Kernel PCA, ICA
PCA, Kernel PCA, ICA Learning Representations. Dimensionality Reduction. Maria-Florina Balcan 04/08/2015 Big & High-Dimensional Data High-Dimensions = Lot of Features Document classification Features per
More informationMachine Learning. CUNY Graduate Center, Spring Lectures 11-12: Unsupervised Learning 1. Professor Liang Huang.
Machine Learning CUNY Graduate Center, Spring 2013 Lectures 11-12: Unsupervised Learning 1 (Clustering: k-means, EM, mixture models) Professor Liang Huang huang@cs.qc.cuny.edu http://acl.cs.qc.edu/~lhuang/teaching/machine-learning
More informationPrincipal Component Analysis -- PCA (also called Karhunen-Loeve transformation)
Principal Component Analysis -- PCA (also called Karhunen-Loeve transformation) PCA transforms the original input space into a lower dimensional space, by constructing dimensions that are linear combinations
More informationIntroduction to Machine Learning
10-701 Introduction to Machine Learning PCA Slides based on 18-661 Fall 2018 PCA Raw data can be Complex, High-dimensional To understand a phenomenon we measure various related quantities If we knew what
More informationMachine Learning - MT & 14. PCA and MDS
Machine Learning - MT 2016 13 & 14. PCA and MDS Varun Kanade University of Oxford November 21 & 23, 2016 Announcements Sheet 4 due this Friday by noon Practical 3 this week (continue next week if necessary)
More informationDimensionality Reduction
Lecture 5 1 Outline 1. Overview a) What is? b) Why? 2. Principal Component Analysis (PCA) a) Objectives b) Explaining variability c) SVD 3. Related approaches a) ICA b) Autoencoders 2 Example 1: Sportsball
More informationWhat is Principal Component Analysis?
What is Principal Component Analysis? Principal component analysis (PCA) Reduce the dimensionality of a data set by finding a new set of variables, smaller than the original set of variables Retains most
More informationUnsupervised learning: beyond simple clustering and PCA
Unsupervised learning: beyond simple clustering and PCA Liza Rebrova Self organizing maps (SOM) Goal: approximate data points in R p by a low-dimensional manifold Unlike PCA, the manifold does not have
More informationLECTURE :ICA. Rita Osadchy. Based on Lecture Notes by A. Ng
LECURE :ICA Rita Osadchy Based on Lecture Notes by A. Ng Cocktail Party Person 1 2 s 1 Mike 2 s 3 Person 3 1 Mike 1 s 2 Person 2 3 Mike 3 microphone signals are mied speech signals 1 2 3 ( t) ( t) ( t)
More informationExpectation Maximization
Expectation Maximization Machine Learning CSE546 Carlos Guestrin University of Washington November 13, 2014 1 E.M.: The General Case E.M. widely used beyond mixtures of Gaussians The recipe is the same
More informationDimensionality reduction
Dimensionality Reduction PCA continued Machine Learning CSE446 Carlos Guestrin University of Washington May 22, 2013 Carlos Guestrin 2005-2013 1 Dimensionality reduction n Input data may have thousands
More informationPCA and LDA. Man-Wai MAK
PCA and LDA Man-Wai MAK Dept. of Electronic and Information Engineering, The Hong Kong Polytechnic University enmwmak@polyu.edu.hk http://www.eie.polyu.edu.hk/ mwmak References: S.J.D. Prince,Computer
More informationPCA and LDA. Man-Wai MAK
PCA and LDA Man-Wai MAK Dept. of Electronic and Information Engineering, The Hong Kong Polytechnic University enmwmak@polyu.edu.hk http://www.eie.polyu.edu.hk/ mwmak References: S.J.D. Prince,Computer
More informationPrincipal Component Analysis (PCA)
Principal Component Analysis (PCA) Salvador Dalí, Galatea of the Spheres CSC411/2515: Machine Learning and Data Mining, Winter 2018 Michael Guerzhoy and Lisa Zhang Some slides from Derek Hoiem and Alysha
More informationStatistical Machine Learning
Statistical Machine Learning Christoph Lampert Spring Semester 2015/2016 // Lecture 12 1 / 36 Unsupervised Learning Dimensionality Reduction 2 / 36 Dimensionality Reduction Given: data X = {x 1,..., x
More informationLecture 24: Principal Component Analysis. Aykut Erdem May 2016 Hacettepe University
Lecture 4: Principal Component Analysis Aykut Erdem May 016 Hacettepe University This week Motivation PCA algorithms Applications PCA shortcomings Autoencoders Kernel PCA PCA Applications Data Visualization
More informationCS4495/6495 Introduction to Computer Vision. 8B-L2 Principle Component Analysis (and its use in Computer Vision)
CS4495/6495 Introduction to Computer Vision 8B-L2 Principle Component Analysis (and its use in Computer Vision) Wavelength 2 Wavelength 2 Principal Components Principal components are all about the directions
More informationRobot Image Credit: Viktoriya Sukhanova 123RF.com. Dimensionality Reduction
Robot Image Credit: Viktoriya Sukhanova 13RF.com Dimensionality Reduction Feature Selection vs. Dimensionality Reduction Feature Selection (last time) Select a subset of features. When classifying novel
More informationPCA and admixture models
PCA and admixture models CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar, Alkes Price PCA and admixture models 1 / 57 Announcements HW1
More informationDeriving Principal Component Analysis (PCA)
-0 Mathematical Foundations for Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Deriving Principal Component Analysis (PCA) Matt Gormley Lecture 11 Oct.
More informationPrincipal Component Analysis and Linear Discriminant Analysis
Principal Component Analysis and Linear Discriminant Analysis Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL 60208 http://www.eecs.northwestern.edu/~yingwu 1/29
More informationPrincipal Component Analysis
Principal Component Analysis Introduction Consider a zero mean random vector R n with autocorrelation matri R = E( T ). R has eigenvectors q(1),,q(n) and associated eigenvalues λ(1) λ(n). Let Q = [ q(1)
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 11-1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed
More informationAdvanced Introduction to Machine Learning CMU-10715
Advanced Introduction to Machine Learning CMU-10715 Principal Component Analysis Barnabás Póczos Contents Motivation PCA algorithms Applications Some of these slides are taken from Karl Booksh Research
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab 1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed in the
More informationUnsupervised Machine Learning and Data Mining. DS 5230 / DS Fall Lecture 7. Jan-Willem van de Meent
Unsupervised Machine Learning and Data Mining DS 5230 / DS 4420 - Fall 2018 Lecture 7 Jan-Willem van de Meent DIMENSIONALITY REDUCTION Borrowing from: Percy Liang (Stanford) Dimensionality Reduction Goal:
More informationMachine Learning (Spring 2012) Principal Component Analysis
1-71 Machine Learning (Spring 1) Principal Component Analysis Yang Xu This note is partly based on Chapter 1.1 in Chris Bishop s book on PRML and the lecture slides on PCA written by Carlos Guestrin in
More informationMachine Learning 11. week
Machine Learning 11. week Feature Extraction-Selection Dimension reduction PCA LDA 1 Feature Extraction Any problem can be solved by machine learning methods in case of that the system must be appropriately
More informationMachine Learning 2nd Edition
INTRODUCTION TO Lecture Slides for Machine Learning 2nd Edition ETHEM ALPAYDIN, modified by Leonardo Bobadilla and some parts from http://www.cs.tau.ac.il/~apartzin/machinelearning/ The MIT Press, 2010
More informationPrincipal Component Analysis (PCA)
Principal Component Analysis (PCA) Additional reading can be found from non-assessed exercises (week 8) in this course unit teaching page. Textbooks: Sect. 6.3 in [1] and Ch. 12 in [2] Outline Introduction
More informationECE 521. Lecture 11 (not on midterm material) 13 February K-means clustering, Dimensionality reduction
ECE 521 Lecture 11 (not on midterm material) 13 February 2017 K-means clustering, Dimensionality reduction With thanks to Ruslan Salakhutdinov for an earlier version of the slides Overview K-means clustering
More informationPrincipal Component Analysis
B: Chapter 1 HTF: Chapter 1.5 Principal Component Analysis Barnabás Póczos University of Alberta Nov, 009 Contents Motivation PCA algorithms Applications Face recognition Facial expression recognition
More informationROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015
ROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015 http://intelligentoptimization.org/lionbook Roberto Battiti
More information7. Variable extraction and dimensionality reduction
7. Variable extraction and dimensionality reduction The goal of the variable selection in the preceding chapter was to find least useful variables so that it would be possible to reduce the dimensionality
More informationAnnouncements (repeat) Principal Components Analysis
4/7/7 Announcements repeat Principal Components Analysis CS 5 Lecture #9 April 4 th, 7 PA4 is due Monday, April 7 th Test # will be Wednesday, April 9 th Test #3 is Monday, May 8 th at 8AM Just hour long
More informationMACHINE LEARNING. Methods for feature extraction and reduction of dimensionality: Probabilistic PCA and kernel PCA
1 MACHINE LEARNING Methods for feature extraction and reduction of dimensionality: Probabilistic PCA and kernel PCA 2 Practicals Next Week Next Week, Practical Session on Computer Takes Place in Room GR
More informationDimensionality Reduction Using PCA/LDA. Hongyu Li School of Software Engineering TongJi University Fall, 2014
Dimensionality Reduction Using PCA/LDA Hongyu Li School of Software Engineering TongJi University Fall, 2014 Dimensionality Reduction One approach to deal with high dimensional data is by reducing their
More informationSTA 414/2104: Lecture 8
STA 414/2104: Lecture 8 6-7 March 2017: Continuous Latent Variable Models, Neural networks With thanks to Russ Salakhutdinov, Jimmy Ba and others Outline Continuous latent variable models Background PCA
More informationFace Recognition. Face Recognition. Subspace-Based Face Recognition Algorithms. Application of Face Recognition
ace Recognition Identify person based on the appearance of face CSED441:Introduction to Computer Vision (2017) Lecture10: Subspace Methods and ace Recognition Bohyung Han CSE, POSTECH bhhan@postech.ac.kr
More informationMachine Learning. Principal Components Analysis. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012
Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Principal Components Analysis Le Song Lecture 22, Nov 13, 2012 Based on slides from Eric Xing, CMU Reading: Chap 12.1, CB book 1 2 Factor or Component
More informationLecture 7: Con3nuous Latent Variable Models
CSC2515 Fall 2015 Introduc3on to Machine Learning Lecture 7: Con3nuous Latent Variable Models All lecture slides will be available as.pdf on the course website: http://www.cs.toronto.edu/~urtasun/courses/csc2515/
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 6 1 / 22 Overview
More informationPrincipal Component Analysis
Principal Component Analysis Anders Øland David Christiansen 1 Introduction Principal Component Analysis, or PCA, is a commonly used multi-purpose technique in data analysis. It can be used for feature
More informationData Mining. Dimensionality reduction. Hamid Beigy. Sharif University of Technology. Fall 1395
Data Mining Dimensionality reduction Hamid Beigy Sharif University of Technology Fall 1395 Hamid Beigy (Sharif University of Technology) Data Mining Fall 1395 1 / 42 Outline 1 Introduction 2 Feature selection
More informationHST.582J/6.555J/16.456J
Blind Source Separation: PCA & ICA HST.582J/6.555J/16.456J Gari D. Clifford gari [at] mit. edu http://www.mit.edu/~gari G. D. Clifford 2005-2009 What is BSS? Assume an observation (signal) is a linear
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 informationSTATS 306B: Unsupervised Learning Spring Lecture 12 May 7
STATS 306B: Unsupervised Learning Spring 2014 Lecture 12 May 7 Lecturer: Lester Mackey Scribe: Lan Huong, Snigdha Panigrahi 12.1 Beyond Linear State Space Modeling Last lecture we completed our discussion
More informationUniversität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen PCA. Tobias Scheffer
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen PCA Tobias Scheffer Overview Principal Component Analysis (PCA) Kernel-PCA Fisher Linear Discriminant Analysis t-sne 2 PCA: Motivation
More informationExample: Face Detection
Announcements HW1 returned New attendance policy Face Recognition: Dimensionality Reduction On time: 1 point Five minutes or more late: 0.5 points Absent: 0 points Biometrics CSE 190 Lecture 14 CSE190,
More informationFocus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations.
Previously Focus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations y = Ax Or A simply represents data Notion of eigenvectors,
More informationSTA 414/2104: Lecture 8
STA 414/2104: Lecture 8 6-7 March 2017: Continuous Latent Variable Models, Neural networks Delivered by Mark Ebden With thanks to Russ Salakhutdinov, Jimmy Ba and others Outline Continuous latent variable
More informationMachine Learning. Dimensionality reduction. Hamid Beigy. Sharif University of Technology. Fall 1395
Machine Learning Dimensionality reduction Hamid Beigy Sharif University of Technology Fall 1395 Hamid Beigy (Sharif University of Technology) Machine Learning Fall 1395 1 / 47 Table of contents 1 Introduction
More informationSignal Analysis. Principal Component Analysis
Multi dimensional Signal Analysis Lecture 2E Principal Component Analysis Subspace representation Note! Given avector space V of dimension N a scalar product defined by G 0 a subspace U of dimension M
More informationLinear & Non-Linear Discriminant Analysis! Hugh R. Wilson
Linear & Non-Linear Discriminant Analysis! Hugh R. Wilson PCA Review! Supervised learning! Fisher linear discriminant analysis! Nonlinear discriminant analysis! Research example! Multiple Classes! Unsupervised
More informationPrincipal Component Analysis (PCA) CSC411/2515 Tutorial
Principal Component Analysis (PCA) CSC411/2515 Tutorial Harris Chan Based on previous tutorial slides by Wenjie Luo, Ladislav Rampasek University of Toronto hchan@cs.toronto.edu October 19th, 2017 (UofT)
More informationKernel methods for comparing distributions, measuring dependence
Kernel methods for comparing distributions, measuring dependence Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Principal component analysis Given a set of M centered observations
More informationCOMP 551 Applied Machine Learning Lecture 13: Dimension reduction and feature selection
COMP 551 Applied Machine Learning Lecture 13: Dimension reduction and feature selection Instructor: Herke van Hoof (herke.vanhoof@cs.mcgill.ca) Based on slides by:, Jackie Chi Kit Cheung Class web page:
More information1 Principal Components Analysis
Lecture 3 and 4 Sept. 18 and Sept.20-2006 Data Visualization STAT 442 / 890, CM 462 Lecture: Ali Ghodsi 1 Principal Components Analysis Principal components analysis (PCA) is a very popular technique for
More informationMTTS1 Dimensionality Reduction and Visualization Spring 2014 Jaakko Peltonen
MTTS1 Dimensionality Reduction and Visualization Spring 2014 Jaakko Peltonen Lecture 3: Linear feature extraction Feature extraction feature extraction: (more general) transform the original to (k < d).
More informationSystem 1 (last lecture) : limited to rigidly structured shapes. System 2 : recognition of a class of varying shapes. Need to:
System 2 : Modelling & Recognising Modelling and Recognising Classes of Classes of Shapes Shape : PDM & PCA All the same shape? System 1 (last lecture) : limited to rigidly structured shapes System 2 :
More informationCovariance and Correlation Matrix
Covariance and Correlation Matrix Given sample {x n } N 1, where x Rd, x n = x 1n x 2n. x dn sample mean x = 1 N N n=1 x n, and entries of sample mean are x i = 1 N N n=1 x in sample covariance matrix
More informationCPSC 340: Machine Learning and Data Mining. More PCA Fall 2017
CPSC 340: Machine Learning and Data Mining More PCA Fall 2017 Admin Assignment 4: Due Friday of next week. No class Monday due to holiday. There will be tutorials next week on MAP/PCA (except Monday).
More informationMachine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function.
Bayesian learning: Machine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function. Let y be the true label and y be the predicted
More informationImage Analysis & Retrieval. Lec 14. Eigenface and Fisherface
Image Analysis & Retrieval Lec 14 Eigenface and Fisherface Zhu Li Dept of CSEE, UMKC Office: FH560E, Email: lizhu@umkc.edu, Ph: x 2346. http://l.web.umkc.edu/lizhu Z. Li, Image Analysis & Retrv, Spring
More informationRegularized Discriminant Analysis and Reduced-Rank LDA
Regularized Discriminant Analysis and Reduced-Rank LDA Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Regularized Discriminant Analysis A compromise between LDA and
More informationDecember 20, MAA704, Multivariate analysis. Christopher Engström. Multivariate. analysis. Principal component analysis
.. December 20, 2013 Todays lecture. (PCA) (PLS-R) (LDA) . (PCA) is a method often used to reduce the dimension of a large dataset to one of a more manageble size. The new dataset can then be used to make
More informationMachine Learning (BSMC-GA 4439) Wenke Liu
Machine Learning (BSMC-GA 4439) Wenke Liu 02-01-2018 Biomedical data are usually high-dimensional Number of samples (n) is relatively small whereas number of features (p) can be large Sometimes p>>n Problems
More informationDimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx
More informationUnsupervised Learning: K- Means & PCA
Unsupervised Learning: K- Means & PCA Unsupervised Learning Supervised learning used labeled data pairs (x, y) to learn a func>on f : X Y But, what if we don t have labels? No labels = unsupervised learning
More informationHeeyoul (Henry) Choi. Dept. of Computer Science Texas A&M University
Heeyoul (Henry) Choi Dept. of Computer Science Texas A&M University hchoi@cs.tamu.edu Introduction Speaker Adaptation Eigenvoice Comparison with others MAP, MLLR, EMAP, RMP, CAT, RSW Experiments Future
More informationPRINCIPAL COMPONENT ANALYSIS
PRINCIPAL COMPONENT ANALYSIS Dimensionality Reduction Tzompanaki Katerina Dimensionality Reduction Unsupervised learning Goal: Find hidden patterns in the data. Used for Visualization Data compression
More informationLearning Eigenfunctions: Links with Spectral Clustering and Kernel PCA
Learning Eigenfunctions: Links with Spectral Clustering and Kernel PCA Yoshua Bengio Pascal Vincent Jean-François Paiement University of Montreal April 2, Snowbird Learning 2003 Learning Modal Structures
More informationComputation. For QDA we need to calculate: Lets first consider the case that
Computation For QDA we need to calculate: δ (x) = 1 2 log( Σ ) 1 2 (x µ ) Σ 1 (x µ ) + log(π ) Lets first consider the case that Σ = I,. This is the case where each distribution is spherical, around the
More informationData Mining Techniques
Data Mining Techniques CS 6220 - Section 3 - Fall 2016 Lecture 12 Jan-Willem van de Meent (credit: Yijun Zhao, Percy Liang) DIMENSIONALITY REDUCTION Borrowing from: Percy Liang (Stanford) Linear Dimensionality
More informationUnsupervised Learning
2018 EE448, Big Data Mining, Lecture 7 Unsupervised Learning Weinan Zhang Shanghai Jiao Tong University http://wnzhang.net http://wnzhang.net/teaching/ee448/index.html ML Problem Setting First build and
More information20 Unsupervised Learning and Principal Components Analysis (PCA)
116 Jonathan Richard Shewchuk 20 Unsupervised Learning and Principal Components Analysis (PCA) UNSUPERVISED LEARNING We have sample points, but no labels! No classes, no y-values, nothing to predict. Goal:
More informationL11: Pattern recognition principles
L11: Pattern recognition principles Bayesian decision theory Statistical classifiers Dimensionality reduction Clustering This lecture is partly based on [Huang, Acero and Hon, 2001, ch. 4] Introduction
More informationDimensionality Reduction. CS57300 Data Mining Fall Instructor: Bruno Ribeiro
Dimensionality Reduction CS57300 Data Mining Fall 2016 Instructor: Bruno Ribeiro Goal } Visualize high dimensional data (and understand its Geometry) } Project the data into lower dimensional spaces }
More informationIntroduction PCA classic Generative models Beyond and summary. PCA, ICA and beyond
PCA, ICA and beyond Summer School on Manifold Learning in Image and Signal Analysis, August 17-21, 2009, Hven Technical University of Denmark (DTU) & University of Copenhagen (KU) August 18, 2009 Motivation
More informationCHAPTER 4 PRINCIPAL COMPONENT ANALYSIS-BASED FUSION
59 CHAPTER 4 PRINCIPAL COMPONENT ANALYSIS-BASED FUSION 4. INTRODUCTION Weighted average-based fusion algorithms are one of the widely used fusion methods for multi-sensor data integration. These methods
More informationArtificial Intelligence Module 2. Feature Selection. Andrea Torsello
Artificial Intelligence Module 2 Feature Selection Andrea Torsello We have seen that high dimensional data is hard to classify (curse of dimensionality) Often however, the data does not fill all the space
More informationMaximum variance formulation
12.1. Principal Component Analysis 561 Figure 12.2 Principal component analysis seeks a space of lower dimensionality, known as the principal subspace and denoted by the magenta line, such that the orthogonal
More informationPCA Review. CS 510 February 25 th, 2013
PCA Review CS 510 February 25 th, 2013 Recall the goal: image matching Probe image, registered to gallery Registered Gallery of Images 3/7/13 CS 510, Image Computa5on, Ross Beveridge & Bruce Draper 2 Getting
More informationData Mining. Linear & nonlinear classifiers. Hamid Beigy. Sharif University of Technology. Fall 1396
Data Mining Linear & nonlinear classifiers Hamid Beigy Sharif University of Technology Fall 1396 Hamid Beigy (Sharif University of Technology) Data Mining Fall 1396 1 / 31 Table of contents 1 Introduction
More informationSome Interesting Problems in Pattern Recognition and Image Processing
Some Interesting Problems in Pattern Recognition and Image Processing JEN-MEI CHANG Department of Mathematics and Statistics California State University, Long Beach jchang9@csulb.edu University of Southern
More informationLinear Dimensionality Reduction
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Principal Component Analysis 3 Factor Analysis
More informationSTA 414/2104: Machine Learning
STA 414/2104: Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistics! rsalakhu@cs.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 8 Continuous Latent Variable
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 informationLECTURE NOTE #10 PROF. ALAN YUILLE
LECTURE NOTE #10 PROF. ALAN YUILLE 1. Principle Component Analysis (PCA) One way to deal with the curse of dimensionality is to project data down onto a space of low dimensions, see figure (1). Figure
More informationPrincipal components analysis COMS 4771
Principal components analysis COMS 4771 1. Representation learning Useful representations of data Representation learning: Given: raw feature vectors x 1, x 2,..., x n R d. Goal: learn a useful feature
More informationCSC 411 Lecture 12: Principal Component Analysis
CSC 411 Lecture 12: Principal Component Analysis Roger Grosse, Amir-massoud Farahmand, and Juan Carrasquilla University of Toronto UofT CSC 411: 12-PCA 1 / 23 Overview Today we ll cover the first unsupervised
More informationSingular Value Decomposition. 1 Singular Value Decomposition and the Four Fundamental Subspaces
Singular Value Decomposition This handout is a review of some basic concepts in linear algebra For a detailed introduction, consult a linear algebra text Linear lgebra and its pplications by Gilbert Strang
More informationImage Analysis & Retrieval Lec 14 - Eigenface & Fisherface
CS/EE 5590 / ENG 401 Special Topics, Spring 2018 Image Analysis & Retrieval Lec 14 - Eigenface & Fisherface Zhu Li Dept of CSEE, UMKC http://l.web.umkc.edu/lizhu Office Hour: Tue/Thr 2:30-4pm@FH560E, Contact:
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationDimensionality Reduction
Dimensionality Reduction Le Song Machine Learning I CSE 674, Fall 23 Unsupervised learning Learning from raw (unlabeled, unannotated, etc) data, as opposed to supervised data where a classification of
More informationCS 4495 Computer Vision Principle Component Analysis
CS 4495 Computer Vision Principle Component Analysis (and it s use in Computer Vision) Aaron Bobick School of Interactive Computing Administrivia PS6 is out. Due *** Sunday, Nov 24th at 11:55pm *** PS7
More information