Cheng Soon Ong & Christian Walder. Canberra February June 2018
|
|
- Maud Warner
- 5 years ago
- Views:
Transcription
1 Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 218 Outlines Overview Introduction Linear Algebra Probability Linear Regression 1 Linear Regression 2 Linear Classification 1 Linear Classification 2 Kernel Methods Sparse Kernel Methods Mixture Models and EM 1 Mixture Models and EM 2 Neural Networks 1 Neural Networks 2 Principal Component Analysis Autoencoders Graphical Models 1 Graphical Models 2 Graphical Models 3 Sampling Sequential Data 1 Sequential Data 2 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 824
2 Part XI Mixture Models and EM 1 419of 824
3 Marginalisation Sum rule p(a, B) = C p(a, B, C) Product rule p(a, B) = p(a B)p(B) Why do we optimize the log likelihood? 42of 824
4 Strategy in this course Estimate best predictor = training = learning Given data (x 1, y 1 ),..., (x n, y n ), find a predictor f w ( ). 1 Identify the type of input x and output y data 2 Propose a (linear) mathematical model for f w 3 Design an objective function or likelihood 4 Calculate the optimal parameter (w) 5 Model uncertainty using the Bayesian approach 6 Implement and compute (the algorithm in python) 7 Interpret and diagnose results We will study unsupervised learning this week 421of 824
5 Mixture Models and EM Complex marginal distributions over observed variables can be expressed via more tractable joint distributions over the expanded space of observed and latent variables. Mixture Models can also be used to cluster data. General technique for finding maximum likelihood estimators in latent variable models: expectation-maximisation (EM) algorithm. 422of 824
6 Example - Wallaby Distribution Introduced very recently to show of 824
7 Example - Wallaby Distribution... that already a mixture of three Gaussian can be fun. p(x) = 3 1 N (x 5,.5) N (x 9, 2) + N (x 2, 2) of 824
8 Example - Wallaby Distribution Use µ, σ as latent variables and define a distribution 3 1 if (µ, σ) = (5,.5) 3 p(µ, σ) = 1 if (µ, σ) = (9, 2) 4 1 if (µ, σ) = (2, 2) otherwise. p(x) = = p(x, µ, σ) dµ dσ p(x µ, σ) p(µ, σ) dµ dσ = 3 1 N (x 5,.5) N (x 9, 2) + N (x 2, 2) of 824
9 Given a set of data {x 1,..., x N } where x n R D, n = 1,..., N. Goal: Partition the data into K clusters. 426of 824
10 Given a set of data {x 1,..., x N } where x n R D, n = 1,..., N. Goal: Partition the data into K clusters. Each cluster contains points close to each other. Introduce a prototype µ k R D for each cluster. Goal: Find 1 a set prototypes µ k, k = 1,..., K, each representing a different cluster. 2 an assignment of each data point to exactly one cluster. 427of 824
11 - The Algorithm Start with arbitrary chosen prototypes µ k, k = 1,..., K. 1 Assign each data point to the closest prototype. 2 Calculate new prototypes as the mean of all data points assigned to each of them. In the following, we will formalise this introducing a notation which will be useful later. 428of 824
12 - Notation Binary indicator variables { 1, if data point x n belongs to cluster k r nk =, otherwise using the 1-of-K coding scheme. Define a distortion measure N K J = r nk x n µ k 2 n=1 k=1 Find the values for {r nk } and {µ k } so as to minimise J. 429of 824
13 - Notation Find the values for {r nk } and {µ k } so as to minimise J. But {r nk } depends on {µ k }, and {µ k } depends on {r nk }. 43of 824
14 - Notation Find the values for {r nk } and {µ k } so as to minimise J. But {r nk } depends on {µ k }, and {µ k } depends on {r nk }. Iterate until no further change 1 Minimise J w.r.t. r nk while keeping {µ k } fixed, r nk = { 1, if k = arg min j x n µ j 2, otherwise. n = 1,..., N Expectation step 2 Minimise J w.r.t. {µ k } while keeping r nk fixed, = 2 µ k = N r nk(x n µ k ) n=1 N n=1 rnkxn N n=1 rnk Maximisation step 431of 824
15 - Example 2 (a) 2 (b) 2 (c) (d) 2 (e) 2 (f) of 824
16 - Example 2 (d) 2 (e) 2 (f) (g) 2 (h) 2 (i) of 824
17 - Cost Function 1 J Cost function J after each E step (blue points) and M step (red points). 434of 824
18 - Notes Initial condition crucial for convergence. What happens, if at least one cluster centre is too far from all data points? Complex step: Finding the nearest neighbour. (Use triangle inequality; build K-D trees,...) Generalise to non-euclidean dissimilarity measures V(x n, µ k ) (called K-medoids algorithm), N K J = r nk V(x n, µ k ). n=1 k=1 Online stochastic algorithm 1 Draw data point x n and locate nearest prototype µ k. 2 Update only µ k using decreasing learning rate η n µ new k = µ old k + η n(x n µ old k ). 435of 824
19 - Image Segmentation Segment an image into regions of reasonable homogeneous appearance. Each pixel is a point in R 3 (red, blue, green). (Note that the pixel intensities are bounded in the range [, 1] and therefore this space is strictly speaking not Euclidean). Run K-means on all points of the image until convergence. Replace all pixels with the corresponding mean µ k. Results in an image with a palette only K different colours. There are much better approaches to image segmentation (but it is an active research topic), this here serves only to illustrate K-means. 436of 824
20 Illustrating - Segmentation K = 2 K = 1 K = 3 Original image 437of 824
21 Illustrating - Segmentation 438of 824
22 - Compression Lossy data compression: accept some errors in the reconstruction as trade-off for higher compression. Apply K-means to the data. Store the code-book vectors µ k. Store the data in the form of references (labels) to the code-book. Each data point has a label in the range [1,..., K]. New data points are also compressed by finding the closest code-book vector and then storing only the label. This technique is also called vector quantisation. 439of 824
23 Illustrating - Compression K = 2 K = 3 4.2% 8.3% K = 1 Original image 16.7% 1 % 44of 824
24 Latent variable modeling We have already seen a mixture of two Gaussians for linear classification However in the clustering scenario, we do not observe the class membership Strategy (this is vague) We have a difficult distribution p(x) We introduce a new variable z to get p(x, z) Model p(x, z) = p(x z)p(z) with easy distributions 441of 824
25 A Gaussian mixture distribution is a linear superposition of Gaussians of the form p(x) = K π k N (x µ k, Σ k ). k=1 As p(x) dx = 1, if follows K k=1 π k = 1. Let us write this with the help of a latent variable z. Definition (Latent variables) Latent variables (as opposed to observable variables), are variables that are not directly observed but are rather inferred (through a mathematical model) from other variables that are observed and directly measured. They are also sometimes called hidden variables, model parameters, or hypothetical variables. 442of 824
26 Let z {, 1} K and K k=1 z k = 1. In words, z is a K-dimensional vector in 1-of-K representation. There are exactly K different possible vectors z depending on which of the K entries is 1. Define the joint distribution p(x, z) in terms of a marginal distribution p(z) and a conditional distribution p(x z) as p(x, z) = p(z) p(x z) z x 443of 824
27 Set the marginal distribution to p(z k = 1) = π k where π k 1 together with K k=1 π k = 1. Because z uses 1-of-K coding, we can also write p(z) = K k=1 π zk k. Set the conditional distribution of x given a particular z to p(x z k = 1) = N (x µ k, Σ k ), or K p(x z) = N (x µ k, Σ k ) zk, k=1 444of 824
28 The marginal distribution over x is now found by summing the joint distribution over all possible states of z p(x) = p(z) p(x z) = z z K = π k N (x µ k, Σ k ) k=1 K k=1 π zk k K N (x µ k, Σ k ) zk k=1 The marginal distribution of x is a Gaussian mixture. For several observations x 1,..., x N we need one latent variable z n per observation. What have we gained? Can now work with the joint distribution p(x, z). Will lead to significant simplification later, especially for EM algorithm. 445of 824
29 Conditional probability of z given x by Bayes theorem γ(z k ) = p(z k = 1 x) = p(z k = 1) p(x z k = 1) K j=1 p(z j = 1) p(x z j = 1) = π k N (x µ k, Σ k ) K j=1 π j N (x µ j, Σ j ) γ(z k ) is the responsibility of component k to explain the observation x. 446of 824
30 - Ancestral Sampling Goal: Generate random samples distributed according to the mixture model. 1 Generate a sample ẑ from the distribution p(z). 2 Generate a value x from the conditional distribution p(x ẑ). Example: Mixture of 3 Gaussians, 5 points. 1 (a) 1 (b) 1 (c) Original states of z. Marginal p(x). (R, G, B) - colours mixed according to γ(z nk ). 447of 824
31 - Maximum Likelihood Given N data points, each of dimension D, we have the data matrix X R N D where each row contains one data point. Similarly, we have the matrix of latent variables Z R N K with rows z T n. Assume the data are drawn i.i.d., the distribution for the data can be represented by a graphical model. π z n x n µ Σ N 448of 824
32 - Maximum Likelihood The log of the likelihood function is then { N K } ln p(x π, µ, Σ) = ln π k N (x µ k, Σ k ) n=1 k=1 Significant problem: If a mean µ j sits directly on a data point x n then N (x n x n, σ 2 j I) = 1 1. (2π) 1/2 σ j Here we assumed Σ k = σk 2 I. But problem is general, just think of a main axis transformation for Σ k. Overfitting (in disguise) occuring again with the maximum likelihood approach. Use heuristics to detect this situation and reset the mean of the corresponding component of the mixture. 449of 824
33 - Maximum Likelihood A K component mixture has a total of K! equivalent solutions corresponding to the K! ways of assigning K sets of parameters to K solutions. Also called identifiability problem. Needs to be considered when the parameters discovered by a model are interpreted. Maximising the log likelihood of a Gaussian mixture is more complex then for a single Gaussian. Summation over all K components inside of the logarithm make it harder. Setting the derivatives of the log likelihood to zero does not longer result in a closed form. May use gradient-based optimisation. Or EM algorithm. Stay tuned. 45of 824
Cheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationPattern Recognition and Machine Learning. Bishop Chapter 9: Mixture Models and EM
Pattern Recognition and Machine Learning Chapter 9: Mixture Models and EM Thomas Mensink Jakob Verbeek October 11, 27 Le Menu 9.1 K-means clustering Getting the idea with a simple example 9.2 Mixtures
More informationMixtures of Gaussians. Sargur Srihari
Mixtures of Gaussians Sargur srihari@cedar.buffalo.edu 1 9. Mixture Models and EM 0. Mixture Models Overview 1. K-Means Clustering 2. Mixtures of Gaussians 3. An Alternative View of EM 4. The EM Algorithm
More informationComputer Vision Group Prof. Daniel Cremers. 6. Mixture Models and Expectation-Maximization
Prof. Daniel Cremers 6. Mixture Models and Expectation-Maximization Motivation Often the introduction of latent (unobserved) random variables into a model can help to express complex (marginal) distributions
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 254 Part V
More informationCheng Soon Ong & Christian Walder. Canberra February June 2017
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2017 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 679 Part XIX
More informationLatent Variable Models and Expectation Maximization
Latent Variable Models and Expectation Maximization Oliver Schulte - CMPT 726 Bishop PRML Ch. 9 2 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 5 1 15 2 25 5 1 15 2 25 2 4 6 8 1 12 14 2 4 6 8 1 12 14 5 1 15
More informationParametric Unsupervised Learning Expectation Maximization (EM) Lecture 20.a
Parametric Unsupervised Learning Expectation Maximization (EM) Lecture 20.a Some slides are due to Christopher Bishop Limitations of K-means Hard assignments of data points to clusters small shift of a
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 89 Part II
More informationLatent Variable Models and Expectation Maximization
Latent Variable Models and Expectation Maximization Oliver Schulte - CMPT 726 Bishop PRML Ch. 9 2 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 5 1 15 2 25 5 1 15 2 25 2 4 6 8 1 12 14 2 4 6 8 1 12 14 5 1 15
More informationECE 521. Lecture 11 (not on midterm material) 13 February K-means clustering, Dimensionality reduction
ECE 521 Lecture 11 (not on midterm material) 13 February 2017 K-means clustering, Dimensionality reduction With thanks to Ruslan Salakhutdinov for an earlier version of the slides Overview K-means clustering
More informationExpectation Maximization
Expectation Maximization Bishop PRML Ch. 9 Alireza Ghane c Ghane/Mori 4 6 8 4 6 8 4 6 8 4 6 8 5 5 5 5 5 5 4 6 8 4 4 6 8 4 5 5 5 5 5 5 µ, Σ) α f Learningscale is slightly Parameters is slightly larger larger
More informationMixture Models and EM
Mixture Models and EM Yoan Miche CIS, HUT March 17, 27 Yoan Miche (CIS, HUT) Mixture Models and EM March 17, 27 1 / 23 Mise en Bouche Yoan Miche (CIS, HUT) Mixture Models and EM March 17, 27 2 / 23 Mise
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationClustering and Gaussian Mixtures
Clustering and Gaussian Mixtures Oliver Schulte - CMPT 883 2 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 5 1 15 2 25 5 1 15 2 25 2 4 6 8 1 12 14 2 4 6 8 1 12 14 5 1 15 2 25 5 1 15 2 25 detected tures detected
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationCSCI-567: Machine Learning (Spring 2019)
CSCI-567: Machine Learning (Spring 2019) Prof. Victor Adamchik U of Southern California Mar. 19, 2019 March 19, 2019 1 / 43 Administration March 19, 2019 2 / 43 Administration TA3 is due this week March
More informationProbabilistic & Unsupervised Learning
Probabilistic & Unsupervised Learning Week 2: Latent Variable Models Maneesh Sahani maneesh@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit, and MSc ML/CSML, Dept Computer Science University College
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 305 Part VII
More informationLecture 10. Announcement. Mixture Models II. Topics of This Lecture. This Lecture: Advanced Machine Learning. Recap: GMMs as Latent Variable Models
Advanced Machine Learning Lecture 10 Mixture Models II 30.11.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de/ Announcement Exercise sheet 2 online Sampling Rejection Sampling Importance
More informationLecture 4: Probabilistic Learning
DD2431 Autumn, 2015 1 Maximum Likelihood Methods Maximum A Posteriori Methods Bayesian methods 2 Classification vs Clustering Heuristic Example: K-means Expectation Maximization 3 Maximum Likelihood Methods
More informationLogistic Regression. COMP 527 Danushka Bollegala
Logistic Regression COMP 527 Danushka Bollegala Binary Classification Given an instance x we must classify it to either positive (1) or negative (0) class We can use {1,-1} instead of {1,0} but we will
More informationLecture 4: Probabilistic Learning. Estimation Theory. Classification with Probability Distributions
DD2431 Autumn, 2014 1 2 3 Classification with Probability Distributions Estimation Theory Classification in the last lecture we assumed we new: P(y) Prior P(x y) Lielihood x2 x features y {ω 1,..., ω K
More informationCurve Fitting Re-visited, Bishop1.2.5
Curve Fitting Re-visited, Bishop1.2.5 Maximum Likelihood Bishop 1.2.5 Model Likelihood differentiation p(t x, w, β) = Maximum Likelihood N N ( t n y(x n, w), β 1). (1.61) n=1 As we did in the case of the
More informationClustering. Professor Ameet Talwalkar. Professor Ameet Talwalkar CS260 Machine Learning Algorithms March 8, / 26
Clustering Professor Ameet Talwalkar Professor Ameet Talwalkar CS26 Machine Learning Algorithms March 8, 217 1 / 26 Outline 1 Administration 2 Review of last lecture 3 Clustering Professor Ameet Talwalkar
More informationProbabilistic Reasoning in Deep Learning
Probabilistic Reasoning in Deep Learning Dr Konstantina Palla, PhD palla@stats.ox.ac.uk September 2017 Deep Learning Indaba, Johannesburgh Konstantina Palla 1 / 39 OVERVIEW OF THE TALK Basics of Bayesian
More informationSTA414/2104. Lecture 11: Gaussian Processes. Department of Statistics
STA414/2104 Lecture 11: Gaussian Processes Department of Statistics www.utstat.utoronto.ca Delivered by Mark Ebden with thanks to Russ Salakhutdinov Outline Gaussian Processes Exam review Course evaluations
More informationData Mining Techniques
Data Mining Techniques CS 6220 - Section 2 - Spring 2017 Lecture 6 Jan-Willem van de Meent (credit: Yijun Zhao, Chris Bishop, Andrew Moore, Hastie et al.) Project Project Deadlines 3 Feb: Form teams of
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Expectation Maximization (EM) and Mixture Models Hamid R. Rabiee Jafar Muhammadi, Mohammad J. Hosseini Spring 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2 Agenda Expectation-maximization
More informationClustering with k-means and Gaussian mixture distributions
Clustering with k-means and Gaussian mixture distributions Machine Learning and Object Recognition 2017-2018 Jakob Verbeek Clustering Finding a group structure in the data Data in one cluster similar to
More informationMathematical Formulation of Our Example
Mathematical Formulation of Our Example We define two binary random variables: open and, where is light on or light off. Our question is: What is? Computer Vision 1 Combining Evidence Suppose our robot
More informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
More informationClustering and Gaussian Mixture Models
Clustering and Gaussian Mixture Models Piyush Rai IIT Kanpur Probabilistic Machine Learning (CS772A) Jan 25, 2016 Probabilistic Machine Learning (CS772A) Clustering and Gaussian Mixture Models 1 Recap
More informationRecent Advances in Bayesian Inference Techniques
Recent Advances in Bayesian Inference Techniques Christopher M. Bishop Microsoft Research, Cambridge, U.K. research.microsoft.com/~cmbishop SIAM Conference on Data Mining, April 2004 Abstract Bayesian
More informationIntroduction to Machine Learning
Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 20: Expectation Maximization Algorithm EM for Mixture Models Many figures courtesy Kevin Murphy s
More informationMachine Learning and Bayesian Inference. Unsupervised learning. Can we find regularity in data without the aid of labels?
Machine Learning and Bayesian Inference Dr Sean Holden Computer Laboratory, Room FC6 Telephone extension 6372 Email: sbh11@cl.cam.ac.uk www.cl.cam.ac.uk/ sbh11/ Unsupervised learning Can we find regularity
More informationClustering. CSL465/603 - Fall 2016 Narayanan C Krishnan
Clustering CSL465/603 - Fall 2016 Narayanan C Krishnan ckn@iitrpr.ac.in Supervised vs Unsupervised Learning Supervised learning Given x ", y " "%& ', learn a function f: X Y Categorical output classification
More informationK-Means and Gaussian Mixture Models
K-Means and Gaussian Mixture Models David Rosenberg New York University October 29, 2016 David Rosenberg (New York University) DS-GA 1003 October 29, 2016 1 / 42 K-Means Clustering K-Means Clustering David
More information9/12/17. Types of learning. Modeling data. Supervised learning: Classification. Supervised learning: Regression. Unsupervised learning: Clustering
Types of learning Modeling data Supervised: we know input and targets Goal is to learn a model that, given input data, accurately predicts target data Unsupervised: we know the input only and want to make
More informationIntroduction to Machine Learning Midterm Exam
10-701 Introduction to Machine Learning Midterm Exam Instructors: Eric Xing, Ziv Bar-Joseph 17 November, 2015 There are 11 questions, for a total of 100 points. This exam is open book, open notes, but
More informationData Preprocessing. Cluster Similarity
1 Cluster Similarity Similarity is most often measured with the help of a distance function. The smaller the distance, the more similar the data objects (points). A function d: M M R is a distance on M
More informationA graph contains a set of nodes (vertices) connected by links (edges or arcs)
BOLTZMANN MACHINES Generative Models Graphical Models A graph contains a set of nodes (vertices) connected by links (edges or arcs) In a probabilistic graphical model, each node represents a random variable,
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 informationLatent Variable View of EM. Sargur Srihari
Latent Variable View of EM Sargur srihari@cedar.buffalo.edu 1 Examples of latent variables 1. Mixture Model Joint distribution is p(x,z) We don t have values for z 2. Hidden Markov Model A single time
More informationContrastive Divergence
Contrastive Divergence Training Products of Experts by Minimizing CD Hinton, 2002 Helmut Puhr Institute for Theoretical Computer Science TU Graz June 9, 2010 Contents 1 Theory 2 Argument 3 Contrastive
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 11 Project
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS Parametric Distributions Basic building blocks: Need to determine given Representation: or? Recall Curve Fitting Binary Variables
More informationCOM336: Neural Computing
COM336: Neural Computing http://www.dcs.shef.ac.uk/ sjr/com336/ Lecture 2: Density Estimation Steve Renals Department of Computer Science University of Sheffield Sheffield S1 4DP UK email: s.renals@dcs.shef.ac.uk
More informationBrief Introduction of Machine Learning Techniques for Content Analysis
1 Brief Introduction of Machine Learning Techniques for Content Analysis Wei-Ta Chu 2008/11/20 Outline 2 Overview Gaussian Mixture Model (GMM) Hidden Markov Model (HMM) Support Vector Machine (SVM) Overview
More informationECE521 week 3: 23/26 January 2017
ECE521 week 3: 23/26 January 2017 Outline Probabilistic interpretation of linear regression - Maximum likelihood estimation (MLE) - Maximum a posteriori (MAP) estimation Bias-variance trade-off Linear
More informationLecture 8: Clustering & Mixture Models
Lecture 8: Clustering & Mixture Models C4B Machine Learning Hilary 2011 A. Zisserman K-means algorithm GMM and the EM algorithm plsa clustering K-means algorithm K-means algorithm Partition data into K
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Expectation Maximization (EM) and Mixture Models Hamid R. Rabiee Jafar Muhammadi, Mohammad J. Hosseini Spring 203 http://ce.sharif.edu/courses/9-92/2/ce725-/ Agenda Expectation-maximization
More informationPerformance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project
Performance Comparison of K-Means and Expectation Maximization with Gaussian Mixture Models for Clustering EE6540 Final Project Devin Cornell & Sushruth Sastry May 2015 1 Abstract In this article, we explore
More informationClustering, K-Means, EM Tutorial
Clustering, K-Means, EM Tutorial Kamyar Ghasemipour Parts taken from Shikhar Sharma, Wenjie Luo, and Boris Ivanovic s tutorial slides, as well as lecture notes Organization: Clustering Motivation K-Means
More informationDeep Learning Srihari. Deep Belief Nets. Sargur N. Srihari
Deep Belief Nets Sargur N. Srihari srihari@cedar.buffalo.edu Topics 1. Boltzmann machines 2. Restricted Boltzmann machines 3. Deep Belief Networks 4. Deep Boltzmann machines 5. Boltzmann machines for continuous
More informationIntroduction to Machine Learning. Introduction to ML - TAU 2016/7 1
Introduction to Machine Learning Introduction to ML - TAU 2016/7 1 Course Administration Lecturers: Amir Globerson (gamir@post.tau.ac.il) Yishay Mansour (Mansour@tau.ac.il) Teaching Assistance: Regev Schweiger
More informationMachine Learning for Signal Processing Bayes Classification and Regression
Machine Learning for Signal Processing Bayes Classification and Regression Instructor: Bhiksha Raj 11755/18797 1 Recap: KNN A very effective and simple way of performing classification Simple model: For
More informationThe Particle Filter. PD Dr. Rudolph Triebel Computer Vision Group. Machine Learning for Computer Vision
The Particle Filter Non-parametric implementation of Bayes filter Represents the belief (posterior) random state samples. by a set of This representation is approximate. Can represent distributions that
More informationGaussian Mixture Models, Expectation Maximization
Gaussian Mixture Models, Expectation Maximization Instructor: Jessica Wu Harvey Mudd College The instructor gratefully acknowledges Andrew Ng (Stanford), Andrew Moore (CMU), Eric Eaton (UPenn), David Kauchak
More informationMachine Learning Techniques for Computer Vision
Machine Learning Techniques for Computer Vision Part 2: Unsupervised Learning Microsoft Research Cambridge x 3 1 0.5 0.2 0 0.5 0.3 0 0.5 1 ECCV 2004, Prague x 2 x 1 Overview of Part 2 Mixture models EM
More informationComputer Vision Group Prof. Daniel Cremers. 3. Regression
Prof. Daniel Cremers 3. Regression Categories of Learning (Rep.) Learnin g Unsupervise d Learning Clustering, density estimation Supervised Learning learning from a training data set, inference on the
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 informationExpectation maximization
Expectation maximization Subhransu Maji CMSCI 689: Machine Learning 14 April 2015 Motivation Suppose you are building a naive Bayes spam classifier. After your are done your boss tells you that there is
More informationMachine Learning. Gaussian Mixture Models. Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall
Machine Learning Gaussian Mixture Models Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall 2012 1 The Generative Model POV We think of the data as being generated from some process. We assume
More informationCS181 Midterm 2 Practice Solutions
CS181 Midterm 2 Practice Solutions 1. Convergence of -Means Consider Lloyd s algorithm for finding a -Means clustering of N data, i.e., minimizing the distortion measure objective function J({r n } N n=1,
More informationBayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 2: Bayesian Basics https://people.orie.cornell.edu/andrew/orie6741 Cornell University August 25, 2016 1 / 17 Canonical Machine Learning
More informationClustering with k-means and Gaussian mixture distributions
Clustering with k-means and Gaussian mixture distributions Machine Learning and Category Representation 2012-2013 Jakob Verbeek, ovember 23, 2012 Course website: http://lear.inrialpes.fr/~verbeek/mlcr.12.13
More informationCOMS 4721: Machine Learning for Data Science Lecture 16, 3/28/2017
COMS 4721: Machine Learning for Data Science Lecture 16, 3/28/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University SOFT CLUSTERING VS HARD CLUSTERING
More informationPattern Recognition and Machine Learning. Bishop Chapter 6: Kernel Methods
Pattern Recognition and Machine Learning Chapter 6: Kernel Methods Vasil Khalidov Alex Kläser December 13, 2007 Training Data: Keep or Discard? Parametric methods (linear/nonlinear) so far: learn parameter
More informationLecture 3. Linear Regression II Bastian Leibe RWTH Aachen
Advanced Machine Learning Lecture 3 Linear Regression II 02.11.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de/ leibe@vision.rwth-aachen.de This Lecture: Advanced Machine Learning Regression
More informationClustering with k-means and Gaussian mixture distributions
Clustering with k-means and Gaussian mixture distributions Machine Learning and Category Representation 2014-2015 Jakob Verbeek, ovember 21, 2014 Course website: http://lear.inrialpes.fr/~verbeek/mlcr.14.15
More informationStochastic Variational Inference for Gaussian Process Latent Variable Models using Back Constraints
Stochastic Variational Inference for Gaussian Process Latent Variable Models using Back Constraints Thang D. Bui Richard E. Turner tdb40@cam.ac.uk ret26@cam.ac.uk Computational and Biological Learning
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 informationIntroduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones
Introduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones http://www.mpia.de/homes/calj/mlpr_mpia2008.html 1 1 Last week... supervised and unsupervised methods need adaptive
More informationLatent Variable Models
Latent Variable Models Stefano Ermon, Aditya Grover Stanford University Lecture 5 Stefano Ermon, Aditya Grover (AI Lab) Deep Generative Models Lecture 5 1 / 31 Recap of last lecture 1 Autoregressive models:
More informationGaussian Mixture Models
Gaussian Mixture Models David Rosenberg, Brett Bernstein New York University April 26, 2017 David Rosenberg, Brett Bernstein (New York University) DS-GA 1003 April 26, 2017 1 / 42 Intro Question Intro
More informationSample Exam COMP 9444 NEURAL NETWORKS Solutions
FAMILY NAME OTHER NAMES STUDENT ID SIGNATURE Sample Exam COMP 9444 NEURAL NETWORKS Solutions (1) TIME ALLOWED 3 HOURS (2) TOTAL NUMBER OF QUESTIONS 12 (3) STUDENTS SHOULD ANSWER ALL QUESTIONS (4) QUESTIONS
More informationIntroduction to Machine Learning Midterm, Tues April 8
Introduction to Machine Learning 10-701 Midterm, Tues April 8 [1 point] Name: Andrew ID: Instructions: You are allowed a (two-sided) sheet of notes. Exam ends at 2:45pm Take a deep breath and don t spend
More informationIntroduction to Machine Learning Midterm Exam Solutions
10-701 Introduction to Machine Learning Midterm Exam Solutions Instructors: Eric Xing, Ziv Bar-Joseph 17 November, 2015 There are 11 questions, for a total of 100 points. This exam is open book, open notes,
More informationMachine Learning Lecture 5
Machine Learning Lecture 5 Linear Discriminant Functions 26.10.2017 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de leibe@vision.rwth-aachen.de Course Outline Fundamentals Bayes Decision Theory
More informationBe able to define the following terms and answer basic questions about them:
CS440/ECE448 Section Q Fall 2017 Final Review Be able to define the following terms and answer basic questions about them: Probability o Random variables, axioms of probability o Joint, marginal, conditional
More informationSTA 414/2104: Machine Learning
STA 414/2104: Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistics! rsalakhu@cs.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 9 Sequential Data So far
More informationThe Expectation-Maximization Algorithm
The Expectation-Maximization Algorithm Francisco S. Melo In these notes, we provide a brief overview of the formal aspects concerning -means, EM and their relation. We closely follow the presentation in
More informationIntroduction to Machine Learning
Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 25: Markov Chain Monte Carlo (MCMC) Course Review and Advanced Topics Many figures courtesy Kevin
More informationIntroduction to Graphical Models
Introduction to Graphical Models The 15 th Winter School of Statistical Physics POSCO International Center & POSTECH, Pohang 2018. 1. 9 (Tue.) Yung-Kyun Noh GENERALIZATION FOR PREDICTION 2 Probabilistic
More informationComputer Vision Group Prof. Daniel Cremers. 10a. Markov Chain Monte Carlo
Group Prof. Daniel Cremers 10a. Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative is Markov Chain
More informationMachine Learning. Bayes Basics. Marc Toussaint U Stuttgart. Bayes, probabilities, Bayes theorem & examples
Machine Learning Bayes Basics Bayes, probabilities, Bayes theorem & examples Marc Toussaint U Stuttgart So far: Basic regression & classification methods: Features + Loss + Regularization & CV All kinds
More informationLarge-Scale Feature Learning with Spike-and-Slab Sparse Coding
Large-Scale Feature Learning with Spike-and-Slab Sparse Coding Ian J. Goodfellow, Aaron Courville, Yoshua Bengio ICML 2012 Presented by Xin Yuan January 17, 2013 1 Outline Contributions Spike-and-Slab
More informationStudy Notes on the Latent Dirichlet Allocation
Study Notes on the Latent Dirichlet Allocation Xugang Ye 1. Model Framework A word is an element of dictionary {1,,}. A document is represented by a sequence of words: =(,, ), {1,,}. A corpus is a collection
More informationPATTERN RECOGNITION AND MACHINE LEARNING
PATTERN RECOGNITION AND MACHINE LEARNING Chapter 1. Introduction Shuai Huang April 21, 2014 Outline 1 What is Machine Learning? 2 Curve Fitting 3 Probability Theory 4 Model Selection 5 The curse of dimensionality
More informationLinear Dynamical Systems
Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations
More informationProbability models for machine learning. Advanced topics ML4bio 2016 Alan Moses
Probability models for machine learning Advanced topics ML4bio 2016 Alan Moses What did we cover in this course so far? 4 major areas of machine learning: Clustering Dimensionality reduction Classification
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 9: Expectation Maximiation (EM) Algorithm, Learning in Undirected Graphical Models Some figures courtesy
More informationChris Bishop s PRML Ch. 8: Graphical Models
Chris Bishop s PRML Ch. 8: Graphical Models January 24, 2008 Introduction Visualize the structure of a probabilistic model Design and motivate new models Insights into the model s properties, in particular
More informationGenerative Clustering, Topic Modeling, & Bayesian Inference
Generative Clustering, Topic Modeling, & Bayesian Inference INFO-4604, Applied Machine Learning University of Colorado Boulder December 12-14, 2017 Prof. Michael Paul Unsupervised Naïve Bayes Last week
More information