Approximate SDP solvers, Matrix Factorizations, the Netflix Prize, and PageRank. Mittagseminar Martin Jaggi, Oct

Size: px
Start display at page:

Download "Approximate SDP solvers, Matrix Factorizations, the Netflix Prize, and PageRank. Mittagseminar Martin Jaggi, Oct"

Transcription

1 Approximate SDP solvers, Matrix Factorizations, the Netflix Prize, and PageRank Mittagseminar Martin Jaggi, Oct 6 009

2 Sparse Approximation The Problem f( ) convex min f(x) x R n x 0 T x = min f(x) X S n n X 0 Tr(X) = vectors living in the simplex symetric matrices living in the spectahedron /0

3 The Problem min f(x) x R n x 0 T x = min f(x) X S n n X 0 Tr(X) = The Algorithm x (k+) := ( λ) +λ X (k+) := ( λ) X (k) +λvv T x (k) e i i := arg max f( x (k) ) i i Coordinate Descent v := arg max v T ( f( X (k) ))v v =,largest Eigenvector x (k) λ =/k X (k) = v () v (k) k Sparsity = k Rank = k v () v (k) = UU T /0

4 The Algorithm x (k+) := ( λ) +λ e i X (k+) := ( λ) X (k) +λ vv T i x (k) := arg max f( x (k) ) i i v := arg max v T ( f( X (k) ))v v = The Convergence O After steps the primal-dual error is. O After steps the primal-dual error is. [ Clarkson SODA '08 ] [ Hazan LATIN '08 ] Approximate Eigenvector computation Instead of v := arg max v T Mv v = M := f( X (k) ) it is enough to work with : v T Mv λ max v v = O v 3/0 Such a can be found by doing Lanzcos steps. Alternative: Power method

5 a side note How to solve general Semidefinite Programs? min f(x) X S n n X 0 Tr(X) = Optimization Version: Feasibility Version: min Tr(CX) Find X s.t. Tr(A i X) b i X S n n X 0 i [m] Tr(A i X) b i i [m + ] X S n n X 0 Tr(X) = f(x) := m+ M log Soft Max i= e M(Tr(A ix) b i ) By this trick, Hazan s algorithm is able to satisfy all constraints up to an error. /0

6 Matrix Factorizations and machine learning Y = Customer Movie UV T = The Netflix Prize: Movies Customers Ratings (Observed Entries %) = George Clooney plays in movie j v () v (k) k u () u (k) = Customer i is female 5/0

7 Matrix Factorizations for recommender systems Y u () u (k) v () v (k) [Short IEEE article, k Wikipedia: Netflix Prize] Factor vector Freddy Got Fingered Half Baked Julien Donkey-Boy Kill Bill: Vol. Freddy vs. Jason Natural Born Killers Road Trip I Heart Huckabees Scarface Punch-Drunk Love The Royal Tenenbaums The Longest Yard Being John Malkovich The Fast and the Furious Lost in Translation Belle de Jour Armageddon Catwoman The Wizard of Oz Citizen Kane Coyote Ugly Maid in Manhattan Runaway Bride Stepmom Sister Act Annie Hall Sophie s Choice Moonstruck The Way We Were The Sound of Music The Waltons: Season Factor vector 6/0 Figure 3. The first two vectors from a matrix decomposition of the Netflix Prize data. Selected movies are placed at the appropriate spot based on their factor vectors in two dimensions. The plot reveals distinct genres, including clusters of movies with strong female leads, fraternity humor, and quirky independent films.

8 Matrix Factorizations and machine learning Applications: Customer i Product j (Amazon, Netflix, Migros Cumulus etc...) Customer i Customer j (Symmetry?, k=?) i j Word i Document j (Search engines, Latent Semantic Analysis) many other applications (e.g. dimensionality reduction, clustering) 7/0

9 m Accuracy vs Model complexity n 3 Y UV T =: X 3 Error Model complexity min U,V f(uv T ) s.t. rank(uv T ) = k Low Rank = rank(x) s.t. U Fro + V Fro = t Low Norm = X [ Srebro NIPS '05 ] f(x) := ij S (X Y ) ij 8/0

10 3 3 Low Norm Matrix Factorization f(x) := ij S (X Y ) ij min U,V s.t. is equivalent to f(uv T ) U Fro + V Fro = Tr(UU T )+Tr(VV T ) = Tr(Z) = t n UU T VU T = m UV T 3 =: Z 3 VV T U (U T V T ) V min Z f(z) Z S (n+m) (n+m) Z 0 Tr(Z) =t Perfectly fits for Hazan s Algorithm. 9/0

11 Low Norm Matrix Factorization Have to be careful, principal EV is not always the largest EV! Add a constant to the diagonal in that case. We need the largest Eigenvector of M := f( ) Z (k) f(z) := (Z Y ) ij ij S 3 =: Z 3 Largest Eigenvector of the bipartite weighted graph with adjacency matrix M. Customers n m Movies Hazan s new approximate SDP solver applies to Low Norm n Matrix mfactorization Easy to parallelize (Power method) 0 M = 3 3 Algorithm maintains sparsity structure of the given matrix, needs no additional memory 0 Speed is comparable to existing methods, and much better than generic SDP solvers 0/0

12 Thanks

Matrix Factorization and Collaborative Filtering

Matrix Factorization and Collaborative Filtering 10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Matrix Factorization and Collaborative Filtering MF Readings: (Koren et al., 2009)

More information

A Simple Algorithm for Nuclear Norm Regularized Problems

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:

More information

Data Mining and Matrices

Data Mining and Matrices Data Mining and Matrices 04 Matrix Completion Rainer Gemulla, Pauli Miettinen May 02, 2013 Recommender systems Problem Set of users Set of items (movies, books, jokes, products, stories,...) Feedback (ratings,

More information

Collaborative Filtering

Collaborative Filtering Case Study 4: Collaborative Filtering Collaborative Filtering Matrix Completion Alternating Least Squares Machine Learning/Statistics for Big Data CSE599C1/STAT592, University of Washington Carlos Guestrin

More information

III. Applications in convex optimization

III. Applications in convex optimization III. Applications in convex optimization nonsymmetric interior-point methods partial separability and decomposition partial separability first order methods interior-point methods Conic linear optimization

More information

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:

More information

Recommendation Systems

Recommendation Systems Recommendation Systems Popularity Recommendation Systems Predicting user responses to options Offering news articles based on users interests Offering suggestions on what the user might like to buy/consume

More information

Collaborative Filtering Matrix Completion Alternating Least Squares

Collaborative Filtering Matrix Completion Alternating Least Squares Case Study 4: Collaborative Filtering Collaborative Filtering Matrix Completion Alternating Least Squares Machine Learning for Big Data CSE547/STAT548, University of Washington Sham Kakade May 19, 2016

More information

A Greedy Framework for First-Order Optimization

A Greedy Framework for First-Order Optimization A Greedy Framework for First-Order Optimization Jacob Steinhardt Department of Computer Science Stanford University Stanford, CA 94305 jsteinhardt@cs.stanford.edu Jonathan Huggins Department of EECS Massachusetts

More information

Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs

Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca & Eilyan Bitar School of Electrical and Computer Engineering American Control Conference (ACC) Chicago,

More information

Using SVD to Recommend Movies

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

More information

A direct formulation for sparse PCA using semidefinite programming

A direct formulation for sparse PCA using semidefinite programming A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley A. d Aspremont, INFORMS, Denver,

More information

Maximum Margin Matrix Factorization

Maximum Margin Matrix Factorization Maximum Margin Matrix Factorization Nati Srebro Toyota Technological Institute Chicago Joint work with Noga Alon Tel-Aviv Yonatan Amit Hebrew U Alex d Aspremont Princeton Michael Fink Hebrew U Tommi Jaakkola

More information

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD DATA MINING LECTURE 8 Dimensionality Reduction PCA -- SVD The curse of dimensionality Real data usually have thousands, or millions of dimensions E.g., web documents, where the dimensionality is the vocabulary

More information

Low Rank Matrix Completion Formulation and Algorithm

Low Rank Matrix Completion Formulation and Algorithm 1 2 Low Rank Matrix Completion and Algorithm Jian Zhang Department of Computer Science, ETH Zurich zhangjianthu@gmail.com March 25, 2014 Movie Rating 1 2 Critic A 5 5 Critic B 6 5 Jian 9 8 Kind Guy B 9

More information

Tractable Upper Bounds on the Restricted Isometry Constant

Tractable Upper Bounds on the Restricted Isometry Constant Tractable Upper Bounds on the Restricted Isometry Constant Alex d Aspremont, Francis Bach, Laurent El Ghaoui Princeton University, École Normale Supérieure, U.C. Berkeley. Support from NSF, DHS and Google.

More information

ECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference

ECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference ECE 18-898G: Special Topics in Signal Processing: Sparsity, Structure, and Inference Low-rank matrix recovery via convex relaxations Yuejie Chi Department of Electrical and Computer Engineering Spring

More information

Clustering. Clustering in R d DSE 210

Clustering. Clustering in R d DSE 210 Clustering DSE 20 Clustering in R d Two common uses of clustering: Vector quantization Find a finite set of representatives that provides good coverage of a complex, possibly infinite, high-dimensional

More information

Matrix completion: Fundamental limits and efficient algorithms. Sewoong Oh Stanford University

Matrix completion: Fundamental limits and efficient algorithms. Sewoong Oh Stanford University Matrix completion: Fundamental limits and efficient algorithms Sewoong Oh Stanford University 1 / 35 Low-rank matrix completion Low-rank Data Matrix Sparse Sampled Matrix Complete the matrix from small

More information

The maximal stable set problem : Copositive programming and Semidefinite Relaxations

The maximal stable set problem : Copositive programming and Semidefinite Relaxations The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu

More information

Collaborative Filtering. Radek Pelánek

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

More information

Linear dimensionality reduction for data analysis

Linear dimensionality reduction for data analysis Linear dimensionality reduction for data analysis Nicolas Gillis Joint work with Robert Luce, François Glineur, Stephen Vavasis, Robert Plemmons, Gabriella Casalino The setup Dimensionality reduction for

More information

Lecture 9: September 28

Lecture 9: September 28 0-725/36-725: Convex Optimization Fall 206 Lecturer: Ryan Tibshirani Lecture 9: September 28 Scribes: Yiming Wu, Ye Yuan, Zhihao Li Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These

More information

Introduction to Machine Learning. PCA and Spectral Clustering. Introduction to Machine Learning, Slides: Eran Halperin

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)

More information

Tutorial: PART 2. Online Convex Optimization, A Game- Theoretic Approach to Learning

Tutorial: PART 2. Online Convex Optimization, A Game- Theoretic Approach to Learning Tutorial: PART 2 Online Convex Optimization, A Game- Theoretic Approach to Learning Elad Hazan Princeton University Satyen Kale Yahoo Research Exploiting curvature: logarithmic regret Logarithmic regret

More information

Deep 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 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 information

Recommender Systems. Dipanjan Das Language Technologies Institute Carnegie Mellon University. 20 November, 2007

Recommender Systems. Dipanjan Das Language Technologies Institute Carnegie Mellon University. 20 November, 2007 Recommender Systems Dipanjan Das Language Technologies Institute Carnegie Mellon University 20 November, 2007 Today s Outline What are Recommender Systems? Two approaches Content Based Methods Collaborative

More information

Tutorial: PART 2. Optimization for Machine Learning. Elad Hazan Princeton University. + help from Sanjeev Arora & Yoram Singer

Tutorial: PART 2. Optimization for Machine Learning. Elad Hazan Princeton University. + help from Sanjeev Arora & Yoram Singer Tutorial: PART 2 Optimization for Machine Learning Elad Hazan Princeton University + help from Sanjeev Arora & Yoram Singer Agenda 1. Learning as mathematical optimization Stochastic optimization, ERM,

More information

ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications

ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications Professor M. Chiang Electrical Engineering Department, Princeton University March

More information

Sparse PCA with applications in finance

Sparse PCA with applications in finance Sparse PCA with applications in finance A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon 1 Introduction

More information

Lecture Note 5: Semidefinite Programming for Stability Analysis

Lecture Note 5: Semidefinite Programming for Stability Analysis ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State

More information

Symmetric Factorization for Nonconvex Optimization

Symmetric Factorization for Nonconvex Optimization Symmetric Factorization for Nonconvex Optimization Qinqing Zheng February 24, 2017 1 Overview A growing body of recent research is shedding new light on the role of nonconvex optimization for tackling

More information

CS598 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 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 information

Distance Geometry-Matrix Completion

Distance Geometry-Matrix Completion Christos Konaxis Algs in Struct BioInfo 2010 Outline Outline Tertiary structure Incomplete data Tertiary structure Measure diffrence of matched sets Def. Root Mean Square Deviation RMSD = 1 n x i y i 2,

More information

We describe the generalization of Hazan s algorithm for symmetric programming

We describe the generalization of Hazan s algorithm for symmetric programming ON HAZAN S ALGORITHM FOR SYMMETRIC PROGRAMMING PROBLEMS L. FAYBUSOVICH Abstract. problems We describe the generalization of Hazan s algorithm for symmetric programming Key words. Symmetric programming,

More information

A direct formulation for sparse PCA using semidefinite programming

A direct formulation for sparse PCA using semidefinite programming A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon

More information

Lecture 5 : Projections

Lecture 5 : Projections Lecture 5 : Projections EE227C. Lecturer: Professor Martin Wainwright. Scribe: Alvin Wan Up until now, we have seen convergence rates of unconstrained gradient descent. Now, we consider a constrained minimization

More information

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Kalev Kask

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Kalev Kask Machine Learning and Data Mining Dimensionality Reduction; PCA & SVD Kalev Kask Motivation High-dimensional data Images of faces Text from articles All S&P 500 stocks Can we describe them in a simpler

More information

Algorithms for Collaborative Filtering

Algorithms for Collaborative Filtering Algorithms for Collaborative Filtering or How to Get Half Way to Winning $1million from Netflix Todd Lipcon Advisor: Prof. Philip Klein The Real-World Problem E-commerce sites would like to make personalized

More information

SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices)

SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Chapter 14 SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Today we continue the topic of low-dimensional approximation to datasets and matrices. Last time we saw the singular

More information

Rank minimization via the γ 2 norm

Rank minimization via the γ 2 norm Rank minimization via the γ 2 norm Troy Lee Columbia University Adi Shraibman Weizmann Institute Rank Minimization Problem Consider the following problem min X rank(x) A i, X b i for i = 1,..., k Arises

More information

Recommender systems, matrix factorization, variable selection and social graph data

Recommender systems, matrix factorization, variable selection and social graph data Recommender systems, matrix factorization, variable selection and social graph data Julien Delporte & Stéphane Canu stephane.canu@litislab.eu StatLearn, april 205, Grenoble Road map Model selection for

More information

EE 381V: Large Scale Learning Spring Lecture 16 March 7

EE 381V: Large Scale Learning Spring Lecture 16 March 7 EE 381V: Large Scale Learning Spring 2013 Lecture 16 March 7 Lecturer: Caramanis & Sanghavi Scribe: Tianyang Bai 16.1 Topics Covered In this lecture, we introduced one method of matrix completion via SVD-based

More information

Jeffrey D. Ullman Stanford University

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

More information

A solution approach for linear optimization with completely positive matrices

A solution approach for linear optimization with completely positive matrices A solution approach for linear optimization with completely positive matrices Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with M. Bomze (Wien) and F.

More information

Matrix Factorizations: A Tale of Two Norms

Matrix Factorizations: A Tale of Two Norms Matrix Factorizations: A Tale of Two Norms Nati Srebro Toyota Technological Institute Chicago Maximum Margin Matrix Factorization S, Jason Rennie, Tommi Jaakkola (MIT), NIPS 2004 Rank, Trace-Norm and Max-Norm

More information

CSE 494/598 Lecture-4: Correlation Analysis. **Content adapted from last year s slides

CSE 494/598 Lecture-4: Correlation Analysis. **Content adapted from last year s slides CSE 494/598 Lecture-4: Correlation Analysis LYDIA MANIKONDA HT TP://WWW.PUBLIC.ASU.EDU/~LMANIKON / **Content adapted from last year s slides Announcements Project-1 Due: February 12 th 2016 Analysis report:

More information

Matrix Completion: Fundamental Limits and Efficient Algorithms

Matrix Completion: Fundamental Limits and Efficient Algorithms Matrix Completion: Fundamental Limits and Efficient Algorithms Sewoong Oh PhD Defense Stanford University July 23, 2010 1 / 33 Matrix completion Find the missing entries in a huge data matrix 2 / 33 Example

More information

Collaborative Filtering: A Machine Learning Perspective

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

More information

On the interior of the simplex, we have the Hessian of d(x), Hd(x) is diagonal with ith. µd(w) + w T c. minimize. subject to w T 1 = 1,

On the interior of the simplex, we have the Hessian of d(x), Hd(x) is diagonal with ith. µd(w) + w T c. minimize. subject to w T 1 = 1, Math 30 Winter 05 Solution to Homework 3. Recognizing the convexity of g(x) := x log x, from Jensen s inequality we get d(x) n x + + x n n log x + + x n n where the equality is attained only at x = (/n,...,

More information

1-Bit Matrix Completion

1-Bit Matrix Completion 1-Bit Matrix Completion Mark A. Davenport School of Electrical and Computer Engineering Georgia Institute of Technology Yaniv Plan Mary Wootters Ewout van den Berg Matrix Completion d When is it possible

More information

Graph Partitioning Using Random Walks

Graph Partitioning Using Random Walks Graph Partitioning Using Random Walks A Convex Optimization Perspective Lorenzo Orecchia Computer Science Why Spectral Algorithms for Graph Problems in practice? Simple to implement Can exploit very efficient

More information

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

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

More information

Lecture 20: November 1st

Lecture 20: November 1st 10-725: Optimization Fall 2012 Lecture 20: November 1st Lecturer: Geoff Gordon Scribes: Xiaolong Shen, Alex Beutel Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not

More information

Lecture Notes 10: Matrix Factorization

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

More information

Spectral k-support Norm Regularization

Spectral k-support Norm Regularization Spectral k-support Norm Regularization Andrew McDonald Department of Computer Science, UCL (Joint work with Massimiliano Pontil and Dimitris Stamos) 25 March, 2015 1 / 19 Problem: Matrix Completion Goal:

More information

Bruce Hendrickson Discrete Algorithms & Math Dept. Sandia National Labs Albuquerque, New Mexico Also, CS Department, UNM

Bruce Hendrickson Discrete Algorithms & Math Dept. Sandia National Labs Albuquerque, New Mexico Also, CS Department, UNM Latent Semantic Analysis and Fiedler Retrieval Bruce Hendrickson Discrete Algorithms & Math Dept. Sandia National Labs Albuquerque, New Mexico Also, CS Department, UNM Informatics & Linear Algebra Eigenvectors

More information

Principal Component Analysis (PCA) for Sparse High-Dimensional Data

Principal Component Analysis (PCA) for Sparse High-Dimensional Data AB Principal Component Analysis (PCA) for Sparse High-Dimensional Data Tapani Raiko, Alexander Ilin, and Juha Karhunen Helsinki University of Technology, Finland Adaptive Informatics Research Center Principal

More information

CSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming

CSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming

More information

Lecture: Matrix Completion

Lecture: Matrix Completion 1/56 Lecture: Matrix Completion http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html Acknowledgement: this slides is based on Prof. Jure Leskovec and Prof. Emmanuel Candes s lecture notes Recommendation systems

More information

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Prof. Alexander Ihler

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Prof. Alexander Ihler + Machine Learning and Data Mining Dimensionality Reduction; PCA & SVD Prof. Alexander Ihler Mo#va#on High-dimensional data Images of faces Text from articles All S&P 00 stocks Can we describe them in

More information

Short Course Robust Optimization and Machine Learning. Lecture 4: Optimization in Unsupervised Learning

Short Course Robust Optimization and Machine Learning. Lecture 4: Optimization in Unsupervised Learning Short Course Robust Optimization and Machine Machine Lecture 4: Optimization in Unsupervised Laurent El Ghaoui EECS and IEOR Departments UC Berkeley Spring seminar TRANSP-OR, Zinal, Jan. 16-19, 2012 s

More information

EE613 Machine Learning for Engineers. Kernel methods Support Vector Machines. jean-marc odobez 2015

EE613 Machine Learning for Engineers. Kernel methods Support Vector Machines. jean-marc odobez 2015 EE613 Machine Learning for Engineers Kernel methods Support Vector Machines jean-marc odobez 2015 overview Kernel methods introductions and main elements defining kernels Kernelization of k-nn, K-Means,

More information

ORF 363/COS 323 Final Exam, Fall 2017

ORF 363/COS 323 Final Exam, Fall 2017 Name: Princeton University Instructor: A.A. Ahmadi ORF 363/COS 323 Final Exam, Fall 2017 January 17, 2018 AIs: B. El Khadir, C. Dibek, G. Hall, J. Zhang, J. Ye, S. Uysal 1. Please write out and sign the

More information

Lecture 18. Ramanujan Graphs continued

Lecture 18. Ramanujan Graphs continued Stanford University Winter 218 Math 233A: Non-constructive methods in combinatorics Instructor: Jan Vondrák Lecture date: March 8, 218 Original scribe: László Miklós Lovász Lecture 18 Ramanujan Graphs

More information

Non-negative Matrix Factorization: Algorithms, Extensions and Applications

Non-negative Matrix Factorization: Algorithms, Extensions and Applications Non-negative Matrix Factorization: Algorithms, Extensions and Applications Emmanouil Benetos www.soi.city.ac.uk/ sbbj660/ March 2013 Emmanouil Benetos Non-negative Matrix Factorization March 2013 1 / 25

More information

Big Data Analytics: Optimization and Randomization

Big 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 information

Learning with Matrix Factorizations

Learning with Matrix Factorizations Learning with Matrix Factorizations Nati Srebro Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Adviser: Tommi Jaakkola (MIT) Committee: Alan Willsky (MIT),

More information

Semidefinite Programming

Semidefinite Programming Semidefinite Programming Notes by Bernd Sturmfels for the lecture on June 26, 208, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra The transition from linear algebra to nonlinear algebra has

More information

Using R for Iterative and Incremental Processing

Using R for Iterative and Incremental Processing Using R for Iterative and Incremental Processing Shivaram Venkataraman, Indrajit Roy, Alvin AuYoung, Robert Schreiber UC Berkeley and HP Labs UC BERKELEY Big Data, Complex Algorithms PageRank (Dominant

More information

a Short Introduction

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

More information

Adaptive one-bit matrix completion

Adaptive one-bit matrix completion Adaptive one-bit matrix completion Joseph Salmon Télécom Paristech, Institut Mines-Télécom Joint work with Jean Lafond (Télécom Paristech) Olga Klopp (Crest / MODAL X, Université Paris Ouest) Éric Moulines

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2013 PROBLEM SET 2

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2013 PROBLEM SET 2 STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2013 PROBLEM SET 2 1. You are not allowed to use the svd for this problem, i.e. no arguments should depend on the svd of A or A. Let W be a subspace of C n. The

More information

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Prof. Alexander Ihler

Machine Learning and Data Mining. Dimensionality Reduction; PCA & SVD. Prof. Alexander Ihler + Machine Learning and Data Mining Dimensionality Reduction; PCA & SVD Prof. Alexander Ihler Mo#va#on High-dimensional data Images of faces Text from articles All S&P 500 stocks Can we describe them in

More information

A direct formulation for sparse PCA using semidefinite programming

A direct formulation for sparse PCA using semidefinite programming A direct formulation for sparse PCA using semidefinite programming Alexandre d Aspremont alexandre.daspremont@m4x.org Department of Electrical Engineering and Computer Science Laurent El Ghaoui elghaoui@eecs.berkeley.edu

More information

Preliminaries Overview OPF and Extensions. Convex Optimization. Lecture 8 - Applications in Smart Grids. Instructor: Yuanzhang Xiao

Preliminaries Overview OPF and Extensions. Convex Optimization. Lecture 8 - Applications in Smart Grids. Instructor: Yuanzhang Xiao Convex Optimization Lecture 8 - Applications in Smart Grids Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 32 Today s Lecture 1 Generalized Inequalities and Semidefinite Programming

More information

Problem structure in semidefinite programs arising in control and signal processing

Problem structure in semidefinite programs arising in control and signal processing Problem structure in semidefinite programs arising in control and signal processing Lieven Vandenberghe Electrical Engineering Department, UCLA Joint work with: Mehrdad Nouralishahi, Tae Roh Semidefinite

More information

Distributed Box-Constrained Quadratic Optimization for Dual Linear SVM

Distributed Box-Constrained Quadratic Optimization for Dual Linear SVM Distributed Box-Constrained Quadratic Optimization for Dual Linear SVM Lee, Ching-pei University of Illinois at Urbana-Champaign Joint work with Dan Roth ICML 2015 Outline Introduction Algorithm Experiments

More information

Mini-Batch Primal and Dual Methods for SVMs

Mini-Batch Primal and Dual Methods for SVMs Mini-Batch Primal and Dual Methods for SVMs Peter Richtárik School of Mathematics The University of Edinburgh Coauthors: M. Takáč (Edinburgh), A. Bijral and N. Srebro (both TTI at Chicago) arxiv:1303.2314

More information

10725/36725 Optimization Homework 2 Solutions

10725/36725 Optimization Homework 2 Solutions 10725/36725 Optimization Homework 2 Solutions 1 Convexity (Kevin) 1.1 Sets Let A R n be a closed set with non-empty interior that has a supporting hyperplane at every point on its boundary. (a) Show that

More information

CS 175: Project in Artificial Intelligence. Slides 4: Collaborative Filtering

CS 175: Project in Artificial Intelligence. Slides 4: Collaborative Filtering CS 175: Project in Artificial Intelligence Slides 4: Collaborative Filtering 1 Topic 6: Collaborative Filtering Some slides taken from Prof. Smyth (with slight modifications) 2 Outline General aspects

More information

Lecture notes for quantum semidefinite programming (SDP) solvers

Lecture notes for quantum semidefinite programming (SDP) solvers CMSC 657, Intro to Quantum Information Processing Lecture on November 15 and 0, 018 Fall 018, University of Maryland Prepared by Tongyang Li, Xiaodi Wu Lecture notes for quantum semidefinite programming

More information

Collaborative Filtering

Collaborative Filtering Collaborative Filtering Nicholas Ruozzi University of Texas at Dallas based on the slides of Alex Smola & Narges Razavian Collaborative Filtering Combining information among collaborating entities to make

More information

Robust Principal Component Analysis

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

More information

L26: Advanced dimensionality reduction

L26: Advanced dimensionality reduction L26: Advanced dimensionality reduction The snapshot CA approach Oriented rincipal Components Analysis Non-linear dimensionality reduction (manifold learning) ISOMA Locally Linear Embedding CSCE 666 attern

More information

Rank, Trace-Norm & Max-Norm

Rank, Trace-Norm & Max-Norm Rank, Trace-Norm & Max-Norm as measures of matrix complexity Nati Srebro University of Toronto Adi Shraibman Hebrew University Matrix Learning users movies 2 1 4 5 5 4? 1 3 3 5 2 4? 5 3? 4 1 3 5 2 1? 4

More information

Stochastic Gradient Descent with Only One Projection

Stochastic Gradient Descent with Only One Projection Stochastic Gradient Descent with Only One Projection Mehrdad Mahdavi, ianbao Yang, Rong Jin, Shenghuo Zhu, and Jinfeng Yi Dept. of Computer Science and Engineering, Michigan State University, MI, USA Machine

More information

Homework 4. Convex Optimization /36-725

Homework 4. Convex Optimization /36-725 Homework 4 Convex Optimization 10-725/36-725 Due Friday November 4 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)

More information

Data Mining Lecture 4: Covariance, EVD, PCA & SVD

Data Mining Lecture 4: Covariance, EVD, PCA & SVD Data Mining Lecture 4: Covariance, EVD, PCA & SVD Jo Houghton ECS Southampton February 25, 2019 1 / 28 Variance and Covariance - Expectation A random variable takes on different values due to chance The

More information

Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora

Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 14: SVD, Power method, and Planted Graph problems (+ eigenvalues of random matrices) Lecturer: Sanjeev Arora Scribe: Today we continue the

More information

Nonlinear Dimensionality Reduction

Nonlinear 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 information

Generalized Conditional Gradient and Its Applications

Generalized Conditional Gradient and Its Applications Generalized Conditional Gradient and Its Applications Yaoliang Yu University of Alberta UBC Kelowna, 04/18/13 Y-L. Yu (UofA) GCG and Its Apps. UBC Kelowna, 04/18/13 1 / 25 1 Introduction 2 Generalized

More information

Andriy Mnih and Ruslan Salakhutdinov

Andriy Mnih and Ruslan Salakhutdinov MATRIX FACTORIZATION METHODS FOR COLLABORATIVE FILTERING Andriy Mnih and Ruslan Salakhutdinov University of Toronto, Machine Learning Group 1 What is collaborative filtering? The goal of collaborative

More information

Sparse Matrix Theory and Semidefinite Optimization

Sparse Matrix Theory and Semidefinite Optimization Sparse Matrix Theory and Semidefinite Optimization Lieven Vandenberghe Department of Electrical Engineering University of California, Los Angeles Joint work with Martin S. Andersen and Yifan Sun Third

More information

Maximum Margin Matrix Factorization for Collaborative Ranking

Maximum Margin Matrix Factorization for Collaborative Ranking Maximum Margin Matrix Factorization for Collaborative Ranking Joint work with Quoc Le, Alexandros Karatzoglou and Markus Weimer Alexander J. Smola sml.nicta.com.au Statistical Machine Learning Program

More information

Machine Learning for Signal Processing Sparse and Overcomplete Representations

Machine Learning for Signal Processing Sparse and Overcomplete Representations Machine Learning for Signal Processing Sparse and Overcomplete Representations Abelino Jimenez (slides from Bhiksha Raj and Sourish Chaudhuri) Oct 1, 217 1 So far Weights Data Basis Data Independent ICA

More information

Statistical Learning & Applications. f w (x) =< f w, K x > H = w T x. α i α j < x i x T i, x j x T j. = < α i x i x T i, α j x j x T j > F

Statistical Learning & Applications. f w (x) =< f w, K x > H = w T x. α i α j < x i x T i, x j x T j. = < α i x i x T i, α j x j x T j > F CR2: Statistical Learning & Applications Examples of Kernels and Unsupervised Learning Lecturer: Julien Mairal Scribes: Rémi De Joannis de Verclos & Karthik Srikanta Kernel Inventory Linear Kernel The

More information

Simple sparse matrices we have seen so far include diagonal matrices and tridiagonal matrices, but these are not the only ones.

Simple sparse matrices we have seen so far include diagonal matrices and tridiagonal matrices, but these are not the only ones. A matrix is sparse if most of its entries are zero. Simple sparse matrices we have seen so far include diagonal matrices and tridiagonal matrices, but these are not the only ones. In fact sparse matrices

More information

Uses of duality. Geoff Gordon & Ryan Tibshirani Optimization /

Uses of duality. Geoff Gordon & Ryan Tibshirani Optimization / Uses of duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 1 Remember conjugate functions Given f : R n R, the function is called its conjugate f (y) = max x R n yt x f(x) Conjugates appear

More information