Mathematics of Machine Learning and Signal Recognition COMS E4995 Instructor: Prof. Homayoon Beigi <firstname.lastname@example.org> (email@example.com) Textbooks: Required: H. Beigi, “Fundamentals of Speaker Recognition, Springer, New York, 2011. Reference Books:
K.P. Murphy, “Machine Learning, A Probabilistic Perspective,” The MIT Press, Cambridge, MA, 2012. H. Beigi, “Fundamentals of Speaker Recognition,” Springer, New York, 2nd Edition, 2022.
M. Loève, “Probability Theory,” Springer, New York, 4th Edition, 1977.
P.R. Halmos, “Measure Theory,” Springer, New York, 1974.
I.T. Jolliffe, “Principal Component Analysis,” Springer, New York, 2nd Edition, 2002.
R. Courant and D. Hilbert, “Methods of Mathematical Physics,” John Wiley & Sons, New York, 1989.
C. F. Gerald and P. O. Wheatley, “Applied Numerical Analysis,” Pearson College Div., 7th Edition, New York, 2003.
G.J. McLachlen and T. Krishnan, “The EM Algorithm and Extensions,” John Wiley & Sons, 2nd Edition, New York, 2008.
W.E. Boyce and R.C. DiPrima, “Elementary Differential Equations and Boundary Value Problems,” John Wiley & Sons, 11th Edition, New York, 2017.
P.W. Berg and J.L. McGregor, Holden Day, San Francisco, 1966.
R. Fletcher, “Practical Methods of Optimization,” John Wiley & Sons, 2nd Edition, New York, 2000.
Grading: Homework (20%): - Problems and coding assignments. Midterm (20%): - Coding assignment and Problems. Project Proposal (10%): - 2-page proposal, including state of the art and proposed methodology.
Final Project (50%): 35% - Test/report of the methodology and results. 15% - Code and Results. Course Description: Mathematics of Machine Learning and Signal Recognition provides the background mathematical background for addressing in-depth problems in machine learning, as well as the treatment of signals, especially time-dependent signals, specifically non-stationary time-dependent signals – although spatial signals such as images are also considered. The course will provides the essentials of several mathematical disciplines which are used in the formulation and solution of the problems in the above fields. These disciplines include Linear Algebra and Numerical Methods, Complex Variable Theory, Measure and Probability Theory (as well as statistics), Information Theory, Metrics and Divergences, Linear Ordinary and Separable Partial Differential Equations of Interest, Integral Transforms, Decision Theory, Transformations, Nonlinear Optimization Theory, and Neural Network Learning Theory. The requirements are Advanced Calculus and Linear Algebra. Knowledge of Differential Equations would be helpful. Lectures: Week 1 - Linear Algebra and Numerical Methods Basic Definitions Norms Gram-Schmidt Orthogonalization Ordinary Gram-Schmidt Orthogonalization Modified Gram-Schmidt Orthogonalization Sherman-Morrison Inversion Formula Vector Representation under a Set of Normal Conjugate Direction Stochastic Matrix Linear Equations Week 2 - Complex Variable Theory Complex Variables Limits Continuity and Forms of Discontinuity Convexity and Concavity of Functions Odd, Even and Periodic Functions Differentiation Analyticity Integration Weeks 3 & 4
- Complex Variable Theory (Continued)
Power Series Expansion of Functions
Relations Between Functions
Orthogonality of Functions
- Measure and Probability Theory and Statistics Set Theory Equivalence and Partitions R-Rough Sets (Rough Sets) Fuzzy Sets Measure Theory Measure Multiple Dimensional Spaces
Metric Space Banach Space (Normed Vector Space) Inner Product Space (Dot Product Space) Infinite Dimensional Spaces (Pre-Hilbert and Hilbert) Probability Measure Integration Functions Probability Density Function Densities in the Cartesian Product Space Cumulative Distribution Function Function Spaces Transformations Statistical Moments Discrete Random Variables Combinations of Random Variables Convergence of a Sequence Sufficient Statistics Moment Estimation Estimating the Mean Law of Large Numbers (LLN) Different Types of Mean Estimating the Variance Multi-Variate Normal Distribution Weeks 5 - Information Theory Sources The Relation between Uncertainty and Choice Discrete Sources Entropy or Uncertainty Generalized Entropy Information The Relation between Information and Entropy Discrete Channels Continuous Sources Differential Entropy (Continuous Entropy) Relative Entropy Mutual Information Fisher Information Weeks 6 - Metrics and Divergences Distance (Metric) Distance Between Sequences Distance Between Vectors and Sets of Vectors Hellinger Distance Divergences and Directed Divergences Kullback-Leibler’s Directed Divergence Jeffreys’ Divergence Bhattacharyya Divergence Matsushita Divergence F-Divergence δ -Divergence χ α Directed Divergence Weeks 7 and 8 - Review of Linear Differential Equations (Ordniary and Separable Partial) - Integral Transforms Integral Equations Kernel Functions Hilbert’s Expansion Theorem Eigenvalues and Eigenfunctions of the Kernel Fourier Series Expansion Convergence of the Fourier Series Parseval’s Theorem Wavelet Series Expansion The Laplace Transform Inversion Some Useful Transforms Complex Fourier Transform (Fourier Integral Transform) Translation Scaling Symmetry Table Time and Complex Scaling and Shifting Convolution Correlation Parseval’s Theorem Power Spectral Density One-Sided Power Spectral Density PSD-per-unit-time Wiener-Khintchine Theorem Discrete Fourier Transform (DFT) Inverse Discrete Fourier Transform (IDFT) Periodicity Plancherel and Parseval’s Theorem Power Spectral Density (PSD) Estimation Fast Fourier Transform (FFT) Discrete-Time Fourier Transform (DTFT) Power Spectral Density (PSD) Estimation Complex Short-Time Fourier Transform (STFT) Discrete-Time Short-Time Fourier Transform DTSTFT Discrete Short-Time Fourier Transform DSTFT Discrete Cosine Transform (DCT) Efficient DCT Computation Week 9 - Difference Equations and The z-Transform Difference Equations z-Transform – Definition Translation Scaling Shifting – Time Lag Shifting – Time Lead Complex Translation Initial Value Theorem Final Value Theorem Real Convolution Theorem Inversion - Cepstrum Week 10 - Decision Theory Hypothesis Testing Bayesian Decision Theory Bayesian Classifier Decision Trees - Unsupervised Clustering and Learning Vector Quantization (VQ) Basic Clustering Techniques Estimation using Incomplete Data - Parameter Estimation Maximum Likelihood Estimation (MLE, MLLR, fMLLR) Maximum A-Posteriori (MAP) Estimation Maximum Entropy Estimation Minimum Relative Entropy Estimation Maximum Mutual Information Estimation (MMIE) Model Selection (AIC and BIC) Week 11 - Transformation Principal Component Analysis (PCA) Linear Discriminant Analysis (LDA) Factor Analysis (FA) Probabilistic Linear Discriminant Analysis (PLDA) - Hidden Markov Modeling (HMM) Memoryless Models Discrete Markov Chains Markov Models Hidden Markov Models Model Design and States Training and Decoding Gaussian Mixture Models (GMM) Practical Issues Week 12 - Nonlinear Optimization Theory Gradient-Based Optimization The Steepest Descent Technique Newton’s Minimization Technique Quasi-Newton or Large Step Gradient Techniques Conjugate Gradient Methods Gradient-Free Optimization Search Methods Gradient-Free Conjugate Direction Methods The Line Search Sub-Problem Practical Considerations Large-Scale Optimization Numerical Stability Nonsmooth Optimization Constrained Optimization The Lagrangian and Lagrange Multipliers Duality Global Convergence Week 13 - Neural Network Learning Perceptron Feedforward Networks Time-Delay Neural Networks (TDNN) Convolutional Neural Networks (CNN) Recurrent Neural Networks (RNN) Long-Short Term Memory Networks (LSTM) End-to-End Sequence (Encoder/Decoder) Neural Networks Embeddings and Transfer Learning
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