Nabídka témat závěrečných prací
Navržená témata jsou vhodná pro studenty FJFI/ČVUT i MFF/UK.
Podle zvoleného tématu a obtížnosti jsou témata vhodná pro studijní obory AAA, MI, AMSM či APIN.
Téma je možno vypsat i individuálně podle zájmu konkrétního studenta/studentky.
Níže uvedená témata jsou tedy jen příklady již hotových zadání. Práce je možno psát česky i anglicky.
Structured matrices in compressed sensing
typ práce: bakalářská práce, diplomová práce
zaměření: AAA, MI_AMSM, MINF, matematická analýza
klíčová slova: Random matrices, compressed sensing, Johnson-Lindenstrauss Lemma
popis: The student will review some basic properties of random matrices in the area of compressed sensing. Then (s)he will discuss the role of structured random matrices - the speed up of matrix-vector multiplication, the reduced amount of random bits needed, theoretical guarantees for their performance, real-life performance for some specific problems.
literatura: H. Boche, R. Calderbank, and G. Kutyniok, A Survey of Compressed Sensing, First chapter in Compressed Sensing and its Applications, Birkhauser, Springer, 2015
F. Krahmer and R. Ward, New and improved Johnson-Lindenstrauss embeddings via the restricted isometry property, SIAM Journal on Mathematical Analysis, 2011
Impact of delay on pattern formation and stability in the attraction-repulsion model
školitel: Dr. Jan Haškovec, doc. Jan Vybíral
typ práce: bakalářská práce, diplomová práce
zaměření: AAA, MI_AMSM, MINF, APIN, matematická analýza, numerická matematika
klíčová slova: attraction-repulsion model, pattern formation, delay differential equations
popis: See the pdf file for description.
Bases of ReLU Neural Networks
typ práce: bakalářská práce, diplomová práce
zaměření: AAA, MI_MM, MI_AMSM, MINF
klíčová slova: ReLU, Neural Networks, Riesz Basis, Frame
popis: The astonishing performance of neural networks in multivariate problems still lacks satisfactory mathematical explanation. In a recent work of I. Daubechies and her co-authors, they proposed a univariate system of piecewise linear functions, which resemble very much the trigonometric system and which form the so-called Riesz basis. Moreover, these functions are easily reproducible as ReLU Neural Networks. In a follow-up work of C. Schneider and J. Vybiral, this was generalized to the multivariate setting. The task of this work will be to investigate further potential improvements of the recent research, both on theoretical as well as practical side. This includes a) optimization of the Riesz constants of the system b) application of an orthonormalization procedure c) numerical implementation of the proposed NN-architecture and the study of its performance in approximation of multivariate functions.
literatura: C. Schneider and J. Vybiral, Multivariate Riesz basis of ReLU neural networks, Appl. Comput. Harm. Anal. (2024)
I. Daubechies, R. DeVore, S. Foucart, B. Hanin, and G. Petrova, Nonlinear Approximation and (Deep) ReLU Networks, Constr. Appr. 55 (2022), 127-172
P. Beneventano, P. Cheridito, R. Graeber, A. Jentzen, and B. Kuckuck, Deep neural network approximation theory for high-dimensional functions, available at arXiv:2112.14523
Function spaces and regularity of Brownian motion
typ práce: disertační práce
zaměření: AAA, matematická analýza (MFF UK)
klíčová slova: Function spaces, Littlewood-Paley decomposition, Brownian motion, path regularity
popis: The aim of the work is the further study of function spaces, which were recently used to characterize the regularity of paths of Brownian motion. We want to provide a systematic study of their basic and more advanced properties.
literatura: H. Kempka, C. Schneider and J. Vybíral, Path regularity of Brownian motion and Brownian sheet, Constr. Appr. (2024)