A graph based approach to semi-supervised learning
|
|
- Dulcie Phebe Gardner
- 5 years ago
- Views:
Transcription
1 A graph based approach to semi-supervised learning 1 Feb 2011
2 Two papers M. Belkin, P. Niyogi, and V Sindhwani. Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research 1-48, M. Belkin, P. Niyogi. Towards a Theoretical Foundation for Laplacian Based Manifold Methods. Journal of Computer and System Sciences, 2007.
3 What is semi-supervised learning? Prediction, but with the help of unsupervised examples.
4 Why semi-supervised learning?
5 Why semi-supervised learning? Practical reasons: unlabeled data cheap
6 Why semi-supervised learning? Practical reasons: unlabeled data cheap More natural model of human learning
7 An example
8 An example
9 An example
10 An example
11 Semi-supervised learning framework 1 l labeled examples (x, y) generated by distribution P. u unlabeled examples drawn from marginal P X. Mercer kernel K. f = argmin f Hk 1 l l V (x i, y i, f ) + γ f 2 K i=1
12 Semi-supervised learning framework 2 Classical representer theorem: f (x) = l α i K(x i, x) i=1
13 Manifold regularization: assumptions Assumptions: P supported on manifold M P(y x) varies smoothly along geodesics of P X
14 Manifold regularization: assumptions Assumptions: P supported on manifold M P(y x) varies smoothly along geodesics of P X Modified objective: f = argmin f HK 1 l l V (x i, y i, f ) + γ A f 2 K + γ I f 2 I i=1
15 Manifold regularization: known marginal Theorem If P X known and M is a smooth Riemannian manifold, f (x) = l i=1 + α(z)k(x, z)dp X (z) M
16 Manifold regularization: unknown marginal Need to estimate marginal and f I
17 Manifold regularization: unknown marginal Need to estimate marginal and f I Only requires unlabeled data
18 Manifold regularization: unknown marginal Need to estimate marginal and f I Only requires unlabeled data Natural choice: f 2 I = M Mf 2 dp
19 Manifold regularization: unknown marginal Need to estimate marginal and f I Only requires unlabeled data Natural choice: f 2 I = M Mf 2 dp Approximate M with graph
20 Manifold regularization: building the graph Single-linkage clustering Nearest neighbor methods
21 Manifold regularization: building the graph Single-linkage clustering Nearest neighbor methods Use graph laplacian instead of manifold Laplacian
22 Manifold regularization: using the graph Theorem By choosing exponential weights for the edges, the graph Laplacian converges to the manifold Laplacian in probability. f 1 = argmin l f HK l i=1 V (x i, y i, f ) + γ A f 2 K + γ I f T Lf (u+l) 2 L = D W
23 Main result Theorem f (x) = l+u i=1 α ik(x i, x)
24 Regularized least squares Classical RLS: argmin f HK 1 l l i=1 (y i f (x i )) 2 + λ f 2 K
25 Regularized least squares Classical RLS: argmin f HK 1 l l i=1 (y i f (x i )) 2 + λ f 2 K Solution: f (x) = l i=1 α i K(x i, x), α = (K + λli ) 1 Y
26 Regularized least squares 1 Classical RLS: argmin l f HK l i=1 (y i f (x i )) 2 + λ f 2 K Solution: f (x) = l i=1 α i K(x i, x), α = (K + λli ) 1 Y Laplacian RLS: argmin f HK 1 l l i=1 (y i f (x i )) 2 + λ A f 2 K + λ I (u+l) 2 f T Lf
27 Regularized least squares 1 Classical RLS: argmin l f HK l i=1 (y i f (x i )) 2 + λ f 2 K Solution: f (x) = l i=1 α i K(x i, x), α = (K + λli ) 1 Y Laplacian RLS: argmin f HK 1 l l i=1 (y i f (x i )) 2 + λ A f 2 K + Solution: f (x) = l+u i=1 α I K(x, x i), α = (JK + λ A li + λ I l LK) 1 Y (u+l) 2 λ I (u+l) 2 f T Lf
28 Support vector machines Like in regularized least squares, there is a version of the SVM called Laplacian SVM.
29 Two moons dataset
30 Wisconsin breast cancer data 683 samples. Benign or malignant? Clump thickness Uniformity of cell size and shape etc
31 Wisconsin breast cancer data: results
32 Longer term stuff Besides geometric structure, what else can we use? Invariance? Learning the manifold: Simplicial complex instead of graph? Homology. Nice example in natural image statistics (Mumford et al, 2003)
33 Longer term stuff 2 Hickernell, Song, and Zhang. Reproducing kernel Banach spaces with the l 1 norm. Preprint. Reproducing kernel Banach spaces with the l 1 norm II: error analysis for regularized least squares regression. Preprint.
Graphs, Geometry and Semi-supervised Learning
Graphs, Geometry and Semi-supervised Learning Mikhail Belkin The Ohio State University, Dept of Computer Science and Engineering and Dept of Statistics Collaborators: Partha Niyogi, Vikas Sindhwani In
More informationBeyond the Point Cloud: From Transductive to Semi-Supervised Learning
Beyond the Point Cloud: From Transductive to Semi-Supervised Learning Vikas Sindhwani, Partha Niyogi, Mikhail Belkin Andrew B. Goldberg goldberg@cs.wisc.edu Department of Computer Sciences University of
More informationManifold Regularization
Manifold Regularization Vikas Sindhwani Department of Computer Science University of Chicago Joint Work with Mikhail Belkin and Partha Niyogi TTI-C Talk September 14, 24 p.1 The Problem of Learning is
More informationSemi-Supervised Learning of Speech Sounds
Aren Jansen Partha Niyogi Department of Computer Science Interspeech 2007 Objectives 1 Present a manifold learning algorithm based on locality preserving projections for semi-supervised phone classification
More informationUnsupervised Learning Techniques Class 07, 1 March 2006 Andrea Caponnetto
Unsupervised Learning Techniques 9.520 Class 07, 1 March 2006 Andrea Caponnetto About this class Goal To introduce some methods for unsupervised learning: Gaussian Mixtures, K-Means, ISOMAP, HLLE, Laplacian
More informationManifold Regularization
9.520: Statistical Learning Theory and Applications arch 3rd, 200 anifold Regularization Lecturer: Lorenzo Rosasco Scribe: Hooyoung Chung Introduction In this lecture we introduce a class of learning algorithms,
More informationof semi-supervised Gaussian process classifiers.
Semi-supervised Gaussian Process Classifiers Vikas Sindhwani Department of Computer Science University of Chicago Chicago, IL 665, USA vikass@cs.uchicago.edu Wei Chu CCLS Columbia University New York,
More informationWhat is semi-supervised learning?
What is semi-supervised learning? In many practical learning domains, there is a large supply of unlabeled data but limited labeled data, which can be expensive to generate text processing, video-indexing,
More informationSemi-Supervised Learning by Multi-Manifold Separation
Semi-Supervised Learning by Multi-Manifold Separation Xiaojin (Jerry) Zhu Department of Computer Sciences University of Wisconsin Madison Joint work with Andrew Goldberg, Zhiting Xu, Aarti Singh, and Rob
More informationRegularization in Reproducing Kernel Banach Spaces
.... Regularization in Reproducing Kernel Banach Spaces Guohui Song School of Mathematical and Statistical Sciences Arizona State University Comp Math Seminar, September 16, 2010 Joint work with Dr. Fred
More informationJustin Solomon MIT, Spring 2017
Justin Solomon MIT, Spring 2017 http://pngimg.com/upload/hammer_png3886.png You can learn a lot about a shape by hitting it (lightly) with a hammer! What can you learn about its shape from vibration frequencies
More informationUnsupervised dimensionality reduction
Unsupervised dimensionality reduction Guillaume Obozinski Ecole des Ponts - ParisTech SOCN course 2014 Guillaume Obozinski Unsupervised dimensionality reduction 1/30 Outline 1 PCA 2 Kernel PCA 3 Multidimensional
More informationHow to learn from very few examples?
How to learn from very few examples? Dengyong Zhou Department of Empirical Inference Max Planck Institute for Biological Cybernetics Spemannstr. 38, 72076 Tuebingen, Germany Outline Introduction Part A
More informationSemi-Supervised Learning in Reproducing Kernel Hilbert Spaces Using Local Invariances
Semi-Supervised Learning in Reproducing Kernel Hilbert Spaces Using Local Invariances Wee Sun Lee,2, Xinhua Zhang,2, and Yee Whye Teh Department of Computer Science, National University of Singapore. 2
More informationMulti-View Point Cloud Kernels for Semi-Supervised Learning
Multi-View Point Cloud Kernels for Semi-Supervised Learning David S. Rosenberg, Vikas Sindhwani, Peter L. Bartlett, Partha Niyogi May 29, 2009 Scope In semi-supervised learning (SSL), we learn a predictive
More informationHou, Ch. et al. IEEE Transactions on Neural Networks March 2011
Hou, Ch. et al. IEEE Transactions on Neural Networks March 2011 Semi-supervised approach which attempts to incorporate partial information from unlabeled data points Semi-supervised approach which attempts
More informationGraph-Based Semi-Supervised Learning
Graph-Based Semi-Supervised Learning Olivier Delalleau, Yoshua Bengio and Nicolas Le Roux Université de Montréal CIAR Workshop - April 26th, 2005 Graph-Based Semi-Supervised Learning Yoshua Bengio, Olivier
More informationLABELED data is expensive to obtain in terms of both. Laplacian Embedded Regression for Scalable Manifold Regularization
Laplacian Embedded Regression for Scalable Manifold Regularization Lin Chen, Ivor W. Tsang, Dong Xu Abstract Semi-supervised Learning (SSL), as a powerful tool to learn from a limited number of labeled
More informationPredicting Graph Labels using Perceptron. Shuang Song
Predicting Graph Labels using Perceptron Shuang Song shs037@eng.ucsd.edu Online learning over graphs M. Herbster, M. Pontil, and L. Wainer, Proc. 22nd Int. Conf. Machine Learning (ICML'05), 2005 Prediction
More informationNonlinear Dimensionality Reduction. Jose A. Costa
Nonlinear Dimensionality Reduction Jose A. Costa Mathematics of Information Seminar, Dec. Motivation Many useful of signals such as: Image databases; Gene expression microarrays; Internet traffic time
More informationLearning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking p. 1/31
Learning from Labeled and Unlabeled Data: Semi-supervised Learning and Ranking Dengyong Zhou zhou@tuebingen.mpg.de Dept. Schölkopf, Max Planck Institute for Biological Cybernetics, Germany Learning from
More informationLearning with Consistency between Inductive Functions and Kernels
Learning with Consistency between Inductive Functions and Kernels Haixuan Yang Irwin King Michael R. Lyu Department of Computer Science & Engineering The Chinese University of Hong Kong Shatin, N.T., Hong
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 informationAnalysis of Spectral Kernel Design based Semi-supervised Learning
Analysis of Spectral Kernel Design based Semi-supervised Learning Tong Zhang IBM T. J. Watson Research Center Yorktown Heights, NY 10598 Rie Kubota Ando IBM T. J. Watson Research Center Yorktown Heights,
More informationOnline Manifold Regularization: A New Learning Setting and Empirical Study
Online Manifold Regularization: A New Learning Setting and Empirical Study Andrew B. Goldberg 1, Ming Li 2, Xiaojin Zhu 1 1 Computer Sciences, University of Wisconsin Madison, USA. {goldberg,jerryzhu}@cs.wisc.edu
More informationIFT LAPLACIAN APPLICATIONS. Mikhail Bessmeltsev
IFT 6112 09 LAPLACIAN APPLICATIONS http://www-labs.iro.umontreal.ca/~bmpix/teaching/6112/2018/ Mikhail Bessmeltsev Rough Intuition http://pngimg.com/upload/hammer_png3886.png You can learn a lot about
More informationVector-valued Manifold Regularization
Hà Quang Minh minh.haquang@iit.it Italian Institute of Technology, Via Morego 30, Genoa 663, Italy Vikas Sindhwani vsindhw@us.ibm.com Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights,
More informationAn RKHS for Multi-View Learning and Manifold Co-Regularization
Vikas Sindhwani vsindhw@us.ibm.com Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA David S. Rosenberg drosen@stat.berkeley.edu Department of Statistics, University
More informationAn Overview of Outlier Detection Techniques and Applications
Machine Learning Rhein-Neckar Meetup An Overview of Outlier Detection Techniques and Applications Ying Gu connygy@gmail.com 28.02.2016 Anomaly/Outlier Detection What are anomalies/outliers? The set of
More informationGraphs in Machine Learning
Graphs in Machine Learning Michal Valko Inria Lille - Nord Europe, France TA: Pierre Perrault Partially based on material by: Mikhail Belkin, Jerry Zhu, Olivier Chapelle, Branislav Kveton October 30, 2017
More informationNon-linear Dimensionality Reduction
Non-linear Dimensionality Reduction CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Introduction Laplacian Eigenmaps Locally Linear Embedding (LLE)
More informationNeural Networks, Convexity, Kernels and Curses
Neural Networks, Convexity, Kernels and Curses Yoshua Bengio Work done with Nicolas Le Roux, Olivier Delalleau and Hugo Larochelle August 26th 2005 Perspective Curse of Dimensionality Most common non-parametric
More informationNonlinear Dimensionality Reduction
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Kernel PCA 2 Isomap 3 Locally Linear Embedding 4 Laplacian Eigenmap
More informationBack to the future: Radial Basis Function networks revisited
Back to the future: Radial Basis Function networks revisited Qichao Que, Mikhail Belkin Department of Computer Science and Engineering Ohio State University Columbus, OH 4310 que, mbelkin@cse.ohio-state.edu
More informationFace Recognition Using Laplacianfaces He et al. (IEEE Trans PAMI, 2005) presented by Hassan A. Kingravi
Face Recognition Using Laplacianfaces He et al. (IEEE Trans PAMI, 2005) presented by Hassan A. Kingravi Overview Introduction Linear Methods for Dimensionality Reduction Nonlinear Methods and Manifold
More informationData dependent operators for the spatial-spectral fusion problem
Data dependent operators for the spatial-spectral fusion problem Wien, December 3, 2012 Joint work with: University of Maryland: J. J. Benedetto, J. A. Dobrosotskaya, T. Doster, K. W. Duke, M. Ehler, A.
More informationDiscrete vs. Continuous: Two Sides of Machine Learning
Discrete vs. Continuous: Two Sides of Machine Learning Dengyong Zhou Department of Empirical Inference Max Planck Institute for Biological Cybernetics Spemannstr. 38, 72076 Tuebingen, Germany Oct. 18,
More informationLearning on Graphs and Manifolds. CMPSCI 689 Sridhar Mahadevan U.Mass Amherst
Learning on Graphs and Manifolds CMPSCI 689 Sridhar Mahadevan U.Mass Amherst Outline Manifold learning is a relatively new area of machine learning (2000-now). Main idea Model the underlying geometry of
More information6.036 midterm review. Wednesday, March 18, 15
6.036 midterm review 1 Topics covered supervised learning labels available unsupervised learning no labels available semi-supervised learning some labels available - what algorithms have you learned that
More informationSelf-Tuning Semantic Image Segmentation
Self-Tuning Semantic Image Segmentation Sergey Milyaev 1,2, Olga Barinova 2 1 Voronezh State University sergey.milyaev@gmail.com 2 Lomonosov Moscow State University obarinova@graphics.cs.msu.su Abstract.
More informationLaplacian Eigenmaps for Dimensionality Reduction and Data Representation
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation Neural Computation, June 2003; 15 (6):1373-1396 Presentation for CSE291 sp07 M. Belkin 1 P. Niyogi 2 1 University of Chicago, Department
More informationClustering. CSL465/603 - Fall 2016 Narayanan C Krishnan
Clustering CSL465/603 - Fall 2016 Narayanan C Krishnan ckn@iitrpr.ac.in Supervised vs Unsupervised Learning Supervised learning Given x ", y " "%& ', learn a function f: X Y Categorical output classification
More informationKernels A Machine Learning Overview
Kernels A Machine Learning Overview S.V.N. Vishy Vishwanathan vishy@axiom.anu.edu.au National ICT of Australia and Australian National University Thanks to Alex Smola, Stéphane Canu, Mike Jordan and Peter
More informationGlobal vs. Multiscale Approaches
Harmonic Analysis on Graphs Global vs. Multiscale Approaches Weizmann Institute of Science, Rehovot, Israel July 2011 Joint work with Matan Gavish (WIS/Stanford), Ronald Coifman (Yale), ICML 10' Challenge:
More information9/26/17. Ridge regression. What our model needs to do. Ridge Regression: L2 penalty. Ridge coefficients. Ridge coefficients
What our model needs to do regression Usually, we are not just trying to explain observed data We want to uncover meaningful trends And predict future observations Our questions then are Is β" a good estimate
More information9.520 Problem Set 2. Due April 25, 2011
9.50 Problem Set Due April 5, 011 Note: there are five problems in total in this set. Problem 1 In classification problems where the data are unbalanced (there are many more examples of one class than
More informationIterative Laplacian Score for Feature Selection
Iterative Laplacian Score for Feature Selection Linling Zhu, Linsong Miao, and Daoqiang Zhang College of Computer Science and echnology, Nanjing University of Aeronautics and Astronautics, Nanjing 2006,
More informationUnlabeled Data: Now It Helps, Now It Doesn t
institution-logo-filena A. Singh, R. D. Nowak, and X. Zhu. In NIPS, 2008. 1 Courant Institute, NYU April 21, 2015 Outline institution-logo-filena 1 Conflicting Views in Semi-supervised Learning The Cluster
More informationStatistical Learning. Dong Liu. Dept. EEIS, USTC
Statistical Learning Dong Liu Dept. EEIS, USTC Chapter 6. Unsupervised and Semi-Supervised Learning 1. Unsupervised learning 2. k-means 3. Gaussian mixture model 4. Other approaches to clustering 5. Principle
More informationMulti-view Laplacian Support Vector Machines
Multi-view Laplacian Support Vector Machines Shiliang Sun Department of Computer Science and Technology, East China Normal University, Shanghai 200241, China slsun@cs.ecnu.edu.cn Abstract. We propose a
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 informationSemi-Supervised Learning in Gigantic Image Collections. Rob Fergus (New York University) Yair Weiss (Hebrew University) Antonio Torralba (MIT)
Semi-Supervised Learning in Gigantic Image Collections Rob Fergus (New York University) Yair Weiss (Hebrew University) Antonio Torralba (MIT) Gigantic Image Collections What does the world look like? High
More informationOne-class Label Propagation Using Local Cone Based Similarity
One-class Label Propagation Using Local Based Similarity Takumi Kobayashi and Nobuyuki Otsu Abstract In this paper, we propose a novel method of label propagation for one-class learning. For binary (positive/negative)
More informationSpectral Bandits for Smooth Graph Functions with Applications in Recommender Systems
Spectral Bandits for Smooth Graph Functions with Applications in Recommender Systems Tomáš Kocák SequeL team INRIA Lille France Michal Valko SequeL team INRIA Lille France Rémi Munos SequeL team, INRIA
More information10-701/ Recitation : Kernels
10-701/15-781 Recitation : Kernels Manojit Nandi February 27, 2014 Outline Mathematical Theory Banach Space and Hilbert Spaces Kernels Commonly Used Kernels Kernel Theory One Weird Kernel Trick Representer
More informationSemi-supervised Learning
Semi-supervised Learning Introduction Supervised learning: x r, y r R r=1 E.g.x r : image, y r : class labels Semi-supervised learning: x r, y r r=1 R, x u R+U u=r A set of unlabeled data, usually U >>
More informationGraph Quality Judgement: A Large Margin Expedition
Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16) Graph Quality Judgement: A Large Margin Expedition Yu-Feng Li Shao-Bo Wang Zhi-Hua Zhou National Key
More informationThe crucial role of statistics in manifold learning
The crucial role of statistics in manifold learning Tyrus Berry Postdoc, Dept. of Mathematical Sciences, GMU Statistics Seminar GMU Feb., 26 Postdoctoral position supported by NSF ANALYSIS OF POINT CLOUDS
More informationKernel Methods. Foundations of Data Analysis. Torsten Möller. Möller/Mori 1
Kernel Methods Foundations of Data Analysis Torsten Möller Möller/Mori 1 Reading Chapter 6 of Pattern Recognition and Machine Learning by Bishop Chapter 12 of The Elements of Statistical Learning by Hastie,
More informationJoint distribution optimal transportation for domain adaptation
Joint distribution optimal transportation for domain adaptation Changhuang Wan Mechanical and Aerospace Engineering Department The Ohio State University March 8 th, 2018 Joint distribution optimal transportation
More informationGaussian Processes (10/16/13)
STA561: Probabilistic machine learning Gaussian Processes (10/16/13) Lecturer: Barbara Engelhardt Scribes: Changwei Hu, Di Jin, Mengdi Wang 1 Introduction In supervised learning, we observe some inputs
More informationThe Learning Problem and Regularization
9.520 Class 02 February 2011 Computational Learning Statistical Learning Theory Learning is viewed as a generalization/inference problem from usually small sets of high dimensional, noisy data. Learning
More informationSemi-Supervised Classification with Universum
Semi-Supervised Classification with Universum Dan Zhang 1, Jingdong Wang 2, Fei Wang 3, Changshui Zhang 4 1,3,4 State Key Laboratory on Intelligent Technology and Systems, Tsinghua National Laboratory
More informationBeyond Scalar Affinities for Network Analysis or Vector Diffusion Maps and the Connection Laplacian
Beyond Scalar Affinities for Network Analysis or Vector Diffusion Maps and the Connection Laplacian Amit Singer Princeton University Department of Mathematics and Program in Applied and Computational Mathematics
More informationUnsupervised Classification via Convex Absolute Value Inequalities
Unsupervised Classification via Convex Absolute Value Inequalities Olvi Mangasarian University of Wisconsin - Madison University of California - San Diego January 17, 2017 Summary Classify completely unlabeled
More informationMachine Learning Practice Page 2 of 2 10/28/13
Machine Learning 10-701 Practice Page 2 of 2 10/28/13 1. True or False Please give an explanation for your answer, this is worth 1 pt/question. (a) (2 points) No classifier can do better than a naive Bayes
More informationContribution from: Springer Verlag Berlin Heidelberg 2005 ISBN
Contribution from: Mathematical Physics Studies Vol. 7 Perspectives in Analysis Essays in Honor of Lennart Carleson s 75th Birthday Michael Benedicks, Peter W. Jones, Stanislav Smirnov (Eds.) Springer
More informationSemi-supervised Learning using Sparse Eigenfunction Bases
Semi-supervised Learning using Sparse Eigenfunction Bases Kaushik Sinha Dept. of Computer Science and Engineering Ohio State University Columbus, OH 43210 sinhak@cse.ohio-state.edu Mikhail Belkin Dept.
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 informationKernel Methods. Konstantin Tretyakov MTAT Machine Learning
Kernel Methods Konstantin Tretyakov (kt@ut.ee) MTAT.03.227 Machine Learning So far Supervised machine learning Linear models Non-linear models Unsupervised machine learning Generic scaffolding So far Supervised
More informationSpectral Techniques for Clustering
Nicola Rebagliati 1/54 Spectral Techniques for Clustering Nicola Rebagliati 29 April, 2010 Nicola Rebagliati 2/54 Thesis Outline 1 2 Data Representation for Clustering Setting Data Representation and Methods
More informationKernels for Multi task Learning
Kernels for Multi task Learning Charles A Micchelli Department of Mathematics and Statistics State University of New York, The University at Albany 1400 Washington Avenue, Albany, NY, 12222, USA Massimiliano
More informationData-dependent representations: Laplacian Eigenmaps
Data-dependent representations: Laplacian Eigenmaps November 4, 2015 Data Organization and Manifold Learning There are many techniques for Data Organization and Manifold Learning, e.g., Principal Component
More informationSemi Supervised Distance Metric Learning
Semi Supervised Distance Metric Learning wliu@ee.columbia.edu Outline Background Related Work Learning Framework Collaborative Image Retrieval Future Research Background Euclidean distance d( x, x ) =
More informationIntroduction to SVM and RVM
Introduction to SVM and RVM Machine Learning Seminar HUS HVL UIB Yushu Li, UIB Overview Support vector machine SVM First introduced by Vapnik, et al. 1992 Several literature and wide applications Relevance
More informationKernel Methods. Konstantin Tretyakov MTAT Machine Learning
Kernel Methods Konstantin Tretyakov (kt@ut.ee) MTAT.03.227 Machine Learning So far Supervised machine learning Linear models Least squares regression, SVR Fisher s discriminant, Perceptron, Logistic model,
More informationLaplacian Eigenmaps for Dimensionality Reduction and Data Representation
Introduction and Data Representation Mikhail Belkin & Partha Niyogi Department of Electrical Engieering University of Minnesota Mar 21, 2017 1/22 Outline Introduction 1 Introduction 2 3 4 Connections to
More informationResearch Statement on Statistics Jun Zhang
Research Statement on Statistics Jun Zhang (junzhang@galton.uchicago.edu) My interest on statistics generally includes machine learning and statistical genetics. My recent work focus on detection and interpretation
More informationDistance Metric Learning in Data Mining (Part II) Fei Wang and Jimeng Sun IBM TJ Watson Research Center
Distance Metric Learning in Data Mining (Part II) Fei Wang and Jimeng Sun IBM TJ Watson Research Center 1 Outline Part I - Applications Motivation and Introduction Patient similarity application Part II
More informationStatistical Translation, Heat Kernels, and Expected Distances
Statistical Translation, Heat Kernels, and Expected Distances Joshua Dillon School of Elec. and Computer Engineering jvdillon@ecn.purdue.edu Guy Lebanon Department of Statistics, and School of Elec. and
More informationAdvances in Manifold Learning Presented by: Naku Nak l Verm r a June 10, 2008
Advances in Manifold Learning Presented by: Nakul Verma June 10, 008 Outline Motivation Manifolds Manifold Learning Random projection of manifolds for dimension reduction Introduction to random projections
More informationLarge Scale Semi-supervised Linear SVMs. University of Chicago
Large Scale Semi-supervised Linear SVMs Vikas Sindhwani and Sathiya Keerthi University of Chicago SIGIR 2006 Semi-supervised Learning (SSL) Motivation Setting Categorize x-billion documents into commercial/non-commercial.
More informationSimilarity and kernels in machine learning
1/31 Similarity and kernels in machine learning Zalán Bodó Babeş Bolyai University, Cluj-Napoca/Kolozsvár Faculty of Mathematics and Computer Science MACS 2016 Eger, Hungary 2/31 Machine learning Overview
More informationGraphs in Machine Learning
Graphs in Machine Learning Michal Valko INRIA Lille - Nord Europe, France Partially based on material by: Mikhail Belkin, Jerry Zhu, Olivier Chapelle, Branislav Kveton February 10, 2015 MVA 2014/2015 Previous
More informationLearning gradients: prescriptive models
Department of Statistical Science Institute for Genome Sciences & Policy Department of Computer Science Duke University May 11, 2007 Relevant papers Learning Coordinate Covariances via Gradients. Sayan
More informationNearest Neighbor. Machine Learning CSE546 Kevin Jamieson University of Washington. October 26, Kevin Jamieson 2
Nearest Neighbor Machine Learning CSE546 Kevin Jamieson University of Washington October 26, 2017 2017 Kevin Jamieson 2 Some data, Bayes Classifier Training data: True label: +1 True label: -1 Optimal
More informationData Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings
Data Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline
More informationCMPSCI 791BB: Advanced ML: Laplacian Learning
CMPSCI 791BB: Advanced ML: Laplacian Learning Sridhar Mahadevan Outline! Spectral graph operators! Combinatorial graph Laplacian! Normalized graph Laplacian! Random walks! Machine learning on graphs! Clustering!
More informationCS798: Selected topics in Machine Learning
CS798: Selected topics in Machine Learning Support Vector Machine Jakramate Bootkrajang Department of Computer Science Chiang Mai University Jakramate Bootkrajang CS798: Selected topics in Machine Learning
More informationImproved Local Coordinate Coding using Local Tangents
Improved Local Coordinate Coding using Local Tangents Kai Yu NEC Laboratories America, 10081 N. Wolfe Road, Cupertino, CA 95129 Tong Zhang Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854
More informationAnticipating Visual Representations from Unlabeled Data. Carl Vondrick, Hamed Pirsiavash, Antonio Torralba
Anticipating Visual Representations from Unlabeled Data Carl Vondrick, Hamed Pirsiavash, Antonio Torralba Overview Problem Key Insight Methods Experiments Problem: Predict future actions and objects Image
More informationNonlinear Methods. Data often lies on or near a nonlinear low-dimensional curve aka manifold.
Nonlinear Methods Data often lies on or near a nonlinear low-dimensional curve aka manifold. 27 Laplacian Eigenmaps Linear methods Lower-dimensional linear projection that preserves distances between all
More informationLearning. Szeged, March Faculty of Mathematics and Computer Science, Babeş Bolyai University, Cluj-Napoca/Kolozsvár
Faculty of Mathematics and Computer Science, Babeş Bolyai University, Cluj-Napoca/Kolozsvár 15 10 5 Machine. Supervised and 0 5 10 15 10 5 0 5 10 15 20 25 Szeged, March 2012 1/41 2/41 Contents Machine.
More informationData Analysis and Manifold Learning Lecture 7: Spectral Clustering
Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture 7 What is spectral
More informationInstance-based Learning CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2016
Instance-based Learning CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2016 Outline Non-parametric approach Unsupervised: Non-parametric density estimation Parzen Windows Kn-Nearest
More informationRiemannian Metric Learning for Symmetric Positive Definite Matrices
CMSC 88J: Linear Subspaces and Manifolds for Computer Vision and Machine Learning Riemannian Metric Learning for Symmetric Positive Definite Matrices Raviteja Vemulapalli Guide: Professor David W. Jacobs
More informationHigher Order Learning with Graphs
Sameer Agarwal sagarwal@cs.ucsd.edu Kristin Branson kbranson@cs.ucsd.edu Serge Belongie sjb@cs.ucsd.edu Department of Computer Science and Engineering, University of California San Diego, La Jolla, CA
More informationGeometry on Probability Spaces
Geometry on Probability Spaces Steve Smale Toyota Technological Institute at Chicago 427 East 60th Street, Chicago, IL 60637, USA E-mail: smale@math.berkeley.edu Ding-Xuan Zhou Department of Mathematics,
More informationNonlinear Dimensionality Reduction
Nonlinear Dimensionality Reduction Piyush Rai CS5350/6350: Machine Learning October 25, 2011 Recap: Linear Dimensionality Reduction Linear Dimensionality Reduction: Based on a linear projection of the
More informationSeeing stars when there aren't many stars Graph-based semi-supervised learning for sentiment categorization
Seeing stars when there aren't many stars Graph-based semi-supervised learning for sentiment categorization Andrew B. Goldberg, Xiaojin Jerry Zhu Computer Sciences Department University of Wisconsin-Madison
More information