Lecture Schedule (updated throughout the term)
Lecture Materials
- [Notation]
- Unit 1: Introductory Material
- Knowledge Based System(KBS) versus Machine Learning(ML) (Discrete versus Continuous Math)
- Addition/Multiplication with KBS vs ML
- Nearest Neighbors
- Polynomial Interpolation, Overfitting, Underfitting, Regularization
- Noise versus Features
- Monomial Basis and Singular Matrices
- Error Sources
- Polynomial Interpolation (Lagrange/Newton basis functions)
- Representation Theory, CNNs for cloth
- Unit 2: Solving Linear Systems
- Systems of linear equations
- Normalization
- Rank and Solvability
- Square, Diagonal, Upper Triangular, and Lower Triangular Matrices
- Gaussian Elimination
- LU Factorization
- Pivoting
- Sparsity
- Unit 3: Understanding Matrices
- Eigenvalues, eigenvectors, and matrix preconditioning
- Singular Value Decomposition
- Singular Matrices and Condition Number
- Vector and Matrix Norms
- Unit 4: Special Matrices
- Diagonally Dominant Matrices
- Symmetric Positive Definite (SPD) Matrices
- Cholesky Factorization
- Symmetric Approximation
- Unit 5: Iterative Solvers
- Continuous Collision Detection
- Residual
- Line Search, Steepest Descent
- Conjugate Gradient (CG) Method
- Unit 6: Local Approximations
- Taylor Series
- Sampling
- Well-Resolved Functions
- Piecewise Approximation: Constant, Linear, Higher Order Interpolation
- Cubic Splines
- 2D Image Segmentation
- Unit 7: Curse of Dimensionality
- Numerical Integration
- Newton-Cotes Quadrature
- Gaussian Quadrature
- Domain Approximation Errors
- Curse of Dimensionality
- Monte Carlo Methods
- Unit 8: Least Squares
- Eliminating Basis Functions
- Solving Linear Systems
- Weighted Minimization
- Least Squares
- Unit 9: Basic Optimization
- Critical Points
- Classifying Critical Points
- Quadratic Form
- Least Squares
- Unit 10: Solving Least Squares
- Normal Equations
- Condition Number
- Summary
- Understanding Least Squares
- Orthogonal Matrices
- Gram-Schmidt
- QR Factorization
- Householder
- Unit 11: Zero Singular Values
- Underdetermined Systems
- Minimum Norm Solution
- Pseudo-Inverse
- Sum of Rank One Matrices
- Matrix Approximation
- Principal Component Analysis (PCA)
- Rank One Updates
- Condition Number of Eigenproblems
- QR Iteration
- Power Method
- Unit 12: Regularization
- Adding an Identity Matrix
- Full Rank Scenario
- Rank Deficient Scenario
- Initial Guess
- Iterative Approach
- Adding a Diagonal Matrix
- Column Space Search Method
- Unit 13: Optimization
- Function Approximation
- Choice of Norm
- Unit 14: Nonlinear Systems
- Jacobian Matrix
- Linearization
- Iterative Solver
- Line Search with Search Directions
- Unit 15: Root finding
- Fixed Point Iteration
- Convergence Rate
- Newton's Method, Secant Method, Bisection Method, and Mixed Methods
- Nonlinear Systems Problems and Optimization
- Unit 16: 1D Optimization
- Leveraging Root Finding
- Unimodal
- Successive Parabolic Interpolation, Golden Section Search, and Mixed Methods
- Nonlinear Systems Problems and Optimization
- Unit 17: Computing Derivatives
- Differentiability and Smoothness
- Activation Functions
- Heaviside, Sigmoid, Rectifier, Softplus, Leaky Rectifier, Arg/Soft Max
- Inequality Constrained Optimization
- Binary Classification and Soft-margin
- Symbolic Differentiation
- Finite Differences
- Automatic Differentiation
- Dropout
- Function Layers
- Unit 18: Avoiding Derivatives
- Broyden's Method
- Symmetric Rank 1
- DFP (Davidon-Fletcher-Powell)
- BFGS (Broyden-Fletcher-Goldfarb-Shanno)
- L-BFGS (Limited Memory BFGS)
- Nonlinear Least Squares
- Gauss Newton and Weighted Gauss Newton
- Levenberg-Marquardt (Damped Nonlinear Least Squares)
- Gradient/Steepest Descent
- Coordinate Descent
- Unit 19: Descent Methods
- Gradient/Steepest Descent
- Steepest Descent for Quadratic Form
- Steepest Descent for Nonlinear Least Squares
- Unit 20: Momentum Methods
- Ordinary Differential Equations (ODEs)
- Gradient Flow
- Families of Solutions and Initial Value Problems (IVPs)
- Well Posedness, Stability, Acccuracy
- Forward Euler, Runge-Kutta, TVD-RK
- Stability Analysis
- Adaptive Time Stepping and Adaptive Learning Rates
- Implicit Methods, Backward Euler, Trapezoidal Rule
- Implicit Stochastic Gradient Descent (ISGD)
- Momentum, Newton's Second Law, First Order Systems
- Momentum Method, Nesterov Momentum, ADAM
- Newmark Methods