L 2,1 Norm and its Applications
|
|
- Thomas Willis
- 5 years ago
- Views:
Transcription
1 L 2, Norm and its Applications Yale Chang Introduction According to the structure of the constraints, the sparsity can be obtained from three types of regularizers for different purposes.. Flat Sparsity. This types of sparsity is often achieved by l -norm regualarizer. Optimization techniques include LARS, linear gradient search, proximal methods. 2. Structual sparsity, including group features detection, jointly vector sparsity, hierarchical group features, etc. The sparsity is often obtained by l 2 /l -norm. 3. Matrix/tensor sparsity, such as matrix/tensor completion. The typical regularizer is the trace norm which can be solved by Singular Value Decomposition thresholding. 2 Definition Given a matrix M R n m, m i is the i-th row while m j is the j-th column. Then the Frobenius norm of the matrix is defined as M F = n m m i 2 2 = Mij 2 () j= The l 2, norm of the matrix is defined as M 2, = m i 2 = m Mij 2 (2) j= l 2, norm is rotational invariant for rows: MR 2, = M 2,. The l 2 norm can be generalized to l r,p norm[]. ( M r,p = m i p r ) p m = ( M ij r ) p r j= p (3)
2 3 Applications 3. Rotational Invariant L-norm PCA for Robust Subspace Factorization[2] In R -norm, distance in spatial dimensions(attribute dimensions) are measured in L 2, while the summation over different data points uses L. Let X = {x,, x n } be n data points in d- dimensional space. In matrix form X = (x ji ), index j over spatial dimensions and index i sum over data points. R -norm is defined as Formulation of standard PCA is Formulation of R -PCA is X R = d j= J SV D = X UV 2 F = U,V x 2 ji 2 (4) x i Uv i 2 (5) J R P CA = X UV R = U,V x i Uv i (6) A common feature of previous approaches using Frobenius norm and L -norm is that they treat the two indexes i and j in the same way. However, these two indexes have different meaning: i runs through data points, while j = run through the spatial dimensions. In strict matrix format, this subtle distinction is easy to get lost. R -norm captures this subtle distinction. 3.2 Robust(L 2, ) Feature Selection[] W J(W ) = XT W Y 2, + γ W 2, (7) where X = {x,, x n } R d n, W R d c, Y R n c. The residue W T x i y i 2 is not squared and thus outliers have less importance than the squared residual W T x i y i Robust Nonnegative Matrix Factorization[3] The assumption of Gaussian noise leads to the formulation of standard NMF by imposing constraints F 0, G 0. While the assumption of the Laplacian noise leads to the formulation of L 2, NMF. The formulation of standard NMF is X F F 0,G 0 G 2 F (8) where X R d n, F R d q +, G Rq n +. The robust NMF(L 2, NMF is formulated as) X F G 2, (9) F 0,G 0 2
3 3.4 L, Feature Selection[4] W ) + λ W, (0) W R d qj(xt This formulation induce sparsity on the maximum absolute value of the element of each row; thereby, pushing all the elements of each row to zero. Sparse PCA/LDA enforce L norm regularization on W. This would set individual sparsity on the elements of W but would not necessarily achieve feature selection. For L p, type regularization, increasing p increases the sparsity sharing between the elements in each row. Therefore L 2, (p = 2) norm is more suitable for feature selection compared to L,. Moreover, p = promises full sharing of the elements. The advantage of working on the central subspace is that the data may contain dimensions irrelevant to the task, applying Φ(X T W ) avoids those noisy subspaces. The other reason is that the projection W operates on the original features X rather than on the non-linearly transformed features Φ(X), which makes it more amenable for interpreting which of the original features are important. 3.5 Joint Feature Selection and Subspace Learning[5] W R d q W 2, + µtr(w T XLX T W ) () Note the Laplacian matrix L is constructed from the original data matrix X. 3.6 Feature Selection via Joint Embedding and Sparse Regression[6] arg Tr(Y LY T ) + β( W T X Y α W 2, ) (2) W,Y Y T =I q q where Y R q n, L = (I n n S) T (I n n S) is the graph Laplacian of Local Linearity Embedding. W R d q. 3.7 Unsupervised Feature Selection Using Nonnegative Spectral Analysis[7] Tr(F T LF ) + α( X T W F 2 F + β W 2, ) (3) F T F =I q,f 0,W where F R n q, L = I n n D /2 SD /2, W R d q. Note that all the elements of F are nonnegative by definition. However, the optimal F has mixed signs without nonnegative constraints, which violates its definition. To address this problem, it is natural and reasonable to impose a nonnegative constraint into the objective function. When both nonnegative and orthogonal constraints are satisfied, there is only one element in each row of F is greater than zero and all the others are zeros. In that way, the learned F is more accurate, and more capable to provide discriative information. 3.8 Multi-Task Feature Learning[8] A T t= m L(y ti, < a t, U T x ti >) + γ A 2 2, (4) The second term combines the tasks and ensures that common features will be selected across them. 3
4 3.9 High-Order Multi-Task Feature Learning[9] B T Xt T B t Y t 2 F + α B () 2, + β( B () + B (2) ) (5) t= where X t R d n, B t R d c, Y t R n c, B () R d (c T ), B (2) R c (d T ). d is the number of features, n is the number of features, T is the number of time points. c is the number of scores, which is known as observations. In this case, L 2, norm enforce different tasks(time points) will select the same set of features. 3.0 Unsupervised Feature Selection for Linked Social Media Data[0] W s.t. Tr(W T XLX T W ) + β W 2, + αtr ( W T X(I n F F T )X T W ) (6) W T (XX T + λi)w = I c where X R m n,m is the number of features, n is the number of samples. W R m c, which assigns each data point with a pseudo-class label where c is the number of pseudo-class labels. L = D S is a laplacian matrix F = H(H T H) 2 is the weighted social dimension indicator matrix, H R K n is the social dimension indicator matrix, which can be obtained through Modularity Maximization. 3. Multi-view Clustering and Feature Learning via Structured Sparsity[] A core assumption in MKL, as well as many existing graph based multi-view learning methods, is that all features in the same data source are considered as equally important and given the same weight in data fusion, i.e., one weight is learned for one kernel matrix or graph. However, one can expect that the feature-wise importance to different learning tasks can vary significantly. 3.2 Robust Unsupervised Feature Selection[2] References X GF 2, + νtr[g T LG] + α XW G 2, + β W 2, (7) F,G,W s.t. G R n c +, GT G = I c, F R c d +, W Rd c [] F. Nie, H. Huang, X. Cai, and C. H. Ding, Efficient and robust feature selection via joint 2, -norms imization, in Advances in Neural Information Processing Systems, pp , 200. [2] C. Ding, D. Zhou, X. He, and H. Zha, R -pca: rotational invariant l -norm principal component analysis for robust subspace factorization, in Proceedings of the 23rd international conference on Machine learning, pp , ACM,
5 [3] D. Kong, C. Ding, and H. Huang, Robust nonnegative matrix factorization using l2-norm, in Proceedings of the 20th ACM international conference on Information and knowledge management, pp , ACM, 20. [4] M. Masaeli, J. G. Dy, and G. M. Fung, From transformation-based dimensionality reduction to feature selection, in Proceedings of the 27th International Conference on Machine Learning (ICML-0), pp , 200. [5] Q. Gu, Z. Li, and J. Han, Joint feature selection and subspace learning, in IJCAI Proceedings- International Joint Conference on Artificial Intelligence, vol. 22, p. 294, 20. [6] C. Hou, F. Nie, D. Yi, and Y. Wu, Feature selection via joint embedding learning and sparse regression, in IJCAI Proceedings-International Joint Conference on Artificial Intelligence, vol. 22, p. 324, 20. [7] Z. Li, Y. Yang, J. Liu, X. Zhou, and H. Lu, Unsupervised feature selection using nonnegative spectral analysis., in AAAI, 202. [8] A. Evgeniou and M. Pontil, Multi-task feature learning, Advances in neural information processing systems, vol. 9, p. 4, [9] H. Wang, F. Nie, H. Huang, J. Yan, S. Kim, S. Risacher, A. Saykin, and L. Shen, High-order multitask feature learning to identify longitudinal phenotypic markers for alzheimer s disease progression prediction, in Advances in Neural Information Processing Systems, pp , 202. [0] J. Tang and H. Liu, Unsupervised feature selection for linked social media data, in Proceedings of the 8th ACM SIGKDD international conference on Knowledge discovery and data ing, pp , ACM, 202. [] H. Wang, F. Nie, and H. Huang, Multi-view clustering and feature learning via structured sparsity, in Proceedings of the 30th International Conference on Machine Learning (ICML-3), pp , 203. [2] M. Qian and C. Zhai, Robust unsupervised feature selection, in Proceedings of the Twenty-Third international joint conference on Artificial Intelligence, pp , AAAI Press,
Coupled Dictionary Learning for Unsupervised Feature Selection
Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI-16) Coupled Dictionary Learning for Unsupervised Feature Selection Pengfei Zhu 1, Qinghua Hu 1, Changqing Zhang 1, Wangmeng
More informationNonnegative Matrix Factorization Clustering on Multiple Manifolds
Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10) Nonnegative Matrix Factorization Clustering on Multiple Manifolds Bin Shen, Luo Si Department of Computer Science,
More informationGraph-Laplacian PCA: Closed-form Solution and Robustness
2013 IEEE Conference on Computer Vision and Pattern Recognition Graph-Laplacian PCA: Closed-form Solution and Robustness Bo Jiang a, Chris Ding b,a, Bin Luo a, Jin Tang a a School of Computer Science and
More informationGraph based Subspace Segmentation. Canyi Lu National University of Singapore Nov. 21, 2013
Graph based Subspace Segmentation Canyi Lu National University of Singapore Nov. 21, 2013 Content Subspace Segmentation Problem Related Work Sparse Subspace Clustering (SSC) Low-Rank Representation (LRR)
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 informationGI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis. Massimiliano Pontil
GI07/COMPM012: Mathematical Programming and Research Methods (Part 2) 2. Least Squares and Principal Components Analysis Massimiliano Pontil 1 Today s plan SVD and principal component analysis (PCA) Connection
More informationSparse multi-kernel based multi-task learning for joint prediction of clinical scores and biomarker identification in Alzheimer s Disease
Sparse multi-kernel based multi-task learning for joint prediction of clinical scores and biomarker identification in Alzheimer s Disease Peng Cao 1, Xiaoli Liu 1, Jinzhu Yang 1, Dazhe Zhao 1, and Osmar
More informationIntroduction to Machine Learning. PCA and Spectral Clustering. Introduction to Machine Learning, Slides: Eran Halperin
1 Introduction to Machine Learning PCA and Spectral Clustering Introduction to Machine Learning, 2013-14 Slides: Eran Halperin Singular Value Decomposition (SVD) The singular value decomposition (SVD)
More informationAdaptive Affinity Matrix for Unsupervised Metric Learning
Adaptive Affinity Matrix for Unsupervised Metric Learning Yaoyi Li, Junxuan Chen, Yiru Zhao and Hongtao Lu Key Laboratory of Shanghai Education Commission for Intelligent Interaction and Cognitive Engineering,
More informationCholesky Decomposition Rectification for Non-negative Matrix Factorization
Cholesky Decomposition Rectification for Non-negative Matrix Factorization Tetsuya Yoshida Graduate School of Information Science and Technology, Hokkaido University N-14 W-9, Sapporo 060-0814, Japan yoshida@meme.hokudai.ac.jp
More informationRaRE: Social Rank Regulated Large-scale Network Embedding
RaRE: Social Rank Regulated Large-scale Network Embedding Authors: Yupeng Gu 1, Yizhou Sun 1, Yanen Li 2, Yang Yang 3 04/26/2018 The Web Conference, 2018 1 University of California, Los Angeles 2 Snapchat
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 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 informationGROUP-SPARSE SUBSPACE CLUSTERING WITH MISSING DATA
GROUP-SPARSE SUBSPACE CLUSTERING WITH MISSING DATA D Pimentel-Alarcón 1, L Balzano 2, R Marcia 3, R Nowak 1, R Willett 1 1 University of Wisconsin - Madison, 2 University of Michigan - Ann Arbor, 3 University
More informationStatistical and Computational Analysis of Locality Preserving Projection
Statistical and Computational Analysis of Locality Preserving Projection Xiaofei He xiaofei@cs.uchicago.edu Department of Computer Science, University of Chicago, 00 East 58th Street, Chicago, IL 60637
More informationMathematical Methods for Data Analysis
Mathematical Methods for Data Analysis Massimiliano Pontil Istituto Italiano di Tecnologia and Department of Computer Science University College London Massimiliano Pontil Mathematical Methods for Data
More informationA Spectral Regularization Framework for Multi-Task Structure Learning
A Spectral Regularization Framework for Multi-Task Structure Learning Massimiliano Pontil Department of Computer Science University College London (Joint work with A. Argyriou, T. Evgeniou, C.A. Micchelli,
More informationMulti-Task Co-clustering via Nonnegative Matrix Factorization
Multi-Task Co-clustering via Nonnegative Matrix Factorization Saining Xie, Hongtao Lu and Yangcheng He Shanghai Jiao Tong University xiesaining@gmail.com, lu-ht@cs.sjtu.edu.cn, yche.sjtu@gmail.com Abstract
More informationGroup Sparse Non-negative Matrix Factorization for Multi-Manifold Learning
LIU, LU, GU: GROUP SPARSE NMF FOR MULTI-MANIFOLD LEARNING 1 Group Sparse Non-negative Matrix Factorization for Multi-Manifold Learning Xiangyang Liu 1,2 liuxy@sjtu.edu.cn Hongtao Lu 1 htlu@sjtu.edu.cn
More informationA Local Non-Negative Pursuit Method for Intrinsic Manifold Structure Preservation
Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence A Local Non-Negative Pursuit Method for Intrinsic Manifold Structure Preservation Dongdong Chen and Jian Cheng Lv and Zhang Yi
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 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 informationMTTTS16 Learning from Multiple Sources
MTTTS16 Learning from Multiple Sources 5 ECTS credits Autumn 2018, University of Tampere Lecturer: Jaakko Peltonen Lecture 6: Multitask learning with kernel methods and nonparametric models On this lecture:
More informationSparse representation classification and positive L1 minimization
Sparse representation classification and positive L1 minimization Cencheng Shen Joint Work with Li Chen, Carey E. Priebe Applied Mathematics and Statistics Johns Hopkins University, August 5, 2014 Cencheng
More informationLearning to Learn and Collaborative Filtering
Appearing in NIPS 2005 workshop Inductive Transfer: Canada, December, 2005. 10 Years Later, Whistler, Learning to Learn and Collaborative Filtering Kai Yu, Volker Tresp Siemens AG, 81739 Munich, Germany
More informationLearning Bound for Parameter Transfer Learning
Learning Bound for Parameter Transfer Learning Wataru Kumagai Faculty of Engineering Kanagawa University kumagai@kanagawa-u.ac.jp Abstract We consider a transfer-learning problem by using the parameter
More informationLinear Methods for Regression. Lijun Zhang
Linear Methods for Regression Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Introduction Linear Regression Models and Least Squares Subset Selection Shrinkage Methods Methods Using Derived
More informationDimension Reduction Techniques. Presented by Jie (Jerry) Yu
Dimension Reduction Techniques Presented by Jie (Jerry) Yu Outline Problem Modeling Review of PCA and MDS Isomap Local Linear Embedding (LLE) Charting Background Advances in data collection and storage
More informationDeep Learning Basics Lecture 7: Factor Analysis. Princeton University COS 495 Instructor: Yingyu Liang
Deep Learning Basics Lecture 7: Factor Analysis Princeton University COS 495 Instructor: Yingyu Liang Supervised v.s. Unsupervised Math formulation for supervised learning Given training data x i, y i
More informationSTATISTICAL LEARNING SYSTEMS
STATISTICAL LEARNING SYSTEMS LECTURE 8: UNSUPERVISED LEARNING: FINDING STRUCTURE IN DATA Institute of Computer Science, Polish Academy of Sciences Ph. D. Program 2013/2014 Principal Component Analysis
More informationCSC 576: Variants of Sparse Learning
CSC 576: Variants of Sparse Learning Ji Liu Department of Computer Science, University of Rochester October 27, 205 Introduction Our previous note basically suggests using l norm to enforce sparsity in
More informationProtein Expression Molecular Pattern Discovery by Nonnegative Principal Component Analysis
Protein Expression Molecular Pattern Discovery by Nonnegative Principal Component Analysis Xiaoxu Han and Joseph Scazzero Department of Mathematics and Bioinformatics Program Department of Accounting and
More informationMulti-Task Clustering using Constrained Symmetric Non-Negative Matrix Factorization
Multi-Task Clustering using Constrained Symmetric Non-Negative Matrix Factorization Samir Al-Stouhi Chandan K. Reddy Abstract Researchers have attempted to improve the quality of clustering solutions through
More information2.3. Clustering or vector quantization 57
Multivariate Statistics non-negative matrix factorisation and sparse dictionary learning The PCA decomposition is by construction optimal solution to argmin A R n q,h R q p X AH 2 2 under constraint :
More informationData Mining. Dimensionality reduction. Hamid Beigy. Sharif University of Technology. Fall 1395
Data Mining Dimensionality reduction Hamid Beigy Sharif University of Technology Fall 1395 Hamid Beigy (Sharif University of Technology) Data Mining Fall 1395 1 / 42 Outline 1 Introduction 2 Feature selection
More informationDeep Learning: Approximation of Functions by Composition
Deep Learning: Approximation of Functions by Composition Zuowei Shen Department of Mathematics National University of Singapore Outline 1 A brief introduction of approximation theory 2 Deep learning: approximation
More informationA Least Squares Formulation for Canonical Correlation Analysis
A Least Squares Formulation for Canonical Correlation Analysis Liang Sun, Shuiwang Ji, and Jieping Ye Department of Computer Science and Engineering Arizona State University Motivation Canonical Correlation
More informationParaGraphE: A Library for Parallel Knowledge Graph Embedding
ParaGraphE: A Library for Parallel Knowledge Graph Embedding Xiao-Fan Niu, Wu-Jun Li National Key Laboratory for Novel Software Technology Department of Computer Science and Technology, Nanjing University,
More informationMachine Learning (BSMC-GA 4439) Wenke Liu
Machine Learning (BSMC-GA 4439) Wenke Liu 02-01-2018 Biomedical data are usually high-dimensional Number of samples (n) is relatively small whereas number of features (p) can be large Sometimes p>>n Problems
More informationMatrix Support Functional and its Applications
Matrix Support Functional and its Applications James V Burke Mathematics, University of Washington Joint work with Yuan Gao (UW) and Tim Hoheisel (McGill), CORS, Banff 2016 June 1, 2016 Connections What
More informationCS598 Machine Learning in Computational Biology (Lecture 5: Matrix - part 2) Professor Jian Peng Teaching Assistant: Rongda Zhu
CS598 Machine Learning in Computational Biology (Lecture 5: Matrix - part 2) Professor Jian Peng Teaching Assistant: Rongda Zhu Feature engineering is hard 1. Extract informative features from domain knowledge
More informationLinear Dimensionality Reduction
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Principal Component Analysis 3 Factor Analysis
More informationScalable Subspace Clustering
Scalable Subspace Clustering René Vidal Center for Imaging Science, Laboratory for Computational Sensing and Robotics, Institute for Computational Medicine, Department of Biomedical Engineering, Johns
More informationSparse Subspace Clustering
Sparse Subspace Clustering Based on Sparse Subspace Clustering: Algorithm, Theory, and Applications by Elhamifar and Vidal (2013) Alex Gutierrez CSCI 8314 March 2, 2017 Outline 1 Motivation and Background
More informationLecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26
Principal Component Analysis Brett Bernstein CDS at NYU April 25, 2017 Brett Bernstein (CDS at NYU) Lecture 13 April 25, 2017 1 / 26 Initial Question Intro Question Question Let S R n n be symmetric. 1
More informationMachine Learning. B. Unsupervised Learning B.2 Dimensionality Reduction. Lars Schmidt-Thieme, Nicolas Schilling
Machine Learning B. Unsupervised Learning B.2 Dimensionality Reduction Lars Schmidt-Thieme, Nicolas Schilling Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University
More informationMulti-Task Learning and Matrix Regularization
Multi-Task Learning and Matrix Regularization Andreas Argyriou Department of Computer Science University College London Collaborators T. Evgeniou (INSEAD) R. Hauser (University of Oxford) M. Herbster (University
More informationRegularized Estimation of High Dimensional Covariance Matrices. Peter Bickel. January, 2008
Regularized Estimation of High Dimensional Covariance Matrices Peter Bickel Cambridge January, 2008 With Thanks to E. Levina (Joint collaboration, slides) I. M. Johnstone (Slides) Choongsoon Bae (Slides)
More informationTighter Low-rank Approximation via Sampling the Leveraged Element
Tighter Low-rank Approximation via Sampling the Leveraged Element Srinadh Bhojanapalli The University of Texas at Austin bsrinadh@utexas.edu Prateek Jain Microsoft Research, India prajain@microsoft.com
More informationDISCUSSION OF INFLUENTIAL FEATURE PCA FOR HIGH DIMENSIONAL CLUSTERING. By T. Tony Cai and Linjun Zhang University of Pennsylvania
Submitted to the Annals of Statistics DISCUSSION OF INFLUENTIAL FEATURE PCA FOR HIGH DIMENSIONAL CLUSTERING By T. Tony Cai and Linjun Zhang University of Pennsylvania We would like to congratulate the
More informationOn the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering
On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering Chris Ding Xiaofeng He Horst D. Simon Abstract Current nonnegative matrix factorization (NMF) deals with X = FG T type. We
More informationMulti-Task Learning and Algorithmic Stability
Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence Multi-Task Learning and Algorithmic Stability Yu Zhang Department of Computer Science, Hong Kong Baptist University The Institute
More informationOrthogonal Nonnegative Matrix Factorization: Multiplicative Updates on Stiefel Manifolds
Orthogonal Nonnegative Matrix Factorization: Multiplicative Updates on Stiefel Manifolds Jiho Yoo and Seungjin Choi Department of Computer Science Pohang University of Science and Technology San 31 Hyoja-dong,
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Matrix Notation Mark Schmidt University of British Columbia Winter 2017 Admin Auditting/registration forms: Submit them at end of class, pick them up end of next class. I need
More informationMULTIPLICATIVE ALGORITHM FOR CORRENTROPY-BASED NONNEGATIVE MATRIX FACTORIZATION
MULTIPLICATIVE ALGORITHM FOR CORRENTROPY-BASED NONNEGATIVE MATRIX FACTORIZATION Ehsan Hosseini Asl 1, Jacek M. Zurada 1,2 1 Department of Electrical and Computer Engineering University of Louisville, Louisville,
More informationLinear Models for Regression. Sargur Srihari
Linear Models for Regression Sargur srihari@cedar.buffalo.edu 1 Topics in Linear Regression What is regression? Polynomial Curve Fitting with Scalar input Linear Basis Function Models Maximum Likelihood
More informationRobustness of Principal Components
PCA for Clustering An objective of principal components analysis is to identify linear combinations of the original variables that are useful in accounting for the variation in those original variables.
More informationRobust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition
Robust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition Gongguo Tang and Arye Nehorai Department of Electrical and Systems Engineering Washington University in St Louis
More informationClassification. The goal: map from input X to a label Y. Y has a discrete set of possible values. We focused on binary Y (values 0 or 1).
Regression and PCA Classification The goal: map from input X to a label Y. Y has a discrete set of possible values We focused on binary Y (values 0 or 1). But we also discussed larger number of classes
More informationMachine Learning. Dimensionality reduction. Hamid Beigy. Sharif University of Technology. Fall 1395
Machine Learning Dimensionality reduction Hamid Beigy Sharif University of Technology Fall 1395 Hamid Beigy (Sharif University of Technology) Machine Learning Fall 1395 1 / 47 Table of contents 1 Introduction
More informationParcimonie en apprentissage statistique
Parcimonie en apprentissage statistique Guillaume Obozinski Ecole des Ponts - ParisTech Journée Parcimonie Fédération Charles Hermite, 23 Juin 2014 Parcimonie en apprentissage 1/44 Classical supervised
More informationOutline. Motivation. Mapping the input space to the feature space Calculating the dot product in the feature space
to The The A s s in to Fabio A. González Ph.D. Depto. de Ing. de Sistemas e Industrial Universidad Nacional de Colombia, Bogotá April 2, 2009 to The The A s s in 1 Motivation Outline 2 The Mapping the
More informationAn Unsupervised Feature Selection Framework for Social Media Data
IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING APRIL 2012 1 An Unsupervised Feature Selection Framework for Social Media Data Jiliang Tang, Huan Liu, Fellow, IEEE Abstract The explosive usage of social
More informationInverse Power Method for Non-linear Eigenproblems
Inverse Power Method for Non-linear Eigenproblems Matthias Hein and Thomas Bühler Anubhav Dwivedi Department of Aerospace Engineering & Mechanics 7th March, 2017 1 / 30 OUTLINE Motivation Non-Linear Eigenproblems
More informationHigh Dimensional Covariance and Precision Matrix Estimation
High Dimensional Covariance and Precision Matrix Estimation Wei Wang Washington University in St. Louis Thursday 23 rd February, 2017 Wei Wang (Washington University in St. Louis) High Dimensional Covariance
More informationStreaming multiscale anomaly detection
Streaming multiscale anomaly detection DATA-ENS Paris and ThalesAlenia Space B Ravi Kiran, Université Lille 3, CRISTaL Joint work with Mathieu Andreux beedotkiran@gmail.com June 20, 2017 (CRISTaL) Streaming
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 informationDiscriminant Uncorrelated Neighborhood Preserving Projections
Journal of Information & Computational Science 8: 14 (2011) 3019 3026 Available at http://www.joics.com Discriminant Uncorrelated Neighborhood Preserving Projections Guoqiang WANG a,, Weijuan ZHANG a,
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 informationData Mining and Matrices
Data Mining and Matrices 05 Semi-Discrete Decomposition Rainer Gemulla, Pauli Miettinen May 16, 2013 Outline 1 Hunting the Bump 2 Semi-Discrete Decomposition 3 The Algorithm 4 Applications SDD alone SVD
More informationLearning Multiple Tasks with a Sparse Matrix-Normal Penalty
Learning Multiple Tasks with a Sparse Matrix-Normal Penalty Yi Zhang and Jeff Schneider NIPS 2010 Presented by Esther Salazar Duke University March 25, 2011 E. Salazar (Reading group) March 25, 2011 1
More informationJure Leskovec Joint work with Jaewon Yang, Julian McAuley
Jure Leskovec (@jure) Joint work with Jaewon Yang, Julian McAuley Given a network, find communities! Sets of nodes with common function, role or property 2 3 Q: How and why do communities form? A: Strength
More informationDiscriminative Direction for Kernel Classifiers
Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering
More informationRobust Laplacian Eigenmaps Using Global Information
Manifold Learning and its Applications: Papers from the AAAI Fall Symposium (FS-9-) Robust Laplacian Eigenmaps Using Global Information Shounak Roychowdhury ECE University of Texas at Austin, Austin, TX
More informationLearning Task Grouping and Overlap in Multi-Task Learning
Learning Task Grouping and Overlap in Multi-Task Learning Abhishek Kumar Hal Daumé III Department of Computer Science University of Mayland, College Park 20 May 2013 Proceedings of the 29 th International
More informationSTATS 306B: Unsupervised Learning Spring Lecture 13 May 12
STATS 306B: Unsupervised Learning Spring 2014 Lecture 13 May 12 Lecturer: Lester Mackey Scribe: Jessy Hwang, Minzhe Wang 13.1 Canonical correlation analysis 13.1.1 Recap CCA is a linear dimensionality
More informationDimension Reduction Using Nonnegative Matrix Tri-Factorization in Multi-label Classification
250 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'15 Dimension Reduction Using Nonnegative Matrix Tri-Factorization in Multi-label Classification Keigo Kimura, Mineichi Kudo and Lu Sun Graduate
More informationOn Spectral Basis Selection for Single Channel Polyphonic Music Separation
On Spectral Basis Selection for Single Channel Polyphonic Music Separation Minje Kim and Seungjin Choi Department of Computer Science Pohang University of Science and Technology San 31 Hyoja-dong, Nam-gu
More informationLecture 24: Principal Component Analysis. Aykut Erdem May 2016 Hacettepe University
Lecture 4: Principal Component Analysis Aykut Erdem May 016 Hacettepe University This week Motivation PCA algorithms Applications PCA shortcomings Autoencoders Kernel PCA PCA Applications Data Visualization
More information6. Regularized linear regression
Foundations of Machine Learning École Centrale Paris Fall 2015 6. Regularized linear regression Chloé-Agathe Azencot Centre for Computational Biology, Mines ParisTech chloe agathe.azencott@mines paristech.fr
More informationLecture Notes 10: Matrix Factorization
Optimization-based data analysis Fall 207 Lecture Notes 0: Matrix Factorization Low-rank models. Rank- model Consider the problem of modeling a quantity y[i, j] that depends on two indices i and j. To
More informationAutomatic Rank Determination in Projective Nonnegative Matrix Factorization
Automatic Rank Determination in Projective Nonnegative Matrix Factorization Zhirong Yang, Zhanxing Zhu, and Erkki Oja Department of Information and Computer Science Aalto University School of Science and
More informationA Randomized Approach for Crowdsourcing in the Presence of Multiple Views
A Randomized Approach for Crowdsourcing in the Presence of Multiple Views Presenter: Yao Zhou joint work with: Jingrui He - 1 - Roadmap Motivation Proposed framework: M2VW Experimental results Conclusion
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 information25 : Graphical induced structured input/output models
10-708: Probabilistic Graphical Models 10-708, Spring 2016 25 : Graphical induced structured input/output models Lecturer: Eric P. Xing Scribes: Raied Aljadaany, Shi Zong, Chenchen Zhu Disclaimer: A large
More informationLarge-scale Collaborative Prediction Using a Nonparametric Random Effects Model
Large-scale Collaborative Prediction Using a Nonparametric Random Effects Model Kai Yu Joint work with John Lafferty and Shenghuo Zhu NEC Laboratories America, Carnegie Mellon University First Prev Page
More informationEmpirical Discriminative Tensor Analysis for Crime Forecasting
Empirical Discriminative Tensor Analysis for Crime Forecasting Yang Mu 1, Wei Ding 1, Melissa Morabito 2, Dacheng Tao 3, 1 Department of Computer Science, University of Massachusetts Boston,100 Morrissey
More informationMultiple Similarities Based Kernel Subspace Learning for Image Classification
Multiple Similarities Based Kernel Subspace Learning for Image Classification Wang Yan, Qingshan Liu, Hanqing Lu, and Songde Ma National Laboratory of Pattern Recognition, Institute of Automation, Chinese
More informationBig Data Analytics: Optimization and Randomization
Big Data Analytics: Optimization and Randomization Tianbao Yang Tutorial@ACML 2015 Hong Kong Department of Computer Science, The University of Iowa, IA, USA Nov. 20, 2015 Yang Tutorial for ACML 15 Nov.
More informationAn Algorithm for Transfer Learning in a Heterogeneous Environment
An Algorithm for Transfer Learning in a Heterogeneous Environment Andreas Argyriou 1, Andreas Maurer 2, and Massimiliano Pontil 1 1 Department of Computer Science University College London Malet Place,
More informationStructure in Data. A major objective in data analysis is to identify interesting features or structure in the data.
Structure in Data A major objective in data analysis is to identify interesting features or structure in the data. The graphical methods are very useful in discovering structure. There are basically two
More informationNote on Algorithm Differences Between Nonnegative Matrix Factorization And Probabilistic Latent Semantic Indexing
Note on Algorithm Differences Between Nonnegative Matrix Factorization And Probabilistic Latent Semantic Indexing 1 Zhong-Yuan Zhang, 2 Chris Ding, 3 Jie Tang *1, Corresponding Author School of Statistics,
More informationRandom Subspace NMF for Unsupervised Transfer Learning
Random Subspace NMF for Unsupervised Transfer Learning Ievgen Redko & Younès Bennani Université Paris 13 - Institut Galilée - Sorbonne Paris Cité Laboratoire d'informatique de Paris-Nord - CNRS (UMR 7030)
More informationAdvanced Introduction to Machine Learning CMU-10715
Advanced Introduction to Machine Learning CMU-10715 Principal Component Analysis Barnabás Póczos Contents Motivation PCA algorithms Applications Some of these slides are taken from Karl Booksh Research
More informationSupport Vector Machine (SVM) and Kernel Methods
Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2014 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More informationCOMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017
COMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University PRINCIPAL COMPONENT ANALYSIS DIMENSIONALITY
More informationPrincipal Component Analysis
B: Chapter 1 HTF: Chapter 1.5 Principal Component Analysis Barnabás Póczos University of Alberta Nov, 009 Contents Motivation PCA algorithms Applications Face recognition Facial expression recognition
More informationLecture 18: Multiclass Support Vector Machines
Fall, 2017 Outlines Overview of Multiclass Learning Traditional Methods for Multiclass Problems One-vs-rest approaches Pairwise approaches Recent development for Multiclass Problems Simultaneous Classification
More informationMemory Efficient Kernel Approximation
Si Si Department of Computer Science University of Texas at Austin ICML Beijing, China June 23, 2014 Joint work with Cho-Jui Hsieh and Inderjit S. Dhillon Outline Background Motivation Low-Rank vs. Block
More information1 Matrix notation and preliminaries from spectral graph theory
Graph clustering (or community detection or graph partitioning) is one of the most studied problems in network analysis. One reason for this is that there are a variety of ways to define a cluster or community.
More information