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Basic concepts and mathematical foundations of learning theory. The topics will be focused on kernel methods which are power tools in learning theory. After the review of Hilbert spaces, the theory of Reproducing Kernel Hilbert Spaces (RKHS) will be introduced, covering regularization techniques, and Tikhonov representations. Applications in machine learning will be described, including Support Vector Machines (SVM), and various neural networks for optimization problems.

MATH-7/8033 - Intelligent Decision Support
Mathematical foundations of decision support systems are addressed. Topics include basic operation research areas of mathematical programming and optimization, such as linear and nonlinear programming, integer and mixed-integer programming, dynamic and approximate dynamic programming, and combinatorial optimization. Mathematical decision theory is addressed, including multistage decisions, Markov chains, decision trees, game theory, and multi-objective optimization. Mathematical models are developed and solved for practical decision support problems, covering business, finance, engineering, cognitive and health applications.

MATH-7/8028  Intelligent Prediction Methods
Theoretical foundations of predictions are described, including deterministic and stochastic systems, autoregressive (AR), ARMA, and Kalman filtering. Recent developments are introduced in cognitive prediction, vector prediction, network prediction, time-lagged recurrent neural networks, and lattice models. Issues of generalization, convergence, and Lyapunov stability of prediction methods are addressed. Applications in image processing, in financial systems and in engineering are discussed.

MATH-7/8047  ADP, Stochastic Optimization & Control
Mathematical foundations of neural networks, learning, nonlinear optimization and control. Exact and approximate optimization of the utility function. Bellman equation, approximate Bellman equation for solving multivariate optimization problems in real time. Partially observable variables, with random noise and tactical objectives varying in time.