Enrollment year
2022/2023
Academic discipline
MAT/09 (OPERATIONAL RESEARCH)
Department
DEPARTMENT OF ECONOMICS AND MANAGEMENT
Curriculum
PERCORSO COMUNE
Period
2nd semester (20/02/2023 - 27/05/2023)
Lesson hours
48 lesson hours
Prerequisites
Basic knowledge of Mathematical Analysis, Linear Algebra, Probability, Numerical Analysis and Programming
Learning outcomes
This course will review the theory and applications of Data Analysis and Numerical Optiization, illustrating the main results and the applications of the theory to practical problems.
Course contents
A) Module of Data Science.
- Elements of geometry, linear algebra, and probability in high dimensional spaces; The Nearest Neighbor problem and data dimension reduction; Random projection and Johnson-Lindenstrauss lemma; Gaussians in high dimension; Data fitting on a spherical Gaussian.
- Singular Values Decomposition (SVD); Best approximation of rank k; SVD applications: Principal Component Analysis (PCA), Spherical Gaussian mixture clustering, Max-Cut Problem.
- Classification: linear separators and kernel method; Overfitting, PAC-Learning Guarantee and Uniform Convergence; Occam's razor and regularization; Support Vector Machines (SVM); VC dimension; k-fold cross validation.
- Clustering: k-means, k-center, k-median; Outliers and initialization strategies.
B) Module of Numerical Optimization.
1. Introduction to Optimization methods. Matlab Optimization Toolbox.
2. Derivative – free methods: Nelder – Mead.
3. Newton method.
4. Descent methods (line search):
- stepsize selection, Wolfe conditions, backtracking.
- Newton direction.
- Quasi – Newton directions(rank 1 update, DFP and BFGS methods)
- Gradient direction.
- Conjugate gradient (methods of Fletcher – Reeves, Polak – Ribiere, Hestenes – Stiefel).
5. Trust – Region methods.
6. Nonlinear Least – Square:
- Gauss – Newton.
- Levenberg - Marquardt.
7. Application to neural networks and Deep Learning.
Teaching methods
Lectures and Matlab laboratory
Reccomended or required readings
Avrim Blum, John Hopcroft, Ravindran Kannan. “Foundations of Data Science”. Cambridge University Press, Jan 23, 2020
Nocedal, Jorge; Wright, Stephen J. Numerical optimization. Second edition. Springer, 2006.
Assessment methods
Final project, presentation and oral examination
Sustainable development goals - Agenda 2030
Università di Pavia - Offerta formativa