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Applied Functional Analysis and Machine Learning*

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Applied Functional Analysis and Machine Learning*

Prof. Gianluigi Pillonetto

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1Dept. of Information Engineering, University of Padova Email: [email protected]

Timetable: Course of 28 hours (2 two-hours lectures per week): Classes on Tuesday 10:30 - 12:30 and Wednesday, 8:30 - 10:30. First lecture on Wednesday November 28th, 2017. Sala Riunioni 318 DEI/G 3-rd floor, via Gradenigo 6).

Course requirements: The classical theory of functions of real variable: limits and continuity, differentiation and Riemann integration, infinite series and uniform convergence. The arith- metic of complex numbers and the basic properties of the complex exponential function. Some elementary set theory. A bit of linear algebra.

Examination and grading: Homework assignments and final test.

Students wishing to take the exam for this course, are kindly requested to contact the Coordina- tor of the PhD program in Mathematical Sciences,

Aim: The course is intended to give a survey of the basic aspects of functional analysis, machine learning, regularization theory and inverse problems.

Course contents:

1. Review of some notions on metric spaces and Lebesgue integration: Metric spaces. Open sets, closed sets, neighborhoods. Convergence, Cauchy sequences, completeness. Com- pletion of metric spaces. Review of the Lebesgue integration theory. Lebesgue spaces.

2. Banach and Hilbert spaces: Normed spaces and Banach spaces. Finite dimensional normed spaces and subspaces. Compactness and finite dimension. Bounded linear op- erators. Linear functionals. The finite dimensional case. Normed spaces of operators and the dual space. Weak topologies. Inner product spaces and Hilbert spaces. Orthog- onal complements and direct sums. Orthonormal sets and sequences. Representation of functionals on Hilbert spaces.

3. Compact linear operators on normed spaces and their spectrum: Spectral properties of bounded linear operators. Compact linear operators on normed spaces. Spectral properties of compact linear operators. Spectral properties of bounded self-adjoint operators, positive operators, operators defined by a kernel. Mercer Kernels and Mercer’s theorem.

4. Reproducing kernel Hilbert spaces, inverse problems and regularization theory: Repre- senter theorem. Reproducing Kernel Hilbert Spaces (RKHS): definition and basic prop- erties. Examples of RKHS. Function estimation problems in RKHS. Tikhonov regular- ization. Primal and dual formulation of loss functions. Regularization networks. Consis- tency/generalization and relationship with Vapnik’s theory and the concept of Vγ dimen- sion. Support vector regression and classification.

References:

[1] W. Rudin. Real and Complex Analysis, McGraw Hill, 2006. . Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, 1978.

[2] G. Wahba. Spline models for observational data. SIAM, 1990.

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[3] C.E. Rasmussen and C.K.I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006.

[4] R.T. Rockafellar. Convex analysis. Princeton University Press, 1996.

Other material and research papers will be available online for download.

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