An Introduction to Spectral Learning
|
|
- Kory Summers
- 5 years ago
- Views:
Transcription
1 An Introduction to Spectral Learning Hanxiao Liu November 8, 2013
2 Outline 1 Method of Moments 2 Learning topic models using spectral properties 3 Anchor words
3 Preliminaries X 1,, X n p (x; θ), θ = (θ 1, θ m ) ˆθ = ˆθ n = w (X 1,, X n ) Maximum Likelihood Estimator (MLE) Bayes Estimator (BE) ˆθ = argmax log L (θ) θ ˆθ = E (θ X) = θp (x θ) π (θ) dθ p (x θ) π (θ) dθ
4 Preliminaries Question What makes a good estimator? MLE is consistent Both the MLE and BE have asymptotic normality ( ) ( ) 1 n ˆθ n θ N 0, I (θ) under mild (regularity) conditions Can be computationally expensive
5 Preliminaries Example (Gamma distribution) p (x i ; α, θ) = 1 Γ (α) θ α xα 1 i ( exp x i θ ) ( ) ( 1 n n L (α, θ) = Γ (α) θ α i=1 ) α 1 ( x i exp ni=1 ) x i MLE is hard to compute due to the existence of Γ (α) θ
6 Method of Moments j-th theoretical moment, j [k] j-th sample moment, j [k] µ j (θ) := E θ ( X j) M j := 1 n X j i n i=1 Plug-in and solve the multivariate polynomial equations M j = µ j (θ) j [k] sometimes can be recast as spectral decomposition
7 Method of Moments Example (Gamma distribution) p (x i ; α, θ) = 1 Γ (α) θ α xα 1 i ( exp x i θ ) 1 n ˆθ = 1 nx n i=1 n i=1 X = E (X i ) = αθ ( ) 2 X i X = Var (Xi ) = αθ 2 ( ) 2 X X i X, ˆα = ˆθ = nx 2 ( ni=1 X i X ) 2
8 Method of Moments lack guarantee about the solution high-order sample moments are hard to estimate To reach a specified accuracy, the required sample size and computational cost is exponential in k (or n)! Question Could we recover the true θ from only low-order moments? Question Could we lower the sample requirement and computational complexity based on some (hopefully mild) assumptions?
9 Learning the Topic Models Papadimitriou et al. (2000) Non-overlapping separation condition (strong) Anandkumar et al. (2012), MoM+SD Full rank assumption (weak) Multinomial Mixture, LDA Arora et al. (2012), MoM+NMF+LP Anchor words (mild) LDA, Correlated Topic Model A more practical algorithm proposed in 2013
10 Learning the Topic Models Suppose there are n documents, k hidden topics, d features M = [µ 1 µ 2... µ k ] R d k, µ j d 1 j [k] w = (w 1,..., w k ), w k 1 P (h = j) = w j j [k] For the v-th word in a document, x v {e 1,... e d } P (x v = e i h = j) = µ i j, j [k], i [d] Goal: Recover the M using low-order moments
11 Learning the Topic Models Construct moment statistics Pairs ij := P (x 1 = e i, x 2 = e j ) Triples ij := P (x 1 = e i, x 2 = e j, x 3 = e t ) Pair = E[x 1 x 2 ] R d d Triples = E[x 1 x 2 x 3 ] R d d d Empirical plug-ins i.e. Pairs ˆ and Triples ˆ could be obtained from data through a straightforward manner We want to establish some equivalence between the empirical moments and parameters of interest
12 Learning the Topic Models Lemma Triples (η) := E[x 1 x 2 x 3, η ] R d d Triples (η) : R d R d d Pairs = M diag (w) M ( ( ) ) Triples (η) = M diag M η diag (w) M The unknown M and w are twisted.
13 Learning the Topic Models Assumption ( Non-degeneracy ) M has full column rank k 1 Find U, V R d k s.t. 2 η R d, define B (η) R k k ( ) 1 ( ) 1 U M and V M exist. B (η) := ( ) ( ) 1 U Triples (η) V U PairsV Lemma (Observable Operator) B (η) = ( ) ( ) ( ) 1 U M diag M η U M
14 Learning the Topic Models Input: Pairs ˆ and Triples ˆ Output: topic-word distributions ˆM Û, ˆV top k left, right eigenvectors of Pairs ˆ a η random sample from range(û ) ( ) ˆξ 1, ˆξ 2,..., ˆξ k right eigenvectors of B (η) b for j 1 to k do ˆµ j Û ˆξ j / 1, Û ˆξ j end return ˆM = [ ˆµ 1 ˆµ 2... ˆµ k ] a Pairs = M diag (w) M b B (η) = ( U M ) diag ( M η ) ( U M ) 1
15 Learning the Topic Models Lemma (Observable Operator) B (η) = ( ) ( ) ( ) 1 U M diag M η U M We hope M η has distinct entries. How to pick η? η e i M η i-th word s distribution over topics Prior knowledge required! Otherwise, η U θ, θ Uniform(S k 1 )
16 Learning the Topic Models SVD is carried out on R k k, k d Only involves trigram statistics i.e. low-order moments Guaranteed to recover the parameters Parameters of more complicated models like LDA can be recovered in the same manner
17 Tensor Decomposition Recall Pairs = M diag (w) M ( ( ) ) Triples (η) = M diag M η diag (w) M Pairs = k w j µ j µ j j k Triples = w j µ j µ j µ j j Symmetric tensor decomposition? µ j need to be orthogonal
18 Tensor Decomposition Whiten Pairs W := UD 1 2 W PairsW = I µ j := w j W µ j We can check that µ j, j [k] are orthonormal vectors Do orthogonal tensor decomposition on Triples (W, W, W ) = k j=1 ( ) 3 k w j W 1 µ j = µ 3 j wj j=1 Then recover µ j from µ j
19 Anchor Words Drawbacks of previous algorithms topics cannot be correlated the bound is weak (comparatively speaking) empirical runtime performance is not satisfactory Alternatively assumptions?
20 Anchor Words Definition (p-separable) M is p-separable if j, i s.t. M ij p and M ij = 0 for j = j Documents do not necessarily contains anchor words Two-fold algorithm 1 Selection: find the anchor word for each topic 2 Recover: recover M based on anchor words Good theoretical guarantees and empirical results
21 Anchor Words 1 1 The illustration is taken from Ankur Moitra s slides,
22 Discussion Summary A brief introduction to MoM Learning topic models by spectral decomposition Anchor words assumption Connections with our work?
Appendix A. Proof to Theorem 1
Appendix A Proof to Theorem In this section, we prove the sample complexity bound given in Theorem The proof consists of three main parts In Appendix A, we prove perturbation lemmas that bound the estimation
More informationEM & Variational Bayes
EM & Variational Bayes Hanxiao Liu September 9, 2014 1 / 19 Outline 1. EM Algorithm 1.1 Introduction 1.2 Example: Mixture of vmfs 2. Variational Bayes 2.1 Introduction 2.2 Example: Bayesian Mixture of
More informationGraphical 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,
More informationECE 598: Representation Learning: Algorithms and Models Fall 2017
ECE 598: Representation Learning: Algorithms and Models Fall 2017 Lecture 1: Tensor Methods in Machine Learning Lecturer: Pramod Viswanathan Scribe: Bharath V Raghavan, Oct 3, 2017 11 Introduction Tensors
More informationEstimating Latent Variable Graphical Models with Moments and Likelihoods
Estimating Latent Variable Graphical Models with Moments and Likelihoods Arun Tejasvi Chaganty Percy Liang Stanford University June 18, 2014 Chaganty, Liang (Stanford University) Moments and Likelihoods
More informationTensor Decompositions for Machine Learning. G. Roeder 1. UBC Machine Learning Reading Group, June University of British Columbia
Network Feature s Decompositions for Machine Learning 1 1 Department of Computer Science University of British Columbia UBC Machine Learning Group, June 15 2016 1/30 Contact information Network Feature
More informationData Mining Techniques
Data Mining Techniques CS 622 - Section 2 - Spring 27 Pre-final Review Jan-Willem van de Meent Feedback Feedback https://goo.gl/er7eo8 (also posted on Piazza) Also, please fill out your TRACE evaluations!
More information1 Mixed effect models and longitudinal data analysis
1 Mixed effect models and longitudinal data analysis Mixed effects models provide a flexible approach to any situation where data have a grouping structure which introduces some kind of correlation between
More informationBayesian Decision and Bayesian Learning
Bayesian Decision and Bayesian Learning Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL 60208 http://www.eecs.northwestern.edu/~yingwu 1 / 30 Bayes Rule p(x ω i
More information1 General problem. 2 Terminalogy. Estimation. Estimate θ. (Pick a plausible distribution from family. ) Or estimate τ = τ(θ).
Estimation February 3, 206 Debdeep Pati General problem Model: {P θ : θ Θ}. Observe X P θ, θ Θ unknown. Estimate θ. (Pick a plausible distribution from family. ) Or estimate τ = τ(θ). Examples: θ = (µ,
More informationLecture 5. Gaussian Models - Part 1. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. November 29, 2016
Lecture 5 Gaussian Models - Part 1 Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza November 29, 2016 Luigi Freda ( La Sapienza University) Lecture 5 November 29, 2016 1 / 42 Outline 1 Basics
More informationDictionary 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
More informationLearning Mixtures of Spherical Gaussians: Moment Methods and Spectral Decompositions
Learning Mixtures of Spherical Gaussians: Moment Methods and Spectral Decompositions [Extended Abstract] ABSTRACT Daniel Hsu Microsoft Research New England dahsu@microsoft.com This work provides a computationally
More informationOrthogonal 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.
More informationStatistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation
Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence
More informationTensor Methods for Feature Learning
Tensor Methods for Feature Learning Anima Anandkumar U.C. Irvine Feature Learning For Efficient Classification Find good transformations of input for improved classification Figures used attributed to
More informationLearning mixtures of spherical Gaussians: moment methods and spectral decompositions
Learning mixtures of spherical Gaussians: moment methods and spectral decompositions Daniel Hsu and Sham M. Kakade Microsoft Research New England October 8, 01 Abstract This work provides a computationally
More information4.2 Estimation on the boundary of the parameter space
Chapter 4 Non-standard inference As we mentioned in Chapter the the log-likelihood ratio statistic is useful in the context of statistical testing because typically it is pivotal (does not depend on any
More informationIntroduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf
1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf 2013-14 We know that X ~ B(n,p), but we do not know p. We get a random sample
More informationSTAT 135 Lab 2 Confidence Intervals, MLE and the Delta Method
STAT 135 Lab 2 Confidence Intervals, MLE and the Delta Method Rebecca Barter February 2, 2015 Confidence Intervals Confidence intervals What is a confidence interval? A confidence interval is calculated
More informationIntroduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Lior Wolf
1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Lior Wolf 2014-15 We know that X ~ B(n,p), but we do not know p. We get a random sample from X, a
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More informationGuaranteed Learning of Latent Variable Models through Spectral and Tensor Methods
Guaranteed Learning of Latent Variable Models through Spectral and Tensor Methods Anima Anandkumar U.C. Irvine Application 1: Clustering Basic operation of grouping data points. Hypothesis: each data point
More informationA Method of Moments for Mixture Models and Hidden Markov Models
JMLR: Workshop and Conference Proceedings vol 23 (2012) 33.1 33.34 25th Annual Conference on Learning Theory A Method of Moments for Mixture Models and Hidden Markov Models Animashree Anandkumar University
More informationDonald Goldfarb IEOR Department Columbia University UCLA Mathematics Department Distinguished Lecture Series May 17 19, 2016
Optimization for Tensor Models Donald Goldfarb IEOR Department Columbia University UCLA Mathematics Department Distinguished Lecture Series May 17 19, 2016 1 Tensors Matrix Tensor: higher-order matrix
More informationLecture 3: More on regularization. Bayesian vs maximum likelihood learning
Lecture 3: More on regularization. Bayesian vs maximum likelihood learning L2 and L1 regularization for linear estimators A Bayesian interpretation of regularization Bayesian vs maximum likelihood fitting
More informationGuaranteed Learning of Latent Variable Models through Tensor Methods
Guaranteed Learning of Latent Variable Models through Tensor Methods Furong Huang University of Maryland furongh@cs.umd.edu ACM SIGMETRICS Tutorial 2018 1/75 Tutorial Topic Learning algorithms for latent
More informationMachine Learning 2017
Machine Learning 2017 Volker Roth Department of Mathematics & Computer Science University of Basel 21st March 2017 Volker Roth (University of Basel) Machine Learning 2017 21st March 2017 1 / 41 Section
More information4. Distributions of Functions of Random Variables
4. Distributions of Functions of Random Variables Setup: Consider as given the joint distribution of X 1,..., X n (i.e. consider as given f X1,...,X n and F X1,...,X n ) Consider k functions g 1 : R n
More informationSTAT 135 Lab 3 Asymptotic MLE and the Method of Moments
STAT 135 Lab 3 Asymptotic MLE and the Method of Moments Rebecca Barter February 9, 2015 Maximum likelihood estimation (a reminder) Maximum likelihood estimation Suppose that we have a sample, X 1, X 2,...,
More informationVolodymyr Kuleshov ú Arun Tejasvi Chaganty ú Percy Liang. May 11, 2015
Tensor Factorization via Matrix Factorization Volodymyr Kuleshov ú Arun Tejasvi Chaganty ú Percy Liang Stanford University May 11, 2015 Kuleshov, Chaganty, Liang (Stanford University) Tensor Factorization
More informationLatent Variable Models and EM algorithm
Latent Variable Models and EM algorithm SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic 3.1 Clustering and Mixture Modelling K-means and hierarchical clustering are non-probabilistic
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 informationDistributed Estimation, Information Loss and Exponential Families. Qiang Liu Department of Computer Science Dartmouth College
Distributed Estimation, Information Loss and Exponential Families Qiang Liu Department of Computer Science Dartmouth College Statistical Learning / Estimation Learning generative models from data Topic
More informationSYSM 6303: Quantitative Introduction to Risk and Uncertainty in Business Lecture 4: Fitting Data to Distributions
SYSM 6303: Quantitative Introduction to Risk and Uncertainty in Business Lecture 4: Fitting Data to Distributions M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu
More informationThe Expectation-Maximization Algorithm
1/29 EM & Latent Variable Models Gaussian Mixture Models EM Theory The Expectation-Maximization Algorithm Mihaela van der Schaar Department of Engineering Science University of Oxford MLE for Latent Variable
More informationCS281 Section 4: Factor Analysis and PCA
CS81 Section 4: Factor Analysis and PCA Scott Linderman At this point we have seen a variety of machine learning models, with a particular emphasis on models for supervised learning. In particular, we
More informationSTAT 730 Chapter 4: Estimation
STAT 730 Chapter 4: Estimation Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Analysis 1 / 23 The likelihood We have iid data, at least initially. Each datum
More informationComputational Lower Bounds for Statistical Estimation Problems
Computational Lower Bounds for Statistical Estimation Problems Ilias Diakonikolas (USC) (joint with Daniel Kane (UCSD) and Alistair Stewart (USC)) Workshop on Local Algorithms, MIT, June 2018 THIS TALK
More informationLearning Topic Models and Latent Bayesian Networks Under Expansion Constraints
Learning Topic Models and Latent Bayesian Networks Under Expansion Constraints Animashree Anandkumar 1, Daniel Hsu 2, Adel Javanmard 3, and Sham M. Kakade 2 1 Department of EECS, University of California,
More informationClustering 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
More informationFourier PCA. Navin Goyal (MSR India), Santosh Vempala (Georgia Tech) and Ying Xiao (Georgia Tech)
Fourier PCA Navin Goyal (MSR India), Santosh Vempala (Georgia Tech) and Ying Xiao (Georgia Tech) Introduction 1. Describe a learning problem. 2. Develop an efficient tensor decomposition. Independent component
More informationGaussian Models (9/9/13)
STA561: Probabilistic machine learning Gaussian Models (9/9/13) Lecturer: Barbara Engelhardt Scribes: Xi He, Jiangwei Pan, Ali Razeen, Animesh Srivastava 1 Multivariate Normal Distribution The multivariate
More informationStat 451 Lecture Notes Numerical Integration
Stat 451 Lecture Notes 03 12 Numerical Integration Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapter 5 in Givens & Hoeting, and Chapters 4 & 18 of Lange 2 Updated: February 11, 2016 1 / 29
More informationKernel methods, kernel SVM and ridge regression
Kernel methods, kernel SVM and ridge regression Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Collaborative Filtering 2 Collaborative Filtering R: rating matrix; U: user factor;
More informationDisentangling Gaussians
Disentangling Gaussians Ankur Moitra, MIT November 6th, 2014 Dean s Breakfast Algorithmic Aspects of Machine Learning 2015 by Ankur Moitra. Note: These are unpolished, incomplete course notes. Developed
More informationComposite Hypotheses and Generalized Likelihood Ratio Tests
Composite Hypotheses and Generalized Likelihood Ratio Tests Rebecca Willett, 06 In many real world problems, it is difficult to precisely specify probability distributions. Our models for data may involve
More informationCS6220 Data Mining Techniques Hidden Markov Models, Exponential Families, and the Forward-backward Algorithm
CS6220 Data Mining Techniques Hidden Markov Models, Exponential Families, and the Forward-backward Algorithm Jan-Willem van de Meent, 19 November 2016 1 Hidden Markov Models A hidden Markov model (HMM)
More informationLatent Variable Models and EM Algorithm
SC4/SM8 Advanced Topics in Statistical Machine Learning Latent Variable Models and EM Algorithm Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/atsml/
More informationLinear Methods for Prediction
Chapter 5 Linear Methods for Prediction 5.1 Introduction We now revisit the classification problem and focus on linear methods. Since our prediction Ĝ(x) will always take values in the discrete set G we
More informationExpectation Maximization and Mixtures of Gaussians
Statistical Machine Learning Notes 10 Expectation Maximiation and Mixtures of Gaussians Instructor: Justin Domke Contents 1 Introduction 1 2 Preliminary: Jensen s Inequality 2 3 Expectation Maximiation
More informationEstimating Covariance Using Factorial Hidden Markov Models
Estimating Covariance Using Factorial Hidden Markov Models João Sedoc 1,2 with: Jordan Rodu 3, Lyle Ungar 1, Dean Foster 1 and Jean Gallier 1 1 University of Pennsylvania Philadelphia, PA joao@cis.upenn.edu
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2009 Prof. Gesine Reinert Our standard situation is that we have data x = x 1, x 2,..., x n, which we view as realisations of random
More informationCS168: The Modern Algorithmic Toolbox Lecture #10: Tensors, and Low-Rank Tensor Recovery
CS168: The Modern Algorithmic Toolbox Lecture #10: Tensors, and Low-Rank Tensor Recovery Tim Roughgarden & Gregory Valiant May 3, 2017 Last lecture discussed singular value decomposition (SVD), and we
More informationNon-convex Robust PCA: Provable Bounds
Non-convex Robust PCA: Provable Bounds Anima Anandkumar U.C. Irvine Joint work with Praneeth Netrapalli, U.N. Niranjan, Prateek Jain and Sujay Sanghavi. Learning with Big Data High Dimensional Regime Missing
More informationLinear Algebra - Part II
Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T
More informationPrincipal Component Analysis
Machine Learning Michaelmas 2017 James Worrell Principal Component Analysis 1 Introduction 1.1 Goals of PCA Principal components analysis (PCA) is a dimensionality reduction technique that can be used
More informationMultivariate Distributions
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Multivariate Distributions We will study multivariate distributions in these notes, focusing 1 in particular on multivariate
More informationSTAT 512 sp 2018 Summary Sheet
STAT 5 sp 08 Summary Sheet Karl B. Gregory Spring 08. Transformations of a random variable Let X be a rv with support X and let g be a function mapping X to Y with inverse mapping g (A = {x X : g(x A}
More informationClustering K-means. Clustering images. Machine Learning CSE546 Carlos Guestrin University of Washington. November 4, 2014.
Clustering K-means Machine Learning CSE546 Carlos Guestrin University of Washington November 4, 2014 1 Clustering images Set of Images [Goldberger et al.] 2 1 K-means Randomly initialize k centers µ (0)
More informationPractice Exam 1. (A) (B) (C) (D) (E) You are given the following data on loss sizes:
Practice Exam 1 1. Losses for an insurance coverage have the following cumulative distribution function: F(0) = 0 F(1,000) = 0.2 F(5,000) = 0.4 F(10,000) = 0.9 F(100,000) = 1 with linear interpolation
More informationIV. Matrix Approximation using Least-Squares
IV. Matrix Approximation using Least-Squares The SVD and Matrix Approximation We begin with the following fundamental question. Let A be an M N matrix with rank R. What is the closest matrix to A that
More informationAn Introduction to Expectation-Maximization
An Introduction to Expectation-Maximization Dahua Lin Abstract This notes reviews the basics about the Expectation-Maximization EM) algorithm, a popular approach to perform model estimation of the generative
More informationEM Algorithm II. September 11, 2018
EM Algorithm II September 11, 2018 Review EM 1/27 (Y obs, Y mis ) f (y obs, y mis θ), we observe Y obs but not Y mis Complete-data log likelihood: l C (θ Y obs, Y mis ) = log { f (Y obs, Y mis θ) Observed-data
More informationj=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. Let u = [u
More informationBANA 7046 Data Mining I Lecture 6. Other Data Mining Algorithms 1
BANA 7046 Data Mining I Lecture 6. Other Data Mining Algorithms 1 Shaobo Li University of Cincinnati 1 Partially based on Hastie, et al. (2009) ESL, and James, et al. (2013) ISLR Data Mining I Lecture
More informationInformation in a Two-Stage Adaptive Optimal Design
Information in a Two-Stage Adaptive Optimal Design Department of Statistics, University of Missouri Designed Experiments: Recent Advances in Methods and Applications DEMA 2011 Isaac Newton Institute for
More informationStat 579: Generalized Linear Models and Extensions
Stat 579: Generalized Linear Models and Extensions Linear Mixed Models for Longitudinal Data Yan Lu April, 2018, week 15 1 / 38 Data structure t1 t2 tn i 1st subject y 11 y 12 y 1n1 Experimental 2nd subject
More informationHigh-dimensional data: Exploratory data analysis
High-dimensional data: Exploratory data analysis Mark van de Wiel mark.vdwiel@vumc.nl Department of Epidemiology and Biostatistics, VUmc & Department of Mathematics, VU University Contributions by Wessel
More informationLogistic Regression. Seungjin Choi
Logistic Regression 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 http://mlg.postech.ac.kr/
More informationEigenvectors and SVD 1
Eigenvectors and SVD 1 Definition Eigenvectors of a square matrix Ax=λx, x=0. Intuition: x is unchanged by A (except for scaling) Examples: axis of rotation, stationary distribution of a Markov chain 2
More informationProbabilistic Time Series Classification
Probabilistic Time Series Classification Y. Cem Sübakan Boğaziçi University 25.06.2013 Y. Cem Sübakan (Boğaziçi University) M.Sc. Thesis Defense 25.06.2013 1 / 54 Problem Statement The goal is to assign
More informationReduced-Rank Hidden Markov Models
Reduced-Rank Hidden Markov Models Sajid M. Siddiqi Byron Boots Geoffrey J. Gordon Carnegie Mellon University ... x 1 x 2 x 3 x τ y 1 y 2 y 3 y τ Sequence of observations: Y =[y 1 y 2 y 3... y τ ] Assume
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationEfficient Spectral Methods for Learning Mixture Models!
Efficient Spectral Methods for Learning Mixture Models Qingqing Huang 2016 February Laboratory for Information & Decision Systems Based on joint works with Munther Dahleh, Rong Ge, Sham Kakade, Greg Valiant.
More informationBayesian Inference. Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Ausín and Mike Wiper Department of Statistics Universidad Carlos III de Madrid Advanced Statistics and Data Mining Summer School
More informationCS839: Probabilistic Graphical Models. Lecture 7: Learning Fully Observed BNs. Theo Rekatsinas
CS839: Probabilistic Graphical Models Lecture 7: Learning Fully Observed BNs Theo Rekatsinas 1 Exponential family: a basic building block For a numeric random variable X p(x ) =h(x)exp T T (x) A( ) = 1
More informationSum-of-Squares Method, Tensor Decomposition, Dictionary Learning
Sum-of-Squares Method, Tensor Decomposition, Dictionary Learning David Steurer Cornell Approximation Algorithms and Hardness, Banff, August 2014 for many problems (e.g., all UG-hard ones): better guarantees
More informationNow consider the case where E(Y) = µ = Xβ and V (Y) = σ 2 G, where G is diagonal, but unknown.
Weighting We have seen that if E(Y) = Xβ and V (Y) = σ 2 G, where G is known, the model can be rewritten as a linear model. This is known as generalized least squares or, if G is diagonal, with trace(g)
More informationBayesian Inference. Chapter 9. Linear models and regression
Bayesian Inference Chapter 9. Linear models and regression M. Concepcion Ausin Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Lecture 12 Dynamical Models CS/CNS/EE 155 Andreas Krause Homework 3 out tonight Start early!! Announcements Project milestones due today Please email to TAs 2 Parameter learning
More informationCS6220: DATA MINING TECHNIQUES
CS6220: DATA MINING TECHNIQUES Matrix Data: Clustering: Part 2 Instructor: Yizhou Sun yzsun@ccs.neu.edu October 19, 2014 Methods to Learn Matrix Data Set Data Sequence Data Time Series Graph & Network
More informationQuadrature for the Finite Free Convolution
Spectral Graph Theory Lecture 23 Quadrature for the Finite Free Convolution Daniel A. Spielman November 30, 205 Disclaimer These notes are not necessarily an accurate representation of what happened in
More informationPMR Learning as Inference
Outline PMR Learning as Inference Probabilistic Modelling and Reasoning Amos Storkey Modelling 2 The Exponential Family 3 Bayesian Sets School of Informatics, University of Edinburgh Amos Storkey PMR Learning
More informationLecture 3. G. Cowan. Lecture 3 page 1. Lectures on Statistical Data Analysis
Lecture 3 1 Probability (90 min.) Definition, Bayes theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests (90 min.) general concepts, test statistics,
More informationLecture 4. Generative Models for Discrete Data - Part 3. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza.
Lecture 4 Generative Models for Discrete Data - Part 3 Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza October 6, 2017 Luigi Freda ( La Sapienza University) Lecture 4 October 6, 2017 1 / 46 Outline
More informationNotes on the framework of Ando and Zhang (2005) 1 Beyond learning good functions: learning good spaces
Notes on the framework of Ando and Zhang (2005 Karl Stratos 1 Beyond learning good functions: learning good spaces 1.1 A single binary classification problem Let X denote the problem domain. Suppose we
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More informationMaster s Written Examination
Master s Written Examination Option: Statistics and Probability Spring 05 Full points may be obtained for correct answers to eight questions Each numbered question (which may have several parts) is worth
More informationCS6220: DATA MINING TECHNIQUES
CS6220: DATA MINING TECHNIQUES Matrix Data: Clustering: Part 2 Instructor: Yizhou Sun yzsun@ccs.neu.edu November 3, 2015 Methods to Learn Matrix Data Text Data Set Data Sequence Data Time Series Graph
More informationLecture 20 May 18, Empirical Bayes Interpretation [Efron & Morris 1973]
Stats 300C: Theory of Statistics Spring 2018 Lecture 20 May 18, 2018 Prof. Emmanuel Candes Scribe: Will Fithian and E. Candes 1 Outline 1. Stein s Phenomenon 2. Empirical Bayes Interpretation of James-Stein
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationStatistical Data Mining and Machine Learning Hilary Term 2016
Statistical Data Mining and Machine Learning Hilary Term 2016 Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/sdmml Naïve Bayes
More informationEIGENVALUES AND SINGULAR VALUE DECOMPOSITION
APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required
More informationReview and continuation from last week Properties of MLEs
Review and continuation from last week Properties of MLEs As we have mentioned, MLEs have a nice intuitive property, and as we have seen, they have a certain equivariance property. We will see later that
More informationSTATS306B STATS306B. Discriminant Analysis. Jonathan Taylor Department of Statistics Stanford University. June 3, 2010
STATS306B Discriminant Analysis Jonathan Taylor Department of Statistics Stanford University June 3, 2010 Spring 2010 Classification Given K classes in R p, represented as densities f i (x), 1 i K classify
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini April 27, 2018 1 / 1 Table of Contents 2 / 1 Linear Algebra Review Read 3.1 and 3.2 from text. 1. Fundamental subspace (rank-nullity, etc.) Im(X ) = ker(x T ) R
More informationMaximum likelihood estimation
Maximum likelihood estimation Guillaume Obozinski Ecole des Ponts - ParisTech Master MVA Maximum likelihood estimation 1/26 Outline 1 Statistical concepts 2 A short review of convex analysis and optimization
More informationInfinitely Imbalanced Logistic Regression
p. 1/1 Infinitely Imbalanced Logistic Regression Art B. Owen Journal of Machine Learning Research, April 2007 Presenter: Ivo D. Shterev p. 2/1 Outline Motivation Introduction Numerical Examples Notation
More informationA Robust Form of Kruskal s Identifiability Theorem
A Robust Form of Kruskal s Identifiability Theorem Aditya Bhaskara (Google NYC) Joint work with Moses Charikar (Princeton University) Aravindan Vijayaraghavan (NYU Northwestern) Background: understanding
More information