Persistentní homologie a neuronové sítě
| Název práce v češtině: | Persistentní homologie a neuronové sítě |
|---|---|
| Název v anglickém jazyce: | Persistent homology and neural networks |
| Klíčová slova: | homologie|neuronové sítě |
| Klíčová slova anglicky: | homology|neural networks |
| Akademický rok vypsání: | 2020/2021 |
| Typ práce: | diplomová práce |
| Jazyk práce: | čeština |
| Ústav: | Informatický ústav Univerzity Karlovy (32-IUUK) |
| Vedoucí / školitel: | doc. Mgr. Robert Šámal, Ph.D. |
| Řešitel: | skrytý - zadáno a potvrzeno stud. odd. |
| Datum přihlášení: | 19.07.2021 |
| Datum zadání: | 19.07.2021 |
| Datum potvrzení stud. oddělením: | 16.08.2021 |
| Datum a čas obhajoby: | 11.02.2022 09:00 |
| Datum odevzdání elektronické podoby: | 07.01.2022 |
| Datum odevzdání tištěné podoby: | 10.01.2022 |
| Datum proběhlé obhajoby: | 11.02.2022 |
| Oponenti: | doc. RNDr. Martin Tancer, Ph.D. |
| Konzultanti: | Bastian Alexander Rieck |
| Zásady pro vypracování |
| Deep learning [2] is used to solve various real-world problems for a decade. Layers of its typical models are differentiable operations between high-dimensional vector spaces. Topology, on the other hand, studies spaces using their connectivity independently of the dimension. Topological characteristics can thus express new additional information about the shape of subspaces or hidden manifolds. The student will pursue the following questions: what are the topological properties of the mentioned high-dimensional spaces, and how descriptive are the topological properties? Computational topology developed persistent homology [1] that computes Betti numbers of a sampled space depending on decreasing spatial resolution. The result is formally a persistence module representable by a persistence diagram. The thesis' aim is to explore steps toward answering the questions using persistent homology and deep learning itself. The publications [3] and [4] will form the base of these steps. |
| Seznam odborné literatury |
| [1] Edelsbrunner, H. and Harer, J. Computational topology: an introduction. American Mathematical Society, 2010.
[2] Goodfellow, I., Bengio, Y., and Courville, A. Deep learning. An Introduction. MIT Press book, 2016. [3] Carlsson, G. and Brüel-Gabrielsson, R. Topological Approaches to Deep Learning. arXiv:1811.01122, 2018 [4] Rieck, B., Togninalli, M., Bock, C., Moor, M., Horn, M., Gumbsch, T., Borgwardt, K. Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology. arXiv:1812.09764, 2018 |