Tensor data analysis Part 2

Tensors and graphical models

Tensors and graphical models Mariya Ishteva with Haesun Park, Le Song Dept. ELEC, VUB Georgia Tech, USA INMA Seminar, May 7, 2013, LLN Outline Tensors Random variables and graphical models Tractable representations

Unfolding Latent Tree Structures using 4th Order Tensors

Mariya Ishteva mariya.ishteva@vub.ac.be ELEC, Vrije Universiteit Brussel, 10 Brussels, Belgium Haesun Park, Le Song {hpark,lsong}@cc.gatech.edu College of Computing, Georgia Institute of Technology, Atlanta,

arxiv: v1 [cs.lg] 3 Oct 2012

Unfolding Latent Tree Structures using 4th Order Tensors Mariya Ishteva, Haesun Park, Le Song College of Computing, Georgia Institute of Technology {mishteva,hpark,lsong}@cc.gatech.edu arxiv:1210.1258v1

Matrix-Product-States/ Tensor-Trains

/ Tensor-Trains November 22, 2016 / Tensor-Trains 1 Matrices What Can We Do With Matrices? Tensors What Can We Do With Tensors? Diagrammatic Notation 2 Singular-Value-Decomposition 3 Curse of Dimensionality

Math 671: Tensor Train decomposition methods

Math 671: Eduardo Corona 1 1 University of Michigan at Ann Arbor December 8, 2016 Table of Contents 1 Preliminaries and goal 2 Unfolding matrices for tensorized arrays The Tensor Train decomposition 3

Graphical Models for Collaborative Filtering

Graphical Models for Collaborative Filtering Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Sequence modeling HMM, Kalman Filter, etc.: Similarity: the same graphical model topology,

https://goo.gl/kfxweg KYOTO UNIVERSITY Statistical Machine Learning Theory Sparsity Hisashi Kashima kashima@i.kyoto-u.ac.jp DEPARTMENT OF INTELLIGENCE SCIENCE AND TECHNOLOGY 1 KYOTO UNIVERSITY Topics:

a Short Introduction

Collaborative Filtering in Recommender Systems: a Short Introduction Norm Matloff Dept. of Computer Science University of California, Davis matloff@cs.ucdavis.edu December 3, 2016 Abstract There is a strong

Structured tensor missing-trace interpolation in the Hierarchical Tucker format Curt Da Silva and Felix J. Herrmann Sept. 26, 2013

Structured tensor missing-trace interpolation in the Hierarchical Tucker format Curt Da Silva and Felix J. Herrmann Sept. 6, 13 SLIM University of British Columbia Motivation 3D seismic experiments - 5D

Learning Decision Trees

Learning Decision Trees Machine Learning Spring 2018 1 This lecture: Learning Decision Trees 1. Representation: What are decision trees? 2. Algorithm: Learning decision trees The ID3 algorithm: A greedy

Lecture 21: Spectral Learning for Graphical Models

10-708: Probabilistic Graphical Models 10-708, Spring 2016 Lecture 21: Spectral Learning for Graphical Models Lecturer: Eric P. Xing Scribes: Maruan Al-Shedivat, Wei-Cheng Chang, Frederick Liu 1 Motivation

26 : Spectral GMs. Lecturer: Eric P. Xing Scribes: Guillermo A Cidre, Abelino Jimenez G.

10-708: Probabilistic Graphical Models, Spring 2015 26 : Spectral GMs Lecturer: Eric P. Xing Scribes: Guillermo A Cidre, Abelino Jimenez G. 1 Introduction A common task in machine learning is to work with

Introduction to the Tensor Train Decomposition and Its Applications in Machine Learning

Introduction to the Tensor Train Decomposition and Its Applications in Machine Learning Anton Rodomanov Higher School of Economics, Russia Bayesian methods research group (http://bayesgroup.ru) 14 March

A Simple Algorithm for Nuclear Norm Regularized Problems

A Simple Algorithm for Nuclear Norm Regularized Problems ICML 00 Martin Jaggi, Marek Sulovský ETH Zurich Matrix Factorizations for recommender systems Y = Customer Movie UV T = u () The Netflix challenge:

Hierarchical Tensor Decomposition of Latent Tree Graphical Models

Le Song, Haesun Park {lsong,hpark}@cc.gatech.edu College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA Mariya Ishteva mariya.ishteva@vub.ac.be ELEC, Vrije Universiteit Brussel,

Low-rank Promoting Transformations and Tensor Interpolation - Applications to Seismic Data Denoising

Low-rank Promoting Transformations and Tensor Interpolation - Applications to Seismic Data Denoising Curt Da Silva and Felix J. Herrmann 2 Dept. of Mathematics 2 Dept. of Earth and Ocean Sciences, University

EECS 275 Matrix Computation

EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 22 1 / 21 Overview

An Introduction to Hierachical (H ) Rank and TT Rank of Tensors with Examples

An Introduction to Hierachical (H ) Rank and TT Rank of Tensors with Examples Lars Grasedyck and Wolfgang Hackbusch Bericht Nr. 329 August 2011 Key words: MSC: hierarchical Tucker tensor rank tensor approximation

Probabilistic Graphical Models

Probabilistic Graphical Models Brown University CSCI 295-P, Spring 213 Prof. Erik Sudderth Lecture 11: Inference & Learning Overview, Gaussian Graphical Models Some figures courtesy Michael Jordan s draft

Unfolding Latent Tree Structures using 4th Order Tensors

Mariya Ishteva mariya.ishteva@vub.ac.be ELEC, Vrije Universiteit Brussel, 1050 Brussels, Belgium Haesun Park, Le Song {hpark,lsong}@cc.gatech.edu College of Computing, Georgia Institute of Technology,

SPARSE signal representations have gained popularity in recent

6958 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 10, OCTOBER 2011 Blind Compressed Sensing Sivan Gleichman and Yonina C. Eldar, Senior Member, IEEE Abstract The fundamental principle underlying

Robust Principal Component Analysis

ELE 538B: Mathematics of High-Dimensional Data Robust Principal Component Analysis Yuxin Chen Princeton University, Fall 2018 Disentangling sparse and low-rank matrices Suppose we are given a matrix M

Using SVD to Recommend Movies

Michael Percy University of California, Santa Cruz Last update: December 12, 2009 Last update: December 12, 2009 1 / Outline 1 Introduction 2 Singular Value Decomposition 3 Experiments 4 Conclusion Last

Tensor networks, TT (Matrix Product States) and Hierarchical Tucker decomposition

Tensor networks, TT (Matrix Product States) and Hierarchical Tucker decomposition R. Schneider (TUB Matheon) John von Neumann Lecture TU Munich, 2012 Setting - Tensors V ν := R n, H d = H := d ν=1 V ν

A 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

Collaborative Filtering: A Machine Learning Perspective

Collaborative Filtering: A Machine Learning Perspective Chapter 6: Dimensionality Reduction Benjamin Marlin Presenter: Chaitanya Desai Collaborative Filtering: A Machine Learning Perspective p.1/18 Topics

TENSOR APPROXIMATION TOOLS FREE OF THE CURSE OF DIMENSIONALITY

TENSOR APPROXIMATION TOOLS FREE OF THE CURSE OF DIMENSIONALITY Eugene Tyrtyshnikov Institute of Numerical Mathematics Russian Academy of Sciences (joint work with Ivan Oseledets) WHAT ARE TENSORS? Tensors

On the convergence of higher-order orthogonality iteration and its extension

On the convergence of higher-order orthogonality iteration and its extension Yangyang Xu IMA, University of Minnesota SIAM Conference LA15, Atlanta October 27, 2015 Best low-multilinear-rank approximation

Dealing with curse and blessing of dimensionality through tensor decompositions

Dealing with curse and blessing of dimensionality through tensor decompositions Lieven De Lathauwer Joint work with Nico Vervliet, Martijn Boussé and Otto Debals June 26, 2017 2 Overview Curse of dimensionality

Lecture 1: Introduction to low-rank tensor representation/approximation. Center for Uncertainty Quantification. Alexander Litvinenko

tifica Lecture 1: Introduction to low-rank tensor representation/approximation Alexander Litvinenko http://sri-uq.kaust.edu.sa/ KAUST Figure : KAUST campus, 5 years old, approx. 7000 people (include 1400

Introduction 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)

A new truncation strategy for the higher-order singular value decomposition

A new truncation strategy for the higher-order singular value decomposition Nick Vannieuwenhoven K.U.Leuven, Belgium Workshop on Matrix Equations and Tensor Techniques RWTH Aachen, Germany November 21,

Overview. Optimization-Based Data Analysis. Carlos Fernandez-Granda

Overview Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda 1/25/2016 Sparsity Denoising Regression Inverse problems Low-rank models Matrix completion

Learning Decision Trees

Learning Decision Trees Machine Learning Fall 2018 Some slides from Tom Mitchell, Dan Roth and others 1 Key issues in machine learning Modeling How to formulate your problem as a machine learning problem?

Structured matrix factorizations. Example: Eigenfaces

Structured matrix factorizations Example: Eigenfaces An extremely large variety of interesting and important problems in machine learning can be formulated as: Given a matrix, find a matrix and a matrix

Tensor networks and deep learning

Tensor networks and deep learning I. Oseledets, A. Cichocki Skoltech, Moscow 26 July 2017 What is a tensor Tensor is d-dimensional array: A(i 1,, i d ) Why tensors Many objects in machine learning can

Robust Low Rank Kernel Embeddings of Multivariate Distributions

Robust Low Rank Kernel Embeddings of Multivariate Distributions Le Song, Bo Dai College of Computing, Georgia Institute of Technology lsong@cc.gatech.edu, bodai@gatech.edu Abstract Kernel embedding of

A randomized block sampling approach to the canonical polyadic decomposition of large-scale tensors

A randomized block sampling approach to the canonical polyadic decomposition of large-scale tensors Nico Vervliet Joint work with Lieven De Lathauwer SIAM AN17, July 13, 2017 2 Classification of hazardous

Rank Selection in Low-rank Matrix Approximations: A Study of Cross-Validation for NMFs

Rank Selection in Low-rank Matrix Approximations: A Study of Cross-Validation for NMFs Bhargav Kanagal Department of Computer Science University of Maryland College Park, MD 277 bhargav@cs.umd.edu Vikas

Orthogonal tensor decomposition

Orthogonal tensor decomposition Daniel Hsu Columbia University Largely based on 2012 arxiv report Tensor decompositions for learning latent variable models, with Anandkumar, Ge, Kakade, and Telgarsky.

Nonlinear Optimization for Optimal Control

Nonlinear Optimization for Optimal Control Pieter Abbeel UC Berkeley EECS Many slides and figures adapted from Stephen Boyd [optional] Boyd and Vandenberghe, Convex Optimization, Chapters 9 11 [optional]

Lecture 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

Final Exam, Machine Learning, Spring 2009

Name: Andrew ID: Final Exam, 10701 Machine Learning, Spring 2009 - The exam is open-book, open-notes, no electronics other than calculators. - The maximum possible score on this exam is 100. You have 3

A Quick Tour of Linear Algebra and Optimization for Machine Learning

A Quick Tour of Linear Algebra and Optimization for Machine Learning Masoud Farivar January 8, 2015 1 / 28 Outline of Part I: Review of Basic Linear Algebra Matrices and Vectors Matrix Multiplication Operators

Probabilistic Low-Rank Matrix Completion with Adaptive Spectral Regularization Algorithms

Probabilistic Low-Rank Matrix Completion with Adaptive Spectral Regularization Algorithms François Caron Department of Statistics, Oxford STATLEARN 2014, Paris April 7, 2014 Joint work with Adrien Todeschini,

PCA and admixture models

PCA and admixture models CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar, Alkes Price PCA and admixture models 1 / 57 Announcements HW1

GI07/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

Online Dictionary Learning with Group Structure Inducing Norms

Online Dictionary Learning with Group Structure Inducing Norms Zoltán Szabó 1, Barnabás Póczos 2, András Lőrincz 1 1 Eötvös Loránd University, Budapest, Hungary 2 Carnegie Mellon University, Pittsburgh,

Rank Determination for Low-Rank Data Completion

Journal of Machine Learning Research 18 017) 1-9 Submitted 7/17; Revised 8/17; Published 9/17 Rank Determination for Low-Rank Data Completion Morteza Ashraphijuo Columbia University New York, NY 1007,

Gaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012

Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature

Support Vector Machines. Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar

Data Mining Support Vector Machines Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar 02/03/2018 Introduction to Data Mining 1 Support Vector Machines Find a linear hyperplane

Machine Learning - MT & 14. PCA and MDS

Machine Learning - MT 2016 13 & 14. PCA and MDS Varun Kanade University of Oxford November 21 & 23, 2016 Announcements Sheet 4 due this Friday by noon Practical 3 this week (continue next week if necessary)

Statistical Performance of Convex Tensor Decomposition

Slides available: h-p://www.ibis.t.u tokyo.ac.jp/ryotat/tensor12kyoto.pdf Statistical Performance of Convex Tensor Decomposition Ryota Tomioka 2012/01/26 @ Kyoto University Perspectives in Informatics

Introduction to Machine Learning

10-701 Introduction to Machine Learning PCA Slides based on 18-661 Fall 2018 PCA Raw data can be Complex, High-dimensional To understand a phenomenon we measure various related quantities If we knew what

CS295: Convex Optimization. Xiaohui Xie Department of Computer Science University of California, Irvine

CS295: Convex Optimization Xiaohui Xie Department of Computer Science University of California, Irvine Course information Prerequisites: multivariate calculus and linear algebra Textbook: Convex Optimization

Matrix Decomposition in Privacy-Preserving Data Mining JUN ZHANG DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF KENTUCKY

Matrix Decomposition in Privacy-Preserving Data Mining JUN ZHANG DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF KENTUCKY OUTLINE Why We Need Matrix Decomposition SVD (Singular Value Decomposition) NMF (Nonnegative

Robustness 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.

Available Ph.D position in Big data processing using sparse tensor representations

Available Ph.D position in Big data processing using sparse tensor representations Research area: advanced mathematical methods applied to big data processing. Keywords: compressed sensing, tensor models,

Truncation Strategy of Tensor Compressive Sensing for Noisy Video Sequences

Journal of Information Hiding and Multimedia Signal Processing c 2016 ISSN 207-4212 Ubiquitous International Volume 7, Number 5, September 2016 Truncation Strategy of Tensor Compressive Sensing for Noisy

Seman&cs with Dense Vectors. Dorota Glowacka

Semancs with Dense Vectors Dorota Glowacka dorota.glowacka@ed.ac.uk Previous lectures: - how to represent a word as a sparse vector with dimensions corresponding to the words in the vocabulary - the values

Probabilistic Latent Semantic Analysis

Probabilistic Latent Semantic Analysis Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr

Regression. Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning)

Linear Regression Regression Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning) Example: Height, Gender, Weight Shoe Size Audio features

Data 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

Lecture: Face Recognition and Feature Reduction

Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 11-1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed

Regression. Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning)

Linear Regression Regression Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning) Example: Height, Gender, Weight Shoe Size Audio features

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St

Structured Lower Rank Approximation by Moody T. Chu (NCSU) joint with Robert E. Funderlic (NCSU) and Robert J. Plemmons (Wake Forest) March 5, 1998 Outline Introduction: Problem Description Diculties Algebraic

Collaborative Filtering. Radek Pelánek

Collaborative Filtering Radek Pelánek 2017 Notes on Lecture the most technical lecture of the course includes some scary looking math, but typically with intuitive interpretation use of standard machine

CS47300: Web Information Search and Management

CS47300: Web Information Search and Management Prof. Chris Clifton 6 September 2017 Material adapted from course created by Dr. Luo Si, now leading Alibaba research group 1 Vector Space Model Disadvantages:

Parallel Numerical Algorithms

Parallel Numerical Algorithms Chapter 6 Matrix Models Section 6.2 Low Rank Approximation Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign CS 554 / CSE 512 Edgar

Reservoir Computing and Echo State Networks

An Introduction to: Reservoir Computing and Echo State Networks Claudio Gallicchio gallicch@di.unipi.it Outline Focus: Supervised learning in domain of sequences Recurrent Neural networks for supervised

Notes on Latent Semantic Analysis

Notes on Latent Semantic Analysis Costas Boulis 1 Introduction One of the most fundamental problems of information retrieval (IR) is to find all documents (and nothing but those) that are semantically

Principal components analysis COMS 4771

Principal components analysis COMS 4771 1. Representation learning Useful representations of data Representation learning: Given: raw feature vectors x 1, x 2,..., x n R d. Goal: learn a useful feature

Machine Learning. Principal Components Analysis. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012

Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Principal Components Analysis Le Song Lecture 22, Nov 13, 2012 Based on slides from Eric Xing, CMU Reading: Chap 12.1, CB book 1 2 Factor or Component

From Matrix to Tensor. Charles F. Van Loan

From Matrix to Tensor Charles F. Van Loan Department of Computer Science January 28, 2016 From Matrix to Tensor From Tensor To Matrix 1 / 68 What is a Tensor? Instead of just A(i, j) it s A(i, j, k) or

Computational Linear Algebra

Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 6: Some Other Stuff PD Dr.

Some tensor decomposition methods for machine learning

Some tensor decomposition methods for machine learning Massimiliano Pontil Istituto Italiano di Tecnologia and University College London 16 August 2016 1 / 36 Outline Problem and motivation Tucker decomposition

Dictionary Learning Using Tensor Methods

Dictionary Learning Using Tensor Methods Anima Anandkumar U.C. Irvine Joint work with Rong Ge, Majid Janzamin and Furong Huang. Feature learning as cornerstone of ML ML Practice Feature learning as cornerstone

Clustering K-means. Machine Learning CSE546. Sham Kakade University of Washington. November 15, Review: PCA Start: unsupervised learning

Clustering K-means Machine Learning CSE546 Sham Kakade University of Washington November 15, 2016 1 Announcements: Project Milestones due date passed. HW3 due on Monday It ll be collaborative HW2 grades

Lecture: Face Recognition and Feature Reduction

Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab 1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed in the

Latent Semantic Analysis. Hongning Wang

Latent Semantic Analysis Hongning Wang CS@UVa VS model in practice Document and query are represented by term vectors Terms are not necessarily orthogonal to each other Synonymy: car v.s. automobile Polysemy:

TENSOR LAYERS FOR COMPRESSION OF DEEP LEARNING NETWORKS. Cris Cecka Senior Research Scientist, NVIDIA GTC 2018

TENSOR LAYERS FOR COMPRESSION OF DEEP LEARNING NETWORKS Cris Cecka Senior Research Scientist, NVIDIA GTC 2018 Tensors Computations and the GPU AGENDA Tensor Networks and Decompositions Tensor Layers in

High-dimensional Statistics

High-dimensional Statistics Pradeep Ravikumar UT Austin Outline 1. High Dimensional Data : Large p, small n 2. Sparsity 3. Group Sparsity 4. Low Rank 1 Curse of Dimensionality Statistical Learning: Given

Kronecker Product Approximation with Multiple Factor Matrices via the Tensor Product Algorithm

Kronecker Product Approximation with Multiple actor Matrices via the Tensor Product Algorithm King Keung Wu, Yeung Yam, Helen Meng and Mehran Mesbahi Department of Mechanical and Automation Engineering,

Bellman s Curse of Dimensionality

Bellman s Curse of Dimensionality n- dimensional state space Number of states grows exponen

Jeffrey D. Ullman Stanford University

Jeffrey D. Ullman Stanford University 2 Often, our data can be represented by an m-by-n matrix. And this matrix can be closely approximated by the product of two matrices that share a small common dimension

CPSC 340: Machine Learning and Data Mining. More PCA Fall 2017

CPSC 340: Machine Learning and Data Mining More PCA Fall 2017 Admin Assignment 4: Due Friday of next week. No class Monday due to holiday. There will be tutorials next week on MAP/PCA (except Monday).

ECE 8201: Low-dimensional Signal Models for High-dimensional Data Analysis

ECE 8201: Low-dimensional Signal Models for High-dimensional Data Analysis Lecture 7: Matrix completion Yuejie Chi The Ohio State University Page 1 Reference Guaranteed Minimum-Rank Solutions of Linear

sublinear time low-rank approximation of positive semidefinite matrices Cameron Musco (MIT) and David P. Woodru (CMU)

sublinear time low-rank approximation of positive semidefinite matrices Cameron Musco (MIT) and David P. Woodru (CMU) 0 overview Our Contributions: 1 overview Our Contributions: A near optimal low-rank

Math 671: Tensor Train decomposition methods II

Math 671: Tensor Train decomposition methods II Eduardo Corona 1 1 University of Michigan at Ann Arbor December 13, 2016 Table of Contents 1 What we ve talked about so far: 2 The Tensor Train decomposition

Clustering. 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

Latent Semantic Analysis. Hongning Wang

Latent Semantic Analysis Hongning Wang CS@UVa Recap: vector space model Represent both doc and query by concept vectors Each concept defines one dimension K concepts define a high-dimensional space Element

CSE 494/598 Lecture-6: Latent Semantic Indexing. **Content adapted from last year s slides

CSE 494/598 Lecture-6: Latent Semantic Indexing LYDIA MANIKONDA HT TP://WWW.PUBLIC.ASU.EDU/~LMANIKON / **Content adapted from last year s slides Announcements Homework-1 and Quiz-1 Project part-2 released

Probabilistic Low-Rank Matrix Completion with Adaptive Spectral Regularization Algorithms

Probabilistic Low-Rank Matrix Completion with Adaptive Spectral Regularization Algorithms Adrien Todeschini Inria Bordeaux JdS 2014, Rennes Aug. 2014 Joint work with François Caron (Univ. Oxford), Marie

Randomized Algorithms

Randomized Algorithms Saniv Kumar, Google Research, NY EECS-6898, Columbia University - Fall, 010 Saniv Kumar 9/13/010 EECS6898 Large Scale Machine Learning 1 Curse of Dimensionality Gaussian Mixture Models

Linear 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

NEW TENSOR DECOMPOSITIONS IN NUMERICAL ANALYSIS AND DATA PROCESSING

NEW TENSOR DECOMPOSITIONS IN NUMERICAL ANALYSIS AND DATA PROCESSING Institute of Numerical Mathematics of Russian Academy of Sciences eugene.tyrtyshnikov@gmail.com 11 October 2012 COLLABORATION MOSCOW:

Singular Value Decomposition

Chapter 5 Singular Value Decomposition We now reach an important Chapter in this course concerned with the Singular Value Decomposition of a matrix A. SVD, as it is commonly referred to, is one of the

NUMERICAL METHODS WITH TENSOR REPRESENTATIONS OF DATA

NUMERICAL METHODS WITH TENSOR REPRESENTATIONS OF DATA Institute of Numerical Mathematics of Russian Academy of Sciences eugene.tyrtyshnikov@gmail.com 2 June 2012 COLLABORATION MOSCOW: I.Oseledets, D.Savostyanov

Sparse & Redundant Signal Representation, and its Role in Image Processing

Sparse & Redundant Signal Representation, and its Role in Michael Elad The CS Department The Technion Israel Institute of technology Haifa 3000, Israel Wave 006 Wavelet and Applications Ecole Polytechnique