Application of Spectral Clustering Algorithm
|
|
- Kathlyn Alexander
- 6 years ago
- Views:
Transcription
1 1/27 Application of Spectral Clustering Algorithm Danielle Middlebrooks Advisor: Kasso Okoudjou Department of Mathematics University of Maryland- College Park Advance Scientific Computing II May 11, 2016
2 2/27 Outline
3 3/27 Background Information Spectral Clustering is technique that makes use of the spectrum of the similarity matrix derived from the data set in order to cluster the data set into different clusters. Implement an algorithm that groups same digits from the MNIST Handwritten digits database in the same cluster. In practice this algorithm and my code will work for any database that wants to group together similar objects.
4 4/27 Motivation Motivated by the N cut problem. min NCut(A 1,..., A k ) := min 1 2 k i=1 W (A i, Āi) vol(a i ) where A is a subset of the vertices V the compliment Ā = V \ A W (A i, A j ) = i A i,j A j w ij vol(a) = i A d i The idea is that the eigenvectors serve as indicator functions in order to easily cluster the database in a reduced dimension.
5 5/27 Implementation Personal Laptop: Macbook Pro. Matlab R2016b 4GB Memory Desktop provided by Norbert Wiener Center Matlab R2015b 128GB Memory
6 6/27 Normalized Laplacian Matrix X i X j 2 Guassian Similarity Function: s(x i, X j ) = e 2σ 2 is a parameter. { W - Adjacency matrix w ij = 1, if s(x i, X j ) > ɛ 0, otherwise D- Degree matrix Unnormalized Laplacian Matrix: L = D W Normalized Laplacian Matrix: L sym = D 1/2 LD 1/2 = I D 1/2 WD 1/2 where σ
7 7/27 Normalized Laplacian Matrix As validation we know the smallest eigenvalue of the Normalized Laplacian will be zero with eigenvector D 1/2 1 To choose the best parameters, we implement the entire algorithm a number of times, changing epsilon each time until we reach some tolerance for the total error σ = 2000 ɛ =
8 8/27 Modified B Matrix Normalized Laplacian Matrix: L sym = D 1/2 LD 1/2 = I D 1/2 WD 1/2 = I B Computing the first p eigenvalues of B using the power method give us the largest eigenvalues in magnitude. Let B mod = B + µi where µ = max(sum(b,2))
9 9/27 Computing first p Eigenvectors Using the Power Method with Deflation on B mod we compute the first p eigenvalues.
10 10/27 Computing first p Eigenvectors By changing convergence criterion and increasing max iterations we obtain λ 1 λ 2 λ 3 λ 4 r 6.90E E E E-09 r =norm( B λ v B λ v,2) (λ, v) came from power method (λ, v ) came from eigs function
11 11/27 Row Normalization Let T R nxk be the eigenvector matrix with norm 1. Set v i,j t i,j = ( p v i,p 2 )1/2 v 11 v 12 v v 1p t 11 t 12 t t 1p v i1 v i2 v i3... v ip t i1 t i2 t i3... t ip v n1 v n2 v n3... v np t n1 t n2 t n3... t np
12 12/27 K-means Clustering Let y i be the ith row of T Randomly select k cluster centroids, z j. Calculate the distance between each y i and z j. Assign the data point to the closest centroid. Recalculate centroids and distances from data points to new centroids. If no data point was reassigned then stop, else reassign data points and repeat.
13 K-means Clustering Assign the original point X i to cluster j if and only if row i of the matrix T was assigned to cluster j. 13/27
14 14/27 Cluster Classification Next we classify each cluster as a particular digit. Digit Cluster Class Run time: 23mins
15 15/27 Results Below is a table of error for each cluster on 2000 Error= Number of incorrect digits in cluster Total number of digits in cluster % 82% 48% 65% 39% 13% 69% 58% 65% 72% Total number of incorrect digits Overall Error= Total number of digits = 59% Overall Error on 1000 images=64% Overall Error on images=49%
16 16/27 Results Cluster 6 Cluster 4 Cluster 3
17 17/27 Addition of New Datapoint- Standard Method Proposition (Nystrom Method) Method for out-of-sample extension Goal: Use a similarity kernel function K(x, y) in order to embed the new data point x in the reduced dimension. Benjio, Y, et al. Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering
18 18/27 Addition of New Datapoint- Another Method? We can determine which cluster a single new datapoint belongs to without re running the entire code. Create a similarity vector, denoted as X sim of 0 s and 1 s Normalize the similarity vector by multiplying it by D 1/2 Compute the projection of the similarity vector onto the eigenvectors of the Normalized Laplacian matrix and normalize. Denoted as C sim that lives in R p. Find the centroid that is closest to C sim
19 19/27 Results Implementation on a random subset of 100 digits. Error Runtime Averaged over 100 digits 61% 12.6sec
20 20/27 Yale Face Database Contains 165 grayscale images of 15 individuals. 11 images per subject, one per different facial expression or configuration. Each image is 32x32 pixels
21 21/27 Results Using 10 subjects and 5 images per subject with σ = 2000 and ɛ = Image Cluster Class Below is a table of error for each cluster classification Error= Number of incorrect faces in cluster Total number of faces in cluster % 33% 60% 83% 0% 66% 44% 40% 60% 66% Total number of incorrect faces Overall Error= Total number of faces = 54%
22 22/27 Results Cluster 5 Cluster 4 Cluster 2
23 23/27 End of October/ Early November: Construct Similarity Graph and Normalized Laplacian matrix. End of November/ Early December: Compute first k eigenvectors validate this. February: Normalize the rows of matrix of eigenvectors and perform dimension reduction. March/April: Cluster the points using k-means and validate this step. End of Spring semester: Implement entire algorithm, optimize and obtain final results.
24 24/27 Conclusion Spectral Clustering is a relatively good clustering technique. Better performance when dataset is sufficiently large. May obtain better results by using a different Normalized Laplacian or different similarity graph.
25 25/27 [1.] Von Cybernetics, U. A Tutorial on Spectral Clustering. Statistics and Computing, 7 (2007) 4. [2.] Shi, J. and Malik J. Normalized cuts and image segmentation. IEEE Transations on Pattern Analysis and Machine Intelligence, 22 (2000) 8. [3.] Chung, Fan. Spectral Graph Theory. N.p.: American Mathematical Society. Regional Conference Series in Mathematics Ser. 92. [4.] Vishnoi, Nisheeth K.Lx = b Laplacian Solvers and their Algorithmic Applications. N.p.: Foundations and Trends in Theoretical Computer Science, [5.] Benjio, Y, Paiement, J, Vincent, P. Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering. 2003
26 26/27 Thank you
27 27/27 Proposition Let K(x i, x j ) denote a kernel function of L sym such that L sym (i, j) = K(x i, x j ). Let (v k, λ k ) be an (eigenvector,eigenvalue) pair that solves L sym v k = λ k v k. Let (f k, λ k ) be an (eigenfunction,eigenvalue) pair that solves Kf k = λ k f k. Then y k (x) is the embedding associated with a new datapoint x. λ k = 1 n λ k n n f k (x) = v ik K(x, x j ) λ k i=1 y k (x) = f k(x) = 1 n v ik K(x, x j ) n λ k i=1
Spectral Clustering on Handwritten Digits Database
October 6, 2015 Spectral Clustering on Handwritten Digits Database Danielle dmiddle1@math.umd.edu Advisor: Kasso Okoudjou kasso@umd.edu Department of Mathematics University of Maryland- College Park Advance
More informationSpectral Clustering. Presented by Eldad Rubinstein Based on a Tutorial by Ulrike von Luxburg TAU Big Data Processing Seminar December 14, 2014
Spectral Clustering Presented by Eldad Rubinstein Based on a Tutorial by Ulrike von Luxburg TAU Big Data Processing Seminar December 14, 2014 What are we going to talk about? Introduction Clustering and
More informationDimension reduction for hyperspectral imaging using laplacian eigenmaps and randomized principal component analysis
Dimension reduction for hyperspectral imaging using laplacian eigenmaps and randomized principal component analysis Yiran Li yl534@math.umd.edu Advisor: Wojtek Czaja wojtek@math.umd.edu 10/17/2014 Abstract
More informationLinear and Non-linear Dimentionality Reduction Applied to Gene Expression Data of Cancer Tissue Samples
Linear and Non-linear Dimentionality Reduction Applied to Gene Expression Data of Cancer Tissue Samples Franck Olivier Ndjakou Njeunje Applied Mathematics, Statistics, and Scientific Computation University
More informationDimension reduction for hyperspectral imaging using laplacian eigenmaps and randomized principal component analysis:midyear Report
Dimension reduction for hyperspectral imaging using laplacian eigenmaps and randomized principal component analysis:midyear Report Yiran Li yl534@math.umd.edu Advisor: Wojtek Czaja wojtek@math.umd.edu
More informationThe clustering in general is the task of grouping a set of objects in such a way that objects
Spectral Clustering: A Graph Partitioning Point of View Yangzihao Wang Computer Science Department, University of California, Davis yzhwang@ucdavis.edu Abstract This course project provide the basic theory
More informationClustering: Classic Methods and Modern Views
Clustering: Classic Methods and Modern Views Marina Meilă University of Washington mmp@stat.washington.edu June 22, 2015 Lorentz Center Workshop on Clusters, Games and Axioms Outline Paradigms for clustering
More informationIntroduction to Machine Learning
Introduction to Machine Learning Clustering Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574 1 / 19 Outline
More informationCS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation
CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation Spring 2005 Ahmed Elgammal Dept of Computer Science CS 534 Segmentation II - 1 Outlines What is Graph cuts Graph-based clustering
More informationVisual Representations for Machine Learning
Visual Representations for Machine Learning Spectral Clustering and Channel Representations Lecture 1 Spectral Clustering: introduction and confusion Michael Felsberg Klas Nordberg The Spectral Clustering
More informationLocally Linear Landmarks for large-scale manifold learning
Locally Linear Landmarks for large-scale manifold learning Max Vladymyrov and Miguel Á. Carreira-Perpiñán Electrical Engineering and Computer Science University of California, Merced http://eecs.ucmerced.edu
More informationNonlinear Dimensionality Reduction Applied to the Classification of Images
onlinear Dimensionality Reduction Applied to the Classification of Images Student: Chae A. Clark (cclark8 [at] math.umd.edu) Advisor: Dr. Kasso A. Okoudjou (kasso [at] math.umd.edu) orbert Wiener Center
More informationBig Data Analytics. Special Topics for Computer Science CSE CSE Feb 11
Big Data Analytics Special Topics for Computer Science CSE 4095-001 CSE 5095-005 Feb 11 Fei Wang Associate Professor Department of Computer Science and Engineering fei_wang@uconn.edu Clustering II Spectral
More informationLarge-Scale Face Manifold Learning
Large-Scale Face Manifold Learning Sanjiv Kumar Google Research New York, NY * Joint work with A. Talwalkar, H. Rowley and M. Mohri 1 Face Manifold Learning 50 x 50 pixel faces R 2500 50 x 50 pixel random
More informationA Weighted Kernel PCA Approach to Graph-Based Image Segmentation
A Weighted Kernel PCA Approach to Graph-Based Image Segmentation Carlos Alzate Johan A. K. Suykens ESAT-SCD-SISTA Katholieke Universiteit Leuven Leuven, Belgium January 25, 2007 International Conference
More informationSpectral Clustering X I AO ZE N G + E L HA M TA BA S SI CS E CL A S S P R ESENTATION MA RCH 1 6,
Spectral Clustering XIAO ZENG + ELHAM TABASSI CSE 902 CLASS PRESENTATION MARCH 16, 2017 1 Presentation based on 1. Von Luxburg, Ulrike. "A tutorial on spectral clustering." Statistics and computing 17.4
More informationDIMENSION REDUCTION FOR HYPERSPECTRAL DATA USING RANDOMIZED PCA AND LAPLACIAN EIGENMAPS
DIMENSION REDUCTION FOR HYPERSPECTRAL DATA USING RANDOMIZED PCA AND LAPLACIAN EIGENMAPS YIRAN LI APPLIED MATHEMATICS, STATISTICS AND SCIENTIFIC COMPUTING ADVISOR: DR. WOJTEK CZAJA, DR. JOHN BENEDETTO DEPARTMENT
More informationClustering. Department Biosysteme Karsten Borgwardt Data Mining Course Basel Fall Semester / 238
Clustering Department Biosysteme Karsten Borgwardt Data Mining Course Basel Fall Semester 2015 163 / 238 What is Clustering? Department Biosysteme Karsten Borgwardt Data Mining Course Basel Fall Semester
More informationA Course in Machine Learning
A Course in Machine Learning Hal Daumé III 13 UNSUPERVISED LEARNING If you have access to labeled training data, you know what to do. This is the supervised setting, in which you have a teacher telling
More informationClustering. SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic
Clustering SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic Clustering is one of the fundamental and ubiquitous tasks in exploratory data analysis a first intuition about the
More informationData fusion and multi-cue data matching using diffusion maps
Data fusion and multi-cue data matching using diffusion maps Stéphane Lafon Collaborators: Raphy Coifman, Andreas Glaser, Yosi Keller, Steven Zucker (Yale University) Part of this work was supported by
More informationAPPROXIMATE SPECTRAL LEARNING USING NYSTROM METHOD. Aleksandar Trokicić
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 31, No 2 (2016), 569 578 APPROXIMATE SPECTRAL LEARNING USING NYSTROM METHOD Aleksandar Trokicić Abstract. Constrained clustering algorithms as an input
More informationCSE 547: Machine Learning for Big Data Spring Problem Set 2. Please read the homework submission policies.
CSE 547: Machine Learning for Big Data Spring 2019 Problem Set 2 Please read the homework submission policies. 1 Principal Component Analysis and Reconstruction (25 points) Let s do PCA and reconstruct
More informationK-means and Hierarchical Clustering
K-means and Hierarchical Clustering Xiaohui Xie University of California, Irvine K-means and Hierarchical Clustering p.1/18 Clustering Given n data points X = {x 1, x 2,, x n }. Clustering is the partitioning
More informationSGN (4 cr) Chapter 11
SGN-41006 (4 cr) Chapter 11 Clustering Jussi Tohka & Jari Niemi Department of Signal Processing Tampere University of Technology February 25, 2014 J. Tohka & J. Niemi (TUT-SGN) SGN-41006 (4 cr) Chapter
More informationIntroduction to spectral clustering
Introduction to spectral clustering Vasileios Zografos zografos@isy.liu.se Klas Nordberg klas@isy.liu.se What this course is Basic introduction into the core ideas of spectral clustering Sufficient to
More informationCS 231A CA Session: Problem Set 4 Review. Kevin Chen May 13, 2016
CS 231A CA Session: Problem Set 4 Review Kevin Chen May 13, 2016 PS4 Outline Problem 1: Viewpoint estimation Problem 2: Segmentation Meanshift segmentation Normalized cut Problem 1: Viewpoint Estimation
More informationClustering in Networks
Clustering in Networks (Spectral Clustering with the Graph Laplacian... a brief introduction) Tom Carter Computer Science CSU Stanislaus http://csustan.csustan.edu/ tom/clustering April 1, 2012 1 Our general
More informationSegmentation: Clustering, Graph Cut and EM
Segmentation: Clustering, Graph Cut and EM Ying Wu Electrical Engineering and Computer Science Northwestern University, Evanston, IL 60208 yingwu@northwestern.edu http://www.eecs.northwestern.edu/~yingwu
More informationCluster Analysis. Mu-Chun Su. Department of Computer Science and Information Engineering National Central University 2003/3/11 1
Cluster Analysis Mu-Chun Su Department of Computer Science and Information Engineering National Central University 2003/3/11 1 Introduction Cluster analysis is the formal study of algorithms and methods
More informationImage Similarities for Learning Video Manifolds. Selen Atasoy MICCAI 2011 Tutorial
Image Similarities for Learning Video Manifolds Selen Atasoy MICCAI 2011 Tutorial Image Spaces Image Manifolds Tenenbaum2000 Roweis2000 Tenenbaum2000 [Tenenbaum2000: J. B. Tenenbaum, V. Silva, J. C. Langford:
More informationSpectral Graph Clustering
Spectral Graph Clustering Benjamin Auffarth Universitat de Barcelona course report for Técnicas Avanzadas de Aprendizaje at Universitat Politècnica de Catalunya January 15, 2007 Abstract Spectral clustering
More informationClustering. So far in the course. Clustering. Clustering. Subhransu Maji. CMPSCI 689: Machine Learning. dist(x, y) = x y 2 2
So far in the course Clustering Subhransu Maji : Machine Learning 2 April 2015 7 April 2015 Supervised learning: learning with a teacher You had training data which was (feature, label) pairs and the goal
More informationSubspace Clustering. Weiwei Feng. December 11, 2015
Subspace Clustering Weiwei Feng December 11, 2015 Abstract Data structure analysis is an important basis of machine learning and data science, which is now widely used in computational visualization problems,
More informationKernel spectral clustering: model representations, sparsity and out-of-sample extensions
Kernel spectral clustering: model representations, sparsity and out-of-sample extensions Johan Suykens and Carlos Alzate K.U. Leuven, ESAT-SCD/SISTA Kasteelpark Arenberg B-3 Leuven (Heverlee), Belgium
More informationClustering. Subhransu Maji. CMPSCI 689: Machine Learning. 2 April April 2015
Clustering Subhransu Maji CMPSCI 689: Machine Learning 2 April 2015 7 April 2015 So far in the course Supervised learning: learning with a teacher You had training data which was (feature, label) pairs
More informationNon Overlapping Communities
Non Overlapping Communities Davide Mottin, Konstantina Lazaridou HassoPlattner Institute Graph Mining course Winter Semester 2016 Acknowledgements Most of this lecture is taken from: http://web.stanford.edu/class/cs224w/slides
More informationEE 701 ROBOT VISION. Segmentation
EE 701 ROBOT VISION Regions and Image Segmentation Histogram-based Segmentation Automatic Thresholding K-means Clustering Spatial Coherence Merging and Splitting Graph Theoretic Segmentation Region Growing
More informationRecognizing Handwritten Digits Using the LLE Algorithm with Back Propagation
Recognizing Handwritten Digits Using the LLE Algorithm with Back Propagation Lori Cillo, Attebury Honors Program Dr. Rajan Alex, Mentor West Texas A&M University Canyon, Texas 1 ABSTRACT. This work is
More informationHyperspectral image segmentation using spatial-spectral graphs
Hyperspectral image segmentation using spatial-spectral graphs David B. Gillis* and Jeffrey H. Bowles Naval Research Laboratory, Remote Sensing Division, Washington, DC 20375 ABSTRACT Spectral graph theory
More informationMy favorite application using eigenvalues: partitioning and community detection in social networks
My favorite application using eigenvalues: partitioning and community detection in social networks Will Hobbs February 17, 2013 Abstract Social networks are often organized into families, friendship groups,
More informationNetwork Traffic Measurements and Analysis
DEIB - Politecnico di Milano Fall, 2017 Introduction Often, we have only a set of features x = x 1, x 2,, x n, but no associated response y. Therefore we are not interested in prediction nor classification,
More informationUnsupervised Clustering of Bitcoin Transaction Data
Unsupervised Clustering of Bitcoin Transaction Data Midyear Report 1 AMSC 663/664 Project Advisor: Dr. Chris Armao By: Stefan Poikonen Bitcoin: A Brief Refresher 2 Bitcoin is a decentralized cryptocurrency
More informationCSCI-B609: A Theorist s Toolkit, Fall 2016 Sept. 6, Firstly let s consider a real world problem: community detection.
CSCI-B609: A Theorist s Toolkit, Fall 016 Sept. 6, 016 Lecture 03: The Sparsest Cut Problem and Cheeger s Inequality Lecturer: Yuan Zhou Scribe: Xuan Dong We will continue studying the spectral graph theory
More informationFrom Pixels to Blobs
From Pixels to Blobs 15-463: Rendering and Image Processing Alexei Efros Today Blobs Need for blobs Extracting blobs Image Segmentation Working with binary images Mathematical Morphology Blob properties
More informationLearning a Manifold as an Atlas Supplementary Material
Learning a Manifold as an Atlas Supplementary Material Nikolaos Pitelis Chris Russell School of EECS, Queen Mary, University of London [nikolaos.pitelis,chrisr,lourdes]@eecs.qmul.ac.uk Lourdes Agapito
More informationSegmentation Computer Vision Spring 2018, Lecture 27
Segmentation http://www.cs.cmu.edu/~16385/ 16-385 Computer Vision Spring 218, Lecture 27 Course announcements Homework 7 is due on Sunday 6 th. - Any questions about homework 7? - How many of you have
More informationBig-data Clustering: K-means vs K-indicators
Big-data Clustering: K-means vs K-indicators Yin Zhang Dept. of Computational & Applied Math. Rice University, Houston, Texas, U.S.A. Joint work with Feiyu Chen & Taiping Zhang (CQU), Liwei Xu (UESTC)
More informationhttp://www.xkcd.com/233/ Text Clustering David Kauchak cs160 Fall 2009 adapted from: http://www.stanford.edu/class/cs276/handouts/lecture17-clustering.ppt Administrative 2 nd status reports Paper review
More informationLocality Preserving Projections (LPP) Abstract
Locality Preserving Projections (LPP) Xiaofei He Partha Niyogi Computer Science Department Computer Science Department The University of Chicago The University of Chicago Chicago, IL 60615 Chicago, IL
More information( ) =cov X Y = W PRINCIPAL COMPONENT ANALYSIS. Eigenvectors of the covariance matrix are the principal components
Review Lecture 14 ! PRINCIPAL COMPONENT ANALYSIS Eigenvectors of the covariance matrix are the principal components 1. =cov X Top K principal components are the eigenvectors with K largest eigenvalues
More informationk-means demo Administrative Machine learning: Unsupervised learning" Assignment 5 out
Machine learning: Unsupervised learning" David Kauchak cs Spring 0 adapted from: http://www.stanford.edu/class/cs76/handouts/lecture7-clustering.ppt http://www.youtube.com/watch?v=or_-y-eilqo Administrative
More informationThe Alternating Direction Method of Multipliers
The Alternating Direction Method of Multipliers With Adaptive Step Size Selection Peter Sutor, Jr. Project Advisor: Professor Tom Goldstein October 8, 2015 1 / 30 Introduction Presentation Outline 1 Convex
More information6.801/866. Segmentation and Line Fitting. T. Darrell
6.801/866 Segmentation and Line Fitting T. Darrell Segmentation and Line Fitting Gestalt grouping Background subtraction K-Means Graph cuts Hough transform Iterative fitting (Next time: Probabilistic segmentation)
More informationMSA220 - Statistical Learning for Big Data
MSA220 - Statistical Learning for Big Data Lecture 13 Rebecka Jörnsten Mathematical Sciences University of Gothenburg and Chalmers University of Technology Clustering Explorative analysis - finding groups
More informationNearest neighbors classifiers
Nearest neighbors classifiers James McInerney Adapted from slides by Daniel Hsu Sept 11, 2017 1 / 25 Housekeeping We received 167 HW0 submissions on Gradescope before midnight Sept 10th. From a random
More informationTELCOM2125: Network Science and Analysis
School of Information Sciences University of Pittsburgh TELCOM2125: Network Science and Analysis Konstantinos Pelechrinis Spring 2015 2 Part 4: Dividing Networks into Clusters The problem l Graph partitioning
More informationImage Segmentation continued Graph Based Methods
Image Segmentation continued Graph Based Methods Previously Images as graphs Fully-connected graph node (vertex) for every pixel link between every pair of pixels, p,q affinity weight w pq for each link
More informationSocial-Network Graphs
Social-Network Graphs Mining Social Networks Facebook, Google+, Twitter Email Networks, Collaboration Networks Identify communities Similar to clustering Communities usually overlap Identify similarities
More informationRobust PDF Table Locator
Robust PDF Table Locator December 17, 2016 1 Introduction Data scientists rely on an abundance of tabular data stored in easy-to-machine-read formats like.csv files. Unfortunately, most government records
More informationClustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin
Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin 2005-2014 1 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin 2005-2014
More informationNonlinear Dimensionality Reduction Applied to the Binary Classification of Images
Nonlinear Dimensionality Reduction Applied to the Binary Classification of Images. Abstract Chae A. Clark cclark8@math.umd.edu University of Maryland, College Park Advisor Dr. Kasso A. Okoudjou kasso@math.umd.edu
More informationSpectral Sparsification in Spectral Clustering
Spectral Sparsification in Spectral Clustering Alireza Chakeri, Hamidreza Farhidzadeh, Lawrence O. Hall Computer Science and Engineering Department University of South Florida Tampa, Florida Email: chakeri@mail.usf.edu,
More informationEfficient Iterative Semi-supervised Classification on Manifold
. Efficient Iterative Semi-supervised Classification on Manifold... M. Farajtabar, H. R. Rabiee, A. Shaban, A. Soltani-Farani Sharif University of Technology, Tehran, Iran. Presented by Pooria Joulani
More informationCS 534: Computer Vision Segmentation and Perceptual Grouping
CS 534: Computer Vision Segmentation and Perceptual Grouping Ahmed Elgammal Dept of Computer Science CS 534 Segmentation - 1 Outlines Mid-level vision What is segmentation Perceptual Grouping Segmentation
More informationTime and Space Efficient Spectral Clustering via Column Sampling
Time and Space Efficient Spectral Clustering via Column Sampling Mu Li Xiao-Chen Lian James T. Kwok Bao-Liang Lu, 3 Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai
More informationFrame based kernel methods for hyperspectral imagery data
Frame based kernel methods for hyperspectral imagery data Norbert Wiener Center Department of Mathematics University of Maryland, College Park Recent Advances in Harmonic Analysis and Elliptic Partial
More informationCOSC 6397 Big Data Analytics. Fuzzy Clustering. Some slides based on a lecture by Prof. Shishir Shah. Edgar Gabriel Spring 2015.
COSC 6397 Big Data Analytics Fuzzy Clustering Some slides based on a lecture by Prof. Shishir Shah Edgar Gabriel Spring 215 Clustering Clustering is a technique for finding similarity groups in data, called
More informationMachine Learning for Data Science (CS4786) Lecture 11
Machine Learning for Data Science (CS4786) Lecture 11 Spectral Clustering Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016fa/ Survey Survey Survey Competition I Out! Preliminary report of
More information10-701/15-781, Fall 2006, Final
-7/-78, Fall 6, Final Dec, :pm-8:pm There are 9 questions in this exam ( pages including this cover sheet). If you need more room to work out your answer to a question, use the back of the page and clearly
More informationClustering. Informal goal. General types of clustering. Applications: Clustering in information search and analysis. Example applications in search
Informal goal Clustering Given set of objects and measure of similarity between them, group similar objects together What mean by similar? What is good grouping? Computation time / quality tradeoff 1 2
More informationClassification and Regression using Linear Networks, Multilayer Perceptrons and Radial Basis Functions
ENEE 739Q SPRING 2002 COURSE ASSIGNMENT 2 REPORT 1 Classification and Regression using Linear Networks, Multilayer Perceptrons and Radial Basis Functions Vikas Chandrakant Raykar Abstract The aim of the
More informationSegmentation and low-level grouping.
Segmentation and low-level grouping. Bill Freeman, MIT 6.869 April 14, 2005 Readings: Mean shift paper and background segmentation paper. Mean shift IEEE PAMI paper by Comanici and Meer, http://www.caip.rutgers.edu/~comanici/papers/msrobustapproach.pdf
More informationADVANCED MACHINE LEARNING MACHINE LEARNING. Kernel for Clustering kernel K-Means
1 MACHINE LEARNING Kernel for Clustering ernel K-Means Outline of Today s Lecture 1. Review principle and steps of K-Means algorithm. Derive ernel version of K-means 3. Exercise: Discuss the geometrical
More informationIntroduction to spectral clustering
Introduction to spectral clustering Denis Hamad LASL ULCO Denis.Hamad@lasl.univ-littoral.fr Philippe Biela HEI LAGIS Philippe.Biela@hei.fr Data Clustering Data clustering Data clustering is an important
More informationImage Segmentation. Selim Aksoy. Bilkent University
Image Segmentation Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Examples of grouping in vision [http://poseidon.csd.auth.gr/lab_research/latest/imgs/s peakdepvidindex_img2.jpg]
More informationThe Pre-Image Problem in Kernel Methods
The Pre-Image Problem in Kernel Methods James Kwok Ivor Tsang Department of Computer Science Hong Kong University of Science and Technology Hong Kong The Pre-Image Problem in Kernel Methods ICML-2003 1
More informationScalable Clustering of Signed Networks Using Balance Normalized Cut
Scalable Clustering of Signed Networks Using Balance Normalized Cut Kai-Yang Chiang,, Inderjit S. Dhillon The 21st ACM International Conference on Information and Knowledge Management (CIKM 2012) Oct.
More informationMachine learning - HT Clustering
Machine learning - HT 2016 10. Clustering Varun Kanade University of Oxford March 4, 2016 Announcements Practical Next Week - No submission Final Exam: Pick up on Monday Material covered next week is not
More informationLecture 6: Unsupervised Machine Learning Dagmar Gromann International Center For Computational Logic
SEMANTIC COMPUTING Lecture 6: Unsupervised Machine Learning Dagmar Gromann International Center For Computational Logic TU Dresden, 23 November 2018 Overview Unsupervised Machine Learning overview Association
More informationGeneralized trace ratio optimization and applications
Generalized trace ratio optimization and applications Mohammed Bellalij, Saïd Hanafi, Rita Macedo and Raca Todosijevic University of Valenciennes, France PGMO Days, 2-4 October 2013 ENSTA ParisTech PGMO
More informationSemi-Supervised Normalized Embeddings for Fusion and Land-Use Classification of Multiple View Data
Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 12-2018 Semi-Supervised Normalized Embeddings for Fusion and Land-Use Classification of Multiple View Data Poppy
More informationCOSC 6339 Big Data Analytics. Fuzzy Clustering. Some slides based on a lecture by Prof. Shishir Shah. Edgar Gabriel Spring 2017.
COSC 6339 Big Data Analytics Fuzzy Clustering Some slides based on a lecture by Prof. Shishir Shah Edgar Gabriel Spring 217 Clustering Clustering is a technique for finding similarity groups in data, called
More informationCS325 Artificial Intelligence Ch. 20 Unsupervised Machine Learning
CS325 Artificial Intelligence Cengiz Spring 2013 Unsupervised Learning Missing teacher No labels, y Just input data, x What can you learn with it? Unsupervised Learning Missing teacher No labels, y Just
More informationClustering and Visualisation of Data
Clustering and Visualisation of Data Hiroshi Shimodaira January-March 28 Cluster analysis aims to partition a data set into meaningful or useful groups, based on distances between data points. In some
More informationCLUSTERING. JELENA JOVANOVIĆ Web:
CLUSTERING JELENA JOVANOVIĆ Email: jeljov@gmail.com Web: http://jelenajovanovic.net OUTLINE What is clustering? Application domains K-Means clustering Understanding it through an example The K-Means algorithm
More informationClustering: K-means and Kernel K-means
Clustering: K-means and Kernel K-means Piyush Rai Machine Learning (CS771A) Aug 31, 2016 Machine Learning (CS771A) Clustering: K-means and Kernel K-means 1 Clustering Usually an unsupervised learning problem
More informationThe Constrained Laplacian Rank Algorithm for Graph-Based Clustering
The Constrained Laplacian Rank Algorithm for Graph-Based Clustering Feiping Nie, Xiaoqian Wang, Michael I. Jordan, Heng Huang Department of Computer Science and Engineering, University of Texas, Arlington
More informationExpectation Maximization!
Expectation Maximization! adapted from: Doug Downey and Bryan Pardo, Northwestern University and http://www.stanford.edu/class/cs276/handouts/lecture17-clustering.ppt Steps in Clustering Select Features
More informationLecture 2: Geometric Graphs, Predictive Graphs and Spectral Analysis
Lecture 2: Geometric Graphs, Predictive Graphs and Spectral Analysis Radu Balan February 8, 2018 Data Graphs Today we discuss construction of dynamical data graphs and spectral analysis. The overarching
More informationCS246: Mining Massive Datasets Jure Leskovec, Stanford University
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu SPAM FARMING 2/11/2013 Jure Leskovec, Stanford C246: Mining Massive Datasets 2 2/11/2013 Jure Leskovec, Stanford
More informationCSE 6242 A / CS 4803 DVA. Feb 12, Dimension Reduction. Guest Lecturer: Jaegul Choo
CSE 6242 A / CS 4803 DVA Feb 12, 2013 Dimension Reduction Guest Lecturer: Jaegul Choo CSE 6242 A / CS 4803 DVA Feb 12, 2013 Dimension Reduction Guest Lecturer: Jaegul Choo Data is Too Big To Do Something..
More informationPlanar Graphs 2, the Colin de Verdière Number
Spectral Graph Theory Lecture 26 Planar Graphs 2, the Colin de Verdière Number Daniel A. Spielman December 4, 2009 26.1 Introduction In this lecture, I will introduce the Colin de Verdière number of a
More informationClustering Algorithms for general similarity measures
Types of general clustering methods Clustering Algorithms for general similarity measures general similarity measure: specified by object X object similarity matrix 1 constructive algorithms agglomerative
More informationImproving Image Segmentation Quality Via Graph Theory
International Symposium on Computers & Informatics (ISCI 05) Improving Image Segmentation Quality Via Graph Theory Xiangxiang Li, Songhao Zhu School of Automatic, Nanjing University of Post and Telecommunications,
More informationLab # 2 - ACS I Part I - DATA COMPRESSION in IMAGE PROCESSING using SVD
Lab # 2 - ACS I Part I - DATA COMPRESSION in IMAGE PROCESSING using SVD Goals. The goal of the first part of this lab is to demonstrate how the SVD can be used to remove redundancies in data; in this example
More informationMATH 567: Mathematical Techniques in Data
Supervised and unsupervised learning Supervised learning problems: MATH 567: Mathematical Techniques in Data (X, Y ) P (X, Y ). Data Science Clustering I is labelled (input/output) with joint density We
More informationCluster Analysis (b) Lijun Zhang
Cluster Analysis (b) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Grid-Based and Density-Based Algorithms Graph-Based Algorithms Non-negative Matrix Factorization Cluster Validation Summary
More informationClustering. Chapter 10 in Introduction to statistical learning
Clustering Chapter 10 in Introduction to statistical learning 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 1 Clustering ² Clustering is the art of finding groups in data (Kaufman and Rousseeuw, 1990). ² What
More informationLecture 11: E-M and MeanShift. CAP 5415 Fall 2007
Lecture 11: E-M and MeanShift CAP 5415 Fall 2007 Review on Segmentation by Clustering Each Pixel Data Vector Example (From Comanciu and Meer) Review of k-means Let's find three clusters in this data These
More information