Applications of Random Matrices in Spectral Computations and Machine Learning
|
|
- Victoria Austin
- 7 years ago
- Views:
Transcription
1 Applications of Random Matrices in Spectral Computations and Machine Learning Dimitris Achlioptas UC Santa Cruz
2 This talk Viewpoint: use randomness to transform the data
3 This talk Viewpoint: use randomness to transform the data Random Projections Fast Spectral Computations Sampling in Kernel PCA
4 The Setting
5 The Setting n d n d n d
6 The Setting n d n d n d
7 The Setting n d n d n d P Output: AP
8 The Johnson-Lindenstrauss lemma
9 The Johnson-Lindenstrauss lemma Algorithm: Projecting onto a random hyperplane (subspace) of dimension succeeds with probability
10 Applications Approximation algorithms [Charikar 02] Hardness of approximation [Trevisan 97] Learning mixtures of Gaussians [Arora, Kannan 01] Approximate nearest-neighbors [Kleinberg 97] Data-stream computations [Alon et al. 99, Indyk 00] Min-cost clustering [Schulman 00]. Information Retrieval (LSI) [Papadimitriou et al. 97]
11 How to pick a random hyperplane
12 How to pick a random hyperplane Take where the are independent random variables [Dasgupta Gupta 99] [Indyk Motwani 99] [Johnson Lindenstrauss 82]
13 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in
14 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in Each column is an unbiased, independent estimator of (via its squared inner product)
15 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in Each column is an unbiased, independent estimator of (via its squared inner product) is the average estimate (since we take the sum)
16 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated
17 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated Without orthonormalization:
18 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated Without orthonormalization: Same thing!
19 Orthonormality: Take #1 Random vectors in high-dimensional Euclidean space are very nearly orthonormal.
20 Orthonormality: Take #1 Random vectors in high-dimensional Euclidean space are very nearly orthonormal. Do they have to be uniformly random? Is the Gaussian distribution magical?
21 JL with binary coins Take where the are independent random variables with
22 JL with binary coins Take where the are independent random variables with Benefits: Much faster in practice ± Only operations (no ) Fewer random bits Derandomization Slightly smaller(!) k
23 JL with binary coins Take where the are independent random variables with Preprocessing with a randomized FFT [Ailon, Chazelle 06]
24 Let s at least look at the data
25 The Setting n d n d n d P Output: AP
26 Spectral Norm: Low Rank Approximations
27 Low Rank Approximations Spectral Norm: Frobenius Norm:
28 Low Rank Approximations Spectral Norm: Frobenius Norm:
29 Low Rank Approximations Spectral Norm: Frobenius Norm:
30 Low Rank Approximations Spectral Norm: Frobenius Norm:
31 Start with a random How to compute A k
32 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x:
33 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x: Synthesize a new candidate by combining the rows of A according to their enthusiasm for x: (This is power iteration on. Also known as PCA.)
34 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x: Synthesize a new candidate by combining the rows of A according to their enthusiasm for x: (This is power iteration on. Also known as PCA.) Project A on subspace orthogonal to x and repeat
35 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise
36 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A.
37 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A. Intuition: The perturbation vectors are nearly orthogonal
38 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A. Intuition: The perturbation vectors are nearly orthogonal No small subspace accommodates many of them
39 Rigorously Lemma: For any matrices A and
40 Rigorously Lemma: For any matrices A and Perspective: For any fixed x we have w.h.p.
41 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix
42 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix Computation-friendly noise:
43 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix Computation-friendly noise: Inject data-dependent noise
44 Quantization
45 Quantization
46 Quantization
47 Quantization
48 Sparsification
49 Sparsification
50 Sparsification
51 Sparsification
52 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint
53 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data
54 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data L2 error can be quadratically better than Nystrom
55 Orthonormality: Take #2 Matrices with independent, 0-mean entries are white noise matrices
56 A scalar analogue
57 A scalar analogue Crude quantization at extremely high rate
58 A scalar analogue Crude quantization at extremely high rate + low-pass filter
59 A scalar analogue Crude quantization at extremely high rate + low-pass filter
60 A scalar analogue Crude quantization at extremely high rate + low-pass filter = 1-bit CD player ( Bitstream )
61 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data L2 error can be quadratically better than Nystrom Useful even for exact computations
62 Accelerating exact computations
63 Kernels
64 Kernels & Support Vector Machines Red and Blue pointclouds Which linear separator (hyperplane)? Maximum margin Optimal can be expressed by inner products with (a few) data points
65 Not always linearly separable
66 Population density
67 Kernel PCA
68 Kernel PCA We can also compute the SVD via the spectrum of n d n n d n
69 Kernel PCA We can also compute the SVD via the spectrum of n d n n d Each entry in AA T n is the inner product of two inputs
70 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function
71 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function Work implicitly in high-dimensional space
72 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function Work implicitly in high-dimensional space Good linear separators in that space
73 From linear to non-linear PCA X-Y p kernel illustrates how the contours of the first 2 components change from straight lines for p=2 to non-linea for p=1.5, 1 and 0.5. From Schölkopf and Smola, Learning with kernels, MIT 2002
74 Kernel PCA with Gaussian Kernel KPCA with Gaussian kernels. The contours follow the cluster densities! First two kernel PCs separate the data nicely. Linear PCA has only 2 components, but kernel PCA has more, since the space dimension is usually large (in this case infinite)
75 Good News: KPCA in brief - Work directly with non-vectorial inputs
76 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03]
77 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03] Bad News: n 2 kernel evaluations are too many.
78 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03] Bad News: n 2 kernel evaluations are too many. Good News: [Shaw-Taylor et al. 03] good generalization rapid spectral decay
79 So, it s enough to sample n n d n d n In practice, 1% of the data is more than enough In theory, we can go down to n polylog(n)
80 Important Features are Preserved
81 Open Problems How general is this stability under noise? For example, does it hold for Support Vector Machines? When can we prove such stability in a black-box fashion, i.e. as with matrices? Can we exploit if for data privacy?
Linear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationUnsupervised Data Mining (Clustering)
Unsupervised Data Mining (Clustering) Javier Béjar KEMLG December 01 Javier Béjar (KEMLG) Unsupervised Data Mining (Clustering) December 01 1 / 51 Introduction Clustering in KDD One of the main tasks in
More informationCS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on
CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d- dimensional subspace Axes of this subspace
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):
More informationRandomized Robust Linear Regression for big data applications
Randomized Robust Linear Regression for big data applications Yannis Kopsinis 1 Dept. of Informatics & Telecommunications, UoA Thursday, Apr 16, 2015 In collaboration with S. Chouvardas, Harris Georgiou,
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationAPPM4720/5720: Fast algorithms for big data. Gunnar Martinsson The University of Colorado at Boulder
APPM4720/5720: Fast algorithms for big data Gunnar Martinsson The University of Colorado at Boulder Course objectives: The purpose of this course is to teach efficient algorithms for processing very large
More informationComparison of Non-linear Dimensionality Reduction Techniques for Classification with Gene Expression Microarray Data
CMPE 59H Comparison of Non-linear Dimensionality Reduction Techniques for Classification with Gene Expression Microarray Data Term Project Report Fatma Güney, Kübra Kalkan 1/15/2013 Keywords: Non-linear
More informationSupervised Feature Selection & Unsupervised Dimensionality Reduction
Supervised Feature Selection & Unsupervised Dimensionality Reduction Feature Subset Selection Supervised: class labels are given Select a subset of the problem features Why? Redundant features much or
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationDistance Metric Learning in Data Mining (Part I) Fei Wang and Jimeng Sun IBM TJ Watson Research Center
Distance Metric Learning in Data Mining (Part I) Fei Wang and Jimeng Sun IBM TJ Watson Research Center 1 Outline Part I - Applications Motivation and Introduction Patient similarity application Part II
More informationSYMMETRIC EIGENFACES MILI I. SHAH
SYMMETRIC EIGENFACES MILI I. SHAH Abstract. Over the years, mathematicians and computer scientists have produced an extensive body of work in the area of facial analysis. Several facial analysis algorithms
More informationNimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff
Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationManifold Learning Examples PCA, LLE and ISOMAP
Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationLearning Tools for Big Data Analytics
Learning Tools for Big Data Analytics Georgios B. Giannakis Acknowledgments: Profs. G. Mateos and K. Slavakis NSF 1343860, 1442686, and MURI-FA9550-10-1-0567 Center for Advanced Signal and Image Sciences
More informationAnalysis of kiva.com Microlending Service! Hoda Eydgahi Julia Ma Andy Bardagjy December 9, 2010 MAS.622j
Analysis of kiva.com Microlending Service! Hoda Eydgahi Julia Ma Andy Bardagjy December 9, 2010 MAS.622j What is Kiva? An organization that allows people to lend small amounts of money via the Internet
More informationEM Clustering Approach for Multi-Dimensional Analysis of Big Data Set
EM Clustering Approach for Multi-Dimensional Analysis of Big Data Set Amhmed A. Bhih School of Electrical and Electronic Engineering Princy Johnson School of Electrical and Electronic Engineering Martin
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More informationLecture Topic: Low-Rank Approximations
Lecture Topic: Low-Rank Approximations Low-Rank Approximations We have seen principal component analysis. The extraction of the first principle eigenvalue could be seen as an approximation of the original
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationSo which is the best?
Manifold Learning Techniques: So which is the best? Todd Wittman Math 8600: Geometric Data Analysis Instructor: Gilad Lerman Spring 2005 Note: This presentation does not contain information on LTSA, which
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationChapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors
Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col
More informationGoing Big in Data Dimensionality:
LUDWIG- MAXIMILIANS- UNIVERSITY MUNICH DEPARTMENT INSTITUTE FOR INFORMATICS DATABASE Going Big in Data Dimensionality: Challenges and Solutions for Mining High Dimensional Data Peer Kröger Lehrstuhl für
More informationNon-negative Matrix Factorization (NMF) in Semi-supervised Learning Reducing Dimension and Maintaining Meaning
Non-negative Matrix Factorization (NMF) in Semi-supervised Learning Reducing Dimension and Maintaining Meaning SAMSI 10 May 2013 Outline Introduction to NMF Applications Motivations NMF as a middle step
More informationDifferential privacy in health care analytics and medical research An interactive tutorial
Differential privacy in health care analytics and medical research An interactive tutorial Speaker: Moritz Hardt Theory Group, IBM Almaden February 21, 2012 Overview 1. Releasing medical data: What could
More informationUnsupervised and supervised dimension reduction: Algorithms and connections
Unsupervised and supervised dimension reduction: Algorithms and connections Jieping Ye Department of Computer Science and Engineering Evolutionary Functional Genomics Center The Biodesign Institute Arizona
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMachine Learning Final Project Spam Email Filtering
Machine Learning Final Project Spam Email Filtering March 2013 Shahar Yifrah Guy Lev Table of Content 1. OVERVIEW... 3 2. DATASET... 3 2.1 SOURCE... 3 2.2 CREATION OF TRAINING AND TEST SETS... 4 2.3 FEATURE
More informationLecture 5 Least-squares
EE263 Autumn 2007-08 Stephen Boyd Lecture 5 Least-squares least-squares (approximate) solution of overdetermined equations projection and orthogonality principle least-squares estimation BLUE property
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationRandom Projection-based Multiplicative Data Perturbation for Privacy Preserving Distributed Data Mining
Random Projection-based Multiplicative Data Perturbation for Privacy Preserving Distributed Data Mining Kun Liu Hillol Kargupta and Jessica Ryan Abstract This paper explores the possibility of using multiplicative
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationx + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationLearning, Sparsity and Big Data
Learning, Sparsity and Big Data M. Magdon-Ismail (Joint Work) January 22, 2014. Out-of-Sample is What Counts NO YES A pattern exists We don t know it We have data to learn it Tested on new cases? Teaching
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More information1 Spectral Methods for Dimensionality
1 Spectral Methods for Dimensionality Reduction Lawrence K. Saul Kilian Q. Weinberger Fei Sha Jihun Ham Daniel D. Lee How can we search for low dimensional structure in high dimensional data? If the data
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationClassifying Large Data Sets Using SVMs with Hierarchical Clusters. Presented by :Limou Wang
Classifying Large Data Sets Using SVMs with Hierarchical Clusters Presented by :Limou Wang Overview SVM Overview Motivation Hierarchical micro-clustering algorithm Clustering-Based SVM (CB-SVM) Experimental
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationMedical Information Management & Mining. You Chen Jan,15, 2013 You.chen@vanderbilt.edu
Medical Information Management & Mining You Chen Jan,15, 2013 You.chen@vanderbilt.edu 1 Trees Building Materials Trees cannot be used to build a house directly. How can we transform trees to building materials?
More informationSupport Vector Machines with Clustering for Training with Very Large Datasets
Support Vector Machines with Clustering for Training with Very Large Datasets Theodoros Evgeniou Technology Management INSEAD Bd de Constance, Fontainebleau 77300, France theodoros.evgeniou@insead.fr Massimiliano
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
More informationPrincipal components analysis
CS229 Lecture notes Andrew Ng Part XI Principal components analysis In our discussion of factor analysis, we gave a way to model data x R n as approximately lying in some k-dimension subspace, where k
More informationText Analytics (Text Mining)
CSE 6242 / CX 4242 Apr 3, 2014 Text Analytics (Text Mining) LSI (uses SVD), Visualization Duen Horng (Polo) Chau Georgia Tech Some lectures are partly based on materials by Professors Guy Lebanon, Jeffrey
More informationClustering Big Data. Efficient Data Mining Technologies. J Singh and Teresa Brooks. June 4, 2015
Clustering Big Data Efficient Data Mining Technologies J Singh and Teresa Brooks June 4, 2015 Hello Bulgaria (http://hello.bg/) A website with thousands of pages... Some pages identical to other pages
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationMultimedia Databases. Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.
Multimedia Databases Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.de 14 Previous Lecture 13 Indexes for Multimedia Data 13.1
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationWavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)
Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationSolutions to Exam in Speech Signal Processing EN2300
Solutions to Exam in Speech Signal Processing EN23 Date: Thursday, Dec 2, 8: 3: Place: Allowed: Grades: Language: Solutions: Q34, Q36 Beta Math Handbook (or corresponding), calculator with empty memory.
More informationKnowledge Discovery from patents using KMX Text Analytics
Knowledge Discovery from patents using KMX Text Analytics Dr. Anton Heijs anton.heijs@treparel.com Treparel Abstract In this white paper we discuss how the KMX technology of Treparel can help searchers
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More informationCompact Summaries for Large Datasets
Compact Summaries for Large Datasets Big Data Graham Cormode University of Warwick G.Cormode@Warwick.ac.uk The case for Big Data in one slide Big data arises in many forms: Medical data: genetic sequences,
More informationA Survey on Pre-processing and Post-processing Techniques in Data Mining
, pp. 99-128 http://dx.doi.org/10.14257/ijdta.2014.7.4.09 A Survey on Pre-processing and Post-processing Techniques in Data Mining Divya Tomar and Sonali Agarwal Indian Institute of Information Technology,
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationData Mining Practical Machine Learning Tools and Techniques
Ensemble learning Data Mining Practical Machine Learning Tools and Techniques Slides for Chapter 8 of Data Mining by I. H. Witten, E. Frank and M. A. Hall Combining multiple models Bagging The basic idea
More informationDifferential Privacy Preserving Spectral Graph Analysis
Differential Privacy Preserving Spectral Graph Analysis Yue Wang, Xintao Wu, and Leting Wu University of North Carolina at Charlotte, {ywang91, xwu, lwu8}@uncc.edu Abstract. In this paper, we focus on
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationLectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal
More informationDATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS
DATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS 1 AND ALGORITHMS Chiara Renso KDD-LAB ISTI- CNR, Pisa, Italy WHAT IS CLUSTER ANALYSIS? Finding groups of objects such that the objects in a group will be similar
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationBig Data Analytics CSCI 4030
High dim. data Graph data Infinite data Machine learning Apps Locality sensitive hashing PageRank, SimRank Filtering data streams SVM Recommen der systems Clustering Community Detection Web advertising
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationSupporting Online Material for
www.sciencemag.org/cgi/content/full/313/5786/504/dc1 Supporting Online Material for Reducing the Dimensionality of Data with Neural Networks G. E. Hinton* and R. R. Salakhutdinov *To whom correspondence
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)
More informationFionn Murtagh, Pedro Contreras International Conference p-adic MATHEMATICAL PHYSICS AND ITS APPLICATIONS. p-adics.2015, September 2015
Constant Time Search and Retrieval in Big Data, with Linear Time and Space Preprocessing, through Randomly Projected Piling and Sparse Ultrametric Coding Fionn Murtagh, Pedro Contreras International Conference
More informationALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationLecture 18 - Clifford Algebras and Spin groups
Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationAn Introduction to Machine Learning
An Introduction to Machine Learning L5: Novelty Detection and Regression Alexander J. Smola Statistical Machine Learning Program Canberra, ACT 0200 Australia Alex.Smola@nicta.com.au Tata Institute, Pune,
More informationConvolution. 1D Formula: 2D Formula: Example on the web: http://www.jhu.edu/~signals/convolve/
Basic Filters (7) Convolution/correlation/Linear filtering Gaussian filters Smoothing and noise reduction First derivatives of Gaussian Second derivative of Gaussian: Laplacian Oriented Gaussian filters
More informationLecture 4 Online and streaming algorithms for clustering
CSE 291: Geometric algorithms Spring 2013 Lecture 4 Online and streaming algorithms for clustering 4.1 On-line k-clustering To the extent that clustering takes place in the brain, it happens in an on-line
More informationA Negative Result Concerning Explicit Matrices With The Restricted Isometry Property
A Negative Result Concerning Explicit Matrices With The Restricted Isometry Property Venkat Chandar March 1, 2008 Abstract In this note, we prove that matrices whose entries are all 0 or 1 cannot achieve
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation
More informationTime Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication
Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 216-8440 This paper
More informationActive Learning SVM for Blogs recommendation
Active Learning SVM for Blogs recommendation Xin Guan Computer Science, George Mason University Ⅰ.Introduction In the DH Now website, they try to review a big amount of blogs and articles and find the
More information