Persistentní homologie a neuronové sítě
| Thesis title in Czech: | Persistentní homologie a neuronové sítě |
|---|---|
| Thesis title in English: | Persistent homology and neural networks |
| Key words: | homologie|neuronové sítě |
| English key words: | homology|neural networks |
| Academic year of topic announcement: | 2020/2021 |
| Thesis type: | diploma thesis |
| Thesis language: | čeština |
| Department: | Computer Science Institute of Charles University (32-IUUK) |
| Supervisor: | doc. Mgr. Robert Šámal, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 19.07.2021 |
| Date of assignment: | 19.07.2021 |
| Confirmed by Study dept. on: | 16.08.2021 |
| Date and time of defence: | 11.02.2022 09:00 |
| Date of electronic submission: | 07.01.2022 |
| Date of submission of printed version: | 10.01.2022 |
| Date of proceeded defence: | 11.02.2022 |
| Opponents: | doc. RNDr. Martin Tancer, Ph.D. |
| Advisors: | Bastian Alexander Rieck |
| Guidelines |
| 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. |
| References |
| [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 |