Joint Factor Analysis for Speaker Verification
|
|
- Dustin Miles
- 6 years ago
- Views:
Transcription
1 Joint Factor Analysis for Speaker Verification Mengke HU ASPITRG Group, ECE Department Drexel University October 12, /37
2 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 2/37
3 Baseline System H0: Target Model + Input Speech Feature Extraction Pre-Processing Σ Λ H1: Background Model 3/37
4 Baseline System 1 Given a speech segment X, we test 2 hypotheses: H 0 : X is from claimed target speaker S (GMM) H 1 : X is not from speaker S, it is from the background (UBM). 2 Decision Rule Score = log p(x TargetModel) p(x UBM) H 0 > < H 1 Threshold Note: Score = log p(x TargetModel) log p(x UBM) 4/37
5 Baseline Experiment 1 Feature Extraction (MFCC) 2 Train the UBM model 3 Obtain Adapted GMM model for target speaker model 4 Test trials against 2 hypotheses 5 Scoring 6 DET(Detection Error Tradeoff) curve false accept VS. false reject Problem How to cancel the channel effect? 5/37
6 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 6/37
7 Session Variation Inter-Speaker Variation: Two utterances are from different speaker 7/37
8 Session Variation Inter-Speaker Variation: Two utterances are from different speaker Inter-Session Variation: Two utterances are from the same speaker Channel effects: Utterances are recorded from different channels Intra-Speaker Variation: Utterances varies with speaker s health or emotional state etc.. 8/37
9 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 9/37
10 Gaussian Mixture Model Review Recall GMM: M s p(s z s ) = N(s;µ s i,σs i )z s,i i=1 p(s) = M s p(z s )p(s z s ) = πi s N(s;µs i,σs i )z s,i z s M s p(z s ) = i=1 π z s,i i i=1 z s is a hidden variable indicate which Gaussian mixture component is active. Remark: {z s,i } i 1...Ms are independent 10/37
11 Hidden Markov Model Graphic Model z 1 z 2 z i 1 z i z i+1 s 1 s 2 s i 1 s i s i+1 P(z n z n 1,...,z 1 ) = P(z n z n 1 ) HMM is often used in speaker recognition. 11/37
12 Hidden Markov Model We have the following joint probability: ( N p(x,z θ) = p(z 1 π) n=2 where A is transition probability matrix and p(z n z n 1,A) = p(z 1 π) = K k=1 p(x n z n,φ) = ) N p(z n z n 1,A) K K m=1 A z n 1,jz n,k jk k=1 j=1 k, π k = 1 π z 1k k K p(x n φ k ) z nk k=1 p(x m z m,φ) 12/37
13 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 13/37
14 Supervector definition Given the GMM mean vector (m c ) F 1, c {1,...,C}, C is the total number of mixture components, F is the dimension of feature vector Supervector is: m CF 1 = (m T 1,...,mT c ) 14/37
15 Speaker and Channel Dependent Supervector M h M h CHANNEL SPACE S C SPEAKER SPACE M h is the speaker-and channel-dependent supervector 15/37
16 Notations S: speaker ID Speaker factors: components of y(s) Channel factors: components of x h (s) Speaker space: affine translating the range of vv by m Channel space: the range of uu Loading matrix for speaker factors and channel factors: v and u h = 1,,H(s): one index from set of recordings for a speaker s C: total number of mixture components for a fixed GMM structure F: dimension of the acoustic feature vectors R C : channel rank R S : speaker rank Σ(s): given speaker s and recording h, the covariance of the observation from GMM d: given speaker s, the covariance of the observation from GMM 16/37
17 Joint Factor Analysis Model JFA model M(s) = m+vy(s) +dz(s) M h (s) = M(s)+ux h (s) m C F : Given a HMM/GMM structure with C mixture components, we concatenate the mean vectors m 1,...,m C together then obtain m M(s): single speaker-dependent supervector M h (s): speaker-and-channel dependent u and v are speaker independent d is a block diagonal matrix z is normal 17/37
18 JFA model M h CHANNEL SPACE M(s) ux h (s) SPEAKER SPACE 18/37
19 Problem Purpose: estimate the hyperparameters Λ = (m,u,v,d,σ). The number of GMM component is large. C = 2048 The dimension of the feature vector is F = 39 C F = = m and Σ Problem Σ is very large and it is not full rank, how to estimate? 19/37
20 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 20/37
21 Principle Components Analysis Technique x 2 x n u 1 x n PCA technique is to find a principal subspace (magenta line), s.t. the variance of the projected points ( x n ) are maximized. x 1 21/37
22 Maximum Variance Formulation Find the principle components for principle subspace Given feature vectors as observations {(x n ) N 1 }, n = 1,...,N, we want to find the principle subspace with M basis, M < N 22/37
23 Maximum Variance Formulation Find the principle components for principle subspace Given feature vectors as observations {(x n ) N 1 }, n = 1,...,N, we want to find the principle subspace with M basis, M < N Sample mean x and sample covariance S: x = 1 N S = 1 N N n=1 x n N (x n x)(x n x) T n=1 Let the M basis for the principle subspace be u 1,...,u M and u T i u = 1, i [M] P N M = [u 1,...,u M ] N M 23/37
24 Maximum Variance Formulation Optimization problem find the 1 st principle component By Lagrange methode: maximize: take derivative maxu T 1 Su 1 u T 1 u 1 = 1 u T 1 Su 1 +λ 1 (1 u T 1 u 1 ) u T 1 Su 1 +λ 1 (1 u T 1 u 1) u 1 = (S+S T )u 1 +λ 1 ( 2u 1 ) = 2Su 1 2λ 1 u 1 = 0 = Su 1 = λ 1 u 1 Solution: λ 1 is the largest eigenvalue of S, the correspond u 1 is the first principle component 24/37
25 Maximum Variance Formulation Find M principle components: find M largest eigenvalues, and their correspond eigenvectors u i, i [M], such that: u T i u i = 1, i [M] u i u j, i j Eigen-decomposition S, find the M largest eigenvalues, decreasing sorted. Then, find the u 1,u 2,...,u M. Remark: [u T 1,u T 2,...,u T M ]T S[u 1,u 2,...,u M ] = P T SP 25/37
26 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 26/37
27 Probabilistic Model PCA model D >> M x D 1 = W D M z M 1 +µ+ǫ x is D-dimension observation vector z is M-dimension hidden variable We are given the following probability distributions: p(z) = N(z 0,I) p(x z) = N(x Wz+µ,σ 2 I) 27/37
28 Probabilistic PCA p(x) is Gaussian p(x) = p(x z)p(z)dz = N(x µ,c), C = WW T +σ 2 I mean and variance of p(x) E[x] = E[Wz+µ+ǫ] = µ cov[x] = E[(Wz+ǫ)(Wz+ǫ) T ] = E[Wzz T W T ]+E[ǫǫ T ] = WW T +σ 2 I 28/37
29 Probabilistic PCA zn σ 2 µ W xn The graph shows for each observation x n is associate with a value of latent variable z n x n can be obtained by marginalization over z n. Using EM algorithm to estimate the parameters in PCA model (Train PCA model) 29/37
30 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 30/37
31 5 steps for JFA Speaker Verification System 1 Train the UBM model 2 Train JFA/PCA model: estimate speaker independent hyperparameters Λ = (m,u,v,d,σ) from a large database in which each speaker is recorded in multiple sessions 3 Adapt Λ from one speaker population to another 4 Enrolling a speaker: estimate the speaker-independent hyperparameters Λ(s) = (m(s), u(s), v(s), d(s), Σ(s)) 5 Test: Given test utterance χ and hypothesized speaker, where X are observations. log P Λ(s)(X) P Λ (X) 31/37
32 Outline 1 Speaker Verification Baseline System Session Variation 2 Joint Factor Analysis Hidden Markov Model Factor Analysis Model Principal Components Analysis (PCA) Probabilistic PCA 3 JFA for Speaker Verification General Steps Hyperparameter estimation 32/37
33 Train the JFA/PCA model Estimate Λ Training set: several speakers with multiple recordings for each speaker Use EM algorithms to estimate Λ Maximum Likelihood Approach (slow) Divergence minimization approach (faster, well initialized) Both algorithm are to fit entire collection of speakers in the training data Total likelihood s P Λ(X(s)), s ranges over the speakers in the training set. It increases from 1 iteration to the next. 33/37
34 Adapt from one speaker population to another Adaptation is necessary since data set is limit. For a given speaker, there are at most 2 recordings. Keep channel space related hyperparameters fixed (u and Σ h ), re-estimate only the speaker space hyperparameters (m,v,d). Remark: Assume channel space related hyperparameters are speaker independent 34/37
35 Enroll a target speaker Estimate Λ(s) Recall JFA model: M(s) = m+vy(s) +dz(s) M h (s) = M(s)+ux h (s) Calculate the posterior distribution M(s) Adjusting the Λ(s) to fit this posterior Adopt minimum divergence approach 35/37
36 Likelihood Function Hyperparameters Λ = (m,u,v,d,σ). P Λ (X(s)) = P Λ (X(s) X)N(X 0,I)dX where: X(s) (observable) is the collections of labeled frames for recording h ( ) T X(s) = X 1 (s),...,x H(s) (s) X(s) (unobservable) is the vector of hidden variables ( ) T X(s) = x 1 (s),...,x H(s),y(s),z(s) N(X 0,I) is the standard Gaussian kernel N(X 0,I) = N(x 1 0,I)...N(x H(s) 0,I)N(y 0,I)N(z 0,I) 36/37
37 Likelihood ratio Given speech data X uttered by speaker t Test H 0 = {t = s} against H 1 = {t s} 1 T log P Λ s (X) P Λ (X) 37/37
i-vector and GMM-UBM Bie Fanhu CSLT, RIIT, THU
i-vector and GMM-UBM Bie Fanhu CSLT, RIIT, THU 2013-11-18 Framework 1. GMM-UBM Feature is extracted by frame. Number of features are unfixed. Gaussian Mixtures are used to fit all the features. The mixtures
More informationSession Variability Compensation in Automatic Speaker Recognition
Session Variability Compensation in Automatic Speaker Recognition Javier González Domínguez VII Jornadas MAVIR Universidad Autónoma de Madrid November 2012 Outline 1. The Inter-session Variability Problem
More informationUncertainty Modeling without Subspace Methods for Text-Dependent Speaker Recognition
Uncertainty Modeling without Subspace Methods for Text-Dependent Speaker Recognition Patrick Kenny, Themos Stafylakis, Md. Jahangir Alam and Marcel Kockmann Odyssey Speaker and Language Recognition Workshop
More informationspeaker recognition using gmm-ubm semester project presentation
speaker recognition using gmm-ubm semester project presentation OBJECTIVES OF THE PROJECT study the GMM-UBM speaker recognition system implement this system with matlab document the code and how it interfaces
More informationHeeyoul (Henry) Choi. Dept. of Computer Science Texas A&M University
Heeyoul (Henry) Choi Dept. of Computer Science Texas A&M University hchoi@cs.tamu.edu Introduction Speaker Adaptation Eigenvoice Comparison with others MAP, MLLR, EMAP, RMP, CAT, RSW Experiments Future
More informationSupport Vector Machines using GMM Supervectors for Speaker Verification
1 Support Vector Machines using GMM Supervectors for Speaker Verification W. M. Campbell, D. E. Sturim, D. A. Reynolds MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02420 Corresponding author e-mail:
More informationAround the Speaker De-Identification (Speaker diarization for de-identification ++) Itshak Lapidot Moez Ajili Jean-Francois Bonastre
Around the Speaker De-Identification (Speaker diarization for de-identification ++) Itshak Lapidot Moez Ajili Jean-Francois Bonastre The 2 Parts HDM based diarization System The homogeneity measure 2 Outline
More informationFront-End Factor Analysis For Speaker Verification
IEEE TRANSACTIONS ON AUDIO, SPEECH AND LANGUAGE PROCESSING Front-End Factor Analysis For Speaker Verification Najim Dehak, Patrick Kenny, Réda Dehak, Pierre Dumouchel, and Pierre Ouellet, Abstract This
More informationECE 661: Homework 10 Fall 2014
ECE 661: Homework 10 Fall 2014 This homework consists of the following two parts: (1) Face recognition with PCA and LDA for dimensionality reduction and the nearest-neighborhood rule for classification;
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 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 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 informationA Small Footprint i-vector Extractor
A Small Footprint i-vector Extractor Patrick Kenny Odyssey Speaker and Language Recognition Workshop June 25, 2012 1 / 25 Patrick Kenny A Small Footprint i-vector Extractor Outline Introduction Review
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 informationAutomatic Speech Recognition (CS753)
Automatic Speech Recognition (CS753) Lecture 21: Speaker Adaptation Instructor: Preethi Jyothi Oct 23, 2017 Speaker variations Major cause of variability in speech is the differences between speakers Speaking
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 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 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 informationIndependent Component Analysis and Unsupervised Learning. Jen-Tzung Chien
Independent Component Analysis and Unsupervised Learning Jen-Tzung Chien TABLE OF CONTENTS 1. Independent Component Analysis 2. Case Study I: Speech Recognition Independent voices Nonparametric likelihood
More informationHidden Markov Models and Gaussian Mixture Models
Hidden Markov Models and Gaussian Mixture Models Hiroshi Shimodaira and Steve Renals Automatic Speech Recognition ASR Lectures 4&5 23&27 January 2014 ASR Lectures 4&5 Hidden Markov Models and Gaussian
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 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 informationIndependent Component Analysis and Unsupervised Learning
Independent Component Analysis and Unsupervised Learning Jen-Tzung Chien National Cheng Kung University TABLE OF CONTENTS 1. Independent Component Analysis 2. Case Study I: Speech Recognition Independent
More informationSpeaker Verification Using Accumulative Vectors with Support Vector Machines
Speaker Verification Using Accumulative Vectors with Support Vector Machines Manuel Aguado Martínez, Gabriel Hernández-Sierra, and José Ramón Calvo de Lara Advanced Technologies Application Center, Havana,
More informationReformulating the HMM as a trajectory model by imposing explicit relationship between static and dynamic features
Reformulating the HMM as a trajectory model by imposing explicit relationship between static and dynamic features Heiga ZEN (Byung Ha CHUN) Nagoya Inst. of Tech., Japan Overview. Research backgrounds 2.
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 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 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 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 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 informationUnsupervised Learning
2018 EE448, Big Data Mining, Lecture 7 Unsupervised Learning Weinan Zhang Shanghai Jiao Tong University http://wnzhang.net http://wnzhang.net/teaching/ee448/index.html ML Problem Setting First build and
More informationData Analysis and Manifold Learning Lecture 6: Probabilistic PCA and Factor Analysis
Data Analysis and Manifold Learning Lecture 6: Probabilistic PCA and Factor Analysis Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture
More informationMaximum variance formulation
12.1. Principal Component Analysis 561 Figure 12.2 Principal component analysis seeks a space of lower dimensionality, known as the principal subspace and denoted by the magenta line, such that the orthogonal
More informationCOMS 4771 Probabilistic Reasoning via Graphical Models. Nakul Verma
COMS 4771 Probabilistic Reasoning via Graphical Models Nakul Verma Last time Dimensionality Reduction Linear vs non-linear Dimensionality Reduction Principal Component Analysis (PCA) Non-linear methods
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 13: Learning in Gaussian Graphical Models, Non-Gaussian Inference, Monte Carlo Methods Some figures
More informationFactor Analysis and Kalman Filtering (11/2/04)
CS281A/Stat241A: Statistical Learning Theory Factor Analysis and Kalman Filtering (11/2/04) Lecturer: Michael I. Jordan Scribes: Byung-Gon Chun and Sunghoon Kim 1 Factor Analysis Factor analysis is used
More informationBayesian Analysis of Speaker Diarization with Eigenvoice Priors
Bayesian Analysis of Speaker Diarization with Eigenvoice Priors Patrick Kenny Centre de recherche informatique de Montréal Patrick.Kenny@crim.ca A year in the lab can save you a day in the library. Panu
More informationCOMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017
COMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University PRINCIPAL COMPONENT ANALYSIS DIMENSIONALITY
More informationProbabilistic Latent Semantic Analysis
Probabilistic Latent Semantic Analysis Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
More informationHidden Markov Models in Language Processing
Hidden Markov Models in Language Processing Dustin Hillard Lecture notes courtesy of Prof. Mari Ostendorf Outline Review of Markov models What is an HMM? Examples General idea of hidden variables: implications
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 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 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 informationPCA and admixture models
PCA and admixture models CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar, Alkes Price PCA and admixture models 1 / 57 Announcements HW1
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 informationSpeaker recognition by means of Deep Belief Networks
Speaker recognition by means of Deep Belief Networks Vasileios Vasilakakis, Sandro Cumani, Pietro Laface, Politecnico di Torino, Italy {first.lastname}@polito.it 1. Abstract Most state of the art speaker
More informationNote Set 5: Hidden Markov Models
Note Set 5: Hidden Markov Models Probabilistic Learning: Theory and Algorithms, CS 274A, Winter 2016 1 Hidden Markov Models (HMMs) 1.1 Introduction Consider observed data vectors x t that are d-dimensional
More informationLinear Dimensionality Reduction
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Principal Component Analysis 3 Factor Analysis
More informationSpeaker Representation and Verification Part II. by Vasileios Vasilakakis
Speaker Representation and Verification Part II by Vasileios Vasilakakis Outline -Approaches of Neural Networks in Speaker/Speech Recognition -Feed-Forward Neural Networks -Training with Back-propagation
More informationLecture 7: Con3nuous Latent Variable Models
CSC2515 Fall 2015 Introduc3on to Machine Learning Lecture 7: Con3nuous Latent Variable Models All lecture slides will be available as.pdf on the course website: http://www.cs.toronto.edu/~urtasun/courses/csc2515/
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 informationHidden Markov Models and Gaussian Mixture Models
Hidden Markov Models and Gaussian Mixture Models Hiroshi Shimodaira and Steve Renals Automatic Speech Recognition ASR Lectures 4&5 25&29 January 2018 ASR Lectures 4&5 Hidden Markov Models and Gaussian
More informationRobust Speaker Identification
Robust Speaker Identification by Smarajit Bose Interdisciplinary Statistical Research Unit Indian Statistical Institute, Kolkata Joint work with Amita Pal and Ayanendranath Basu Overview } } } } } } }
More informationWeighted Finite-State Transducers in Computational Biology
Weighted Finite-State Transducers in Computational Biology Mehryar Mohri Courant Institute of Mathematical Sciences mohri@cims.nyu.edu Joint work with Corinna Cortes (Google Research). 1 This Tutorial
More informationPCA & ICA. CE-717: Machine Learning Sharif University of Technology Spring Soleymani
PCA & ICA CE-717: Machine Learning Sharif University of Technology Spring 2015 Soleymani Dimensionality Reduction: Feature Selection vs. Feature Extraction Feature selection Select a subset of a given
More informationCourse 495: Advanced Statistical Machine Learning/Pattern Recognition
Course 495: Advanced Statistical Machine Learning/Pattern Recognition Goal (Lecture): To present Probabilistic Principal Component Analysis (PPCA) using both Maximum Likelihood (ML) and Expectation Maximization
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 informationp(d θ ) l(θ ) 1.2 x x x
p(d θ ).2 x 0-7 0.8 x 0-7 0.4 x 0-7 l(θ ) -20-40 -60-80 -00 2 3 4 5 6 7 θ ˆ 2 3 4 5 6 7 θ ˆ 2 3 4 5 6 7 θ θ x FIGURE 3.. The top graph shows several training points in one dimension, known or assumed to
More informationHidden Markov Models
CS769 Spring 2010 Advanced Natural Language Processing Hidden Markov Models Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu 1 Part-of-Speech Tagging The goal of Part-of-Speech (POS) tagging is to label each
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 8 Continuous Latent Variable
More informationThe Expectation Maximization or EM algorithm
The Expectation Maximization or EM algorithm Carl Edward Rasmussen November 15th, 2017 Carl Edward Rasmussen The EM algorithm November 15th, 2017 1 / 11 Contents notation, objective the lower bound functional,
More informationMixtures of Gaussians with Sparse Regression Matrices. Constantinos Boulis, Jeffrey Bilmes
Mixtures of Gaussians with Sparse Regression Matrices Constantinos Boulis, Jeffrey Bilmes {boulis,bilmes}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UW Electrical Engineering
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 informationDimensionality Reduction. CS57300 Data Mining Fall Instructor: Bruno Ribeiro
Dimensionality Reduction CS57300 Data Mining Fall 2016 Instructor: Bruno Ribeiro Goal } Visualize high dimensional data (and understand its Geometry) } Project the data into lower dimensional spaces }
More informationManifold Learning for Signal and Visual Processing Lecture 9: Probabilistic PCA (PPCA), Factor Analysis, Mixtures of PPCA
Manifold Learning for Signal and Visual Processing Lecture 9: Probabilistic PCA (PPCA), Factor Analysis, Mixtures of PPCA Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inria.fr http://perception.inrialpes.fr/
More informationLecture 6: April 19, 2002
EE596 Pat. Recog. II: Introduction to Graphical Models Spring 2002 Lecturer: Jeff Bilmes Lecture 6: April 19, 2002 University of Washington Dept. of Electrical Engineering Scribe: Huaning Niu,Özgür Çetin
More informationEigenvoice Speaker Adaptation via Composite Kernel PCA
Eigenvoice Speaker Adaptation via Composite Kernel PCA James T. Kwok, Brian Mak and Simon Ho Department of Computer Science Hong Kong University of Science and Technology Clear Water Bay, Hong Kong [jamesk,mak,csho]@cs.ust.hk
More informationCSC411 Fall 2018 Homework 5
Homework 5 Deadline: Wednesday, Nov. 4, at :59pm. Submission: You need to submit two files:. Your solutions to Questions and 2 as a PDF file, hw5_writeup.pdf, through MarkUs. (If you submit answers to
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 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 7 Approximate
More informationCSC411: Final Review. James Lucas & David Madras. December 3, 2018
CSC411: Final Review James Lucas & David Madras December 3, 2018 Agenda 1. A brief overview 2. Some sample questions Basic ML Terminology The final exam will be on the entire course; however, it will be
More informationAdvanced Introduction to Machine Learning
10-715 Advanced Introduction to Machine Learning Homework 3 Due Nov 12, 10.30 am Rules 1. Homework is due on the due date at 10.30 am. Please hand over your homework at the beginning of class. Please see
More informationL11: Pattern recognition principles
L11: Pattern recognition principles Bayesian decision theory Statistical classifiers Dimensionality reduction Clustering This lecture is partly based on [Huang, Acero and Hon, 2001, ch. 4] Introduction
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 informationComputer Vision Group Prof. Daniel Cremers. 11. Sampling Methods: Markov Chain Monte Carlo
Group Prof. Daniel Cremers 11. Sampling Methods: Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative
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 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 informationIntroduction to Probabilistic Graphical Models: Exercises
Introduction to Probabilistic Graphical Models: Exercises Cédric Archambeau Xerox Research Centre Europe cedric.archambeau@xrce.xerox.com Pascal Bootcamp Marseille, France, July 2010 Exercise 1: basics
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 informationIntroduction PCA classic Generative models Beyond and summary. PCA, ICA and beyond
PCA, ICA and beyond Summer School on Manifold Learning in Image and Signal Analysis, August 17-21, 2009, Hven Technical University of Denmark (DTU) & University of Copenhagen (KU) August 18, 2009 Motivation
More informationHidden Markov Bayesian Principal Component Analysis
Hidden Markov Bayesian Principal Component Analysis M. Alvarez malvarez@utp.edu.co Universidad Tecnológica de Pereira Pereira, Colombia R. Henao rhenao@utp.edu.co Universidad Tecnológica de Pereira Pereira,
More informationCISC 889 Bioinformatics (Spring 2004) Hidden Markov Models (II)
CISC 889 Bioinformatics (Spring 24) Hidden Markov Models (II) a. Likelihood: forward algorithm b. Decoding: Viterbi algorithm c. Model building: Baum-Welch algorithm Viterbi training Hidden Markov models
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 informationJoint Optimization of Segmentation and Appearance Models
Joint Optimization of Segmentation and Appearance Models David Mandle, Sameep Tandon April 29, 2013 David Mandle, Sameep Tandon (Stanford) April 29, 2013 1 / 19 Overview 1 Recap: Image Segmentation 2 Optimization
More informationIntroduction to Machine Learning. PCA and Spectral Clustering. Introduction to Machine Learning, Slides: Eran Halperin
1 Introduction to Machine Learning PCA and Spectral Clustering Introduction to Machine Learning, 2013-14 Slides: Eran Halperin Singular Value Decomposition (SVD) The singular value decomposition (SVD)
More informationExperiments with a Gaussian Merging-Splitting Algorithm for HMM Training for Speech Recognition
Experiments with a Gaussian Merging-Splitting Algorithm for HMM Training for Speech Recognition ABSTRACT It is well known that the expectation-maximization (EM) algorithm, commonly used to estimate hidden
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 295-P, Spring 213 Prof. Erik Sudderth Lecture 11: Inference & Learning Overview, Gaussian Graphical Models Some figures courtesy Michael Jordan s draft
More informationAn Integration of Random Subspace Sampling and Fishervoice for Speaker Verification
Odyssey 2014: The Speaker and Language Recognition Workshop 16-19 June 2014, Joensuu, Finland An Integration of Random Subspace Sampling and Fishervoice for Speaker Verification Jinghua Zhong 1, Weiwu
More informationRandomized Algorithms
Randomized Algorithms Saniv Kumar, Google Research, NY EECS-6898, Columbia University - Fall, 010 Saniv Kumar 9/13/010 EECS6898 Large Scale Machine Learning 1 Curse of Dimensionality Gaussian Mixture Models
More informationLinear Dynamical Systems (Kalman filter)
Linear Dynamical Systems (Kalman filter) (a) Overview of HMMs (b) From HMMs to Linear Dynamical Systems (LDS) 1 Markov Chains with Discrete Random Variables x 1 x 2 x 3 x T Let s assume we have discrete
More informationPrincipal Component Analysis and Linear Discriminant Analysis
Principal Component Analysis and Linear Discriminant Analysis Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL 60208 http://www.eecs.northwestern.edu/~yingwu 1/29
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 informationUniversity of Cambridge. MPhil in Computer Speech Text & Internet Technology. Module: Speech Processing II. Lecture 2: Hidden Markov Models I
University of Cambridge MPhil in Computer Speech Text & Internet Technology Module: Speech Processing II Lecture 2: Hidden Markov Models I o o o o o 1 2 3 4 T 1 b 2 () a 12 2 a 3 a 4 5 34 a 23 b () b ()
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 informationFast speaker diarization based on binary keys. Xavier Anguera and Jean François Bonastre
Fast speaker diarization based on binary keys Xavier Anguera and Jean François Bonastre Outline Introduction Speaker diarization Binary speaker modeling Binary speaker diarization system Experiments Conclusions
More informationMixtures of Gaussians with Sparse Structure
Mixtures of Gaussians with Sparse Structure Costas Boulis 1 Abstract When fitting a mixture of Gaussians to training data there are usually two choices for the type of Gaussians used. Either diagonal or
More informationImage Analysis & Retrieval. Lec 14. Eigenface and Fisherface
Image Analysis & Retrieval Lec 14 Eigenface and Fisherface Zhu Li Dept of CSEE, UMKC Office: FH560E, Email: lizhu@umkc.edu, Ph: x 2346. http://l.web.umkc.edu/lizhu Z. Li, Image Analysis & Retrv, Spring
More informationBayesian Networks BY: MOHAMAD ALSABBAGH
Bayesian Networks BY: MOHAMAD ALSABBAGH Outlines Introduction Bayes Rule Bayesian Networks (BN) Representation Size of a Bayesian Network Inference via BN BN Learning Dynamic BN Introduction Conditional
More informationExpectation Maximization
Expectation Maximization Machine Learning CSE546 Carlos Guestrin University of Washington November 13, 2014 1 E.M.: The General Case E.M. widely used beyond mixtures of Gaussians The recipe is the same
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 information