# Bachelor/Master Theses, Semester Projects, and DAS DS Capstone Projects

If you are interested in one of the following topics, please send an email to Prof. Bölcskei and include your complete transcripts. Please note that we can not respond to requests that do not contain your transcripts.

These projects serve to illustrate the general nature of projects we offer. You are most welcome to inquire directly with Prof. Bölcskei about tailored research projects. Likewise, please contact Prof. Bölcskei in case you are interested in a bachelor thesis project.

Also, we have a list of ongoing and finished theses on our website.

## List of Bachelor Projects (BA)

## List of Semester Projects (SP)

- Why do neural networks generalize?
- Approximation rates of transformer networks
- Acoustic sensing and trajectory estimation of objects flying at supersonic speed (with industry)
- Separability of structured data
- On the metric entropy of dynamical systems
- Learning with general scattering networks
- The "logic" behind recurrent neural networks

## List of Master Projects (MA)

- Learning in indefinite spaces
- Automatic synopsis generation from amendment proposals for German law
- Why do neural networks generalize?
- Approximation rates of transformer networks
- Learning cellular automaton transition rules with transformers
- On the metric entropy of dynamical systems
- Finite-precision neural networks
- The "logic" behind recurrent neural networks

### Learning in indefinite spaces (MA)

In classical learning theory, a symmetric, positive semidefinite and continuous kernel function is used to construct a reproducing kernel Hilbert space, which serves as a hypothesis space for learning algorithms [1].

However, in many applications the kernel function fails to be positive semidefinite [2] which, in turn, leads to so-called (indefinite) Krein spaces [3]. The goal of this project is to develop a theory of learning for reproducing kernel Krein spaces.

Type of project: 100% theory

Prerequisites: Strong mathematical background, measure theory, functional analysis

Supervisor: Erwin Riegler

Professor:
Helmut Bölcskei

References:

[1] F. Cucker and D. X. Zhou, "Learning theory," *ser. Cambridge Monographs on Applied and Computational Mathematics*, Cambridge University Press, 2007.

[2] R. Luss and A. d’Aspremont, "Support vector machine classification with indefinite kernels," *Mathematical Programming Computation*, vol. 1, no. 2-3, pp. 97–118, Oct. 2009.

[3] A. Gheondea, "Reproducing kernel Krein spaces," *Chapter 14 in D. Alpay, Operator Theory*, Springer, 2015.

### Automatic synopsis generation from amendment proposals for German law (MA)

Changes to German law are proposed in the form of amendments, which contain natural language instructions on how to change individual words or sentences within the current law (see e.g. [1]). For laypeople it is difficult to infer, from such proposals, the text of the law after the amendment is accepted, thus reducing the ability of the general public to participate in the legislative process [2]. The goal of this project is to develop a machine learning algorithm that reads the current version of the law as well as the proposed amendment and then produces the associated new version of the law. This will allow to automatically generate a synopsis that compares the previous and proposed versions (see [3] for an example).

Recently significant advances in machine translation and question answering were made using transformer networks that are pretrained on large unsupervised data sets [4, 5, 6]. Machine learning solutions for the specific task at hand have, however, not been studied previously. Significant new contributions will hence be required. In particular, the semi-structured nature of amendments might make it necessary to incorporate a copy mechanism [7, 8, 9]. In this project, you will have the opportunity to, first, make novel contributions to the field of natural language processing and, second, to develop a working algorithm that can be deployed online and used by the general public.

Type of project: 70% implementation/programming, 30% model development

Prerequisites: Experience with deep learning for natural language processing (NLP), knowledge of German

Supervisor: Clemens Hutter, Joseph Rumstadt

Professor:
Helmut Bölcskei

References:

[1]
"Gesetz zur Modernisierung des notariellen Berufsrechts und zur Änderung weiterer Vorschriften."
[Link to Document]

[2] F. Herbert, "Verfassungsblog: On matters constitutional," 2021, doi: 10.17176/20210305-033813-0. [Link to Document]

[3] "Synopse: Gesetz zur Modernisierung des notariellen Berufsrechts und zur Änderung weiterer Vorschriften." [Link to Document]

[4]
A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. Gomez, L. Kaiser, and I. Polosukhin, "Attention is all you need," *Advances in Neural Information Processing Systems*, pp. 5999–6009, 2017.
[Link to Document]

[5]
A. Radford, T. Narasimhan, T. Salimans, and I. Sutskever, "Improving language understanding by generative pre-training," *Preprint*, pp. 1–12, 2018.
[Link to Document]

[6]
J. Devlin, M. Chang, K. Lee, and K. Toutanova, "BERT: Pre-training of deep bidirectional transformers for language understanding," *NAACL HLT 2019 - 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies - Proceedings of the Conference*, vol. 1, pp. 4171–4186, 2019.
[Link to Document]

[7]
J. Gu, Z. Lu, H. Li, and V. Li, "Incorporating copying mechanism in sequence-to-sequence learning," *54th Annual Meeting of the Association for Computational Linguistics, ACL 2016 - Long Papers*, vol. 3, pp. 1631–1640, 2016, doi: 10.18653/v1/p16-1154.
[Link to Document]

[8] A. See, P. Liu, and C. Manning, "Get to the point: Summarization with pointer-generator networks," *ACL 2017 - 55th Annual Meeting of the Association for Computational Linguistics, Proceedings of the Conference (Long Papers)*, vol. 1, pp. 1073–1083, 2017, doi: 10.18653/v1/P17-1099.
[Link to Document]

[9] B. McCann, N. Keskar, C. Xiong, and R. Socher, "The natural language decathlon: Multitask learning as question answering." [Link to Document]

### Why do neural networks generalize? (BA/SP/MA)

Deep ReLU neural networks (DNNs) are known for their high expressivity, resulting from the ability to have the number of linear regions they realize grow exponentially in network depth [1]. Most practical applications employ overparametrized DNNs, where the number of network weights is larger, by orders of magnitude, than the number of training samples. Hence, there are many different parameter choices that can perfectly fit the training data [2]. It is therefore surprising that weights found through gradient descent tend to lead to networks that generalize well, even though there are many other parameter choices that achieve the same training error while leading to poor performance on testing data [3].

It was recently argued in [4] that this happens because minima that lead to good generalization performance have more volume in parameter space and are thus more easily found by optimization procedures. The purpose of this project is to understand what kinds of hypothesis classes lead to such behavior [5, 6]. In this project you will investigate the relationship between volume in parameter space and the shape of decision boundaries of DNNs. Initially, this can be done through simulation experiments, which might then inform a mathematical theory. As a starting point you will study [7] which might be useful, as it uses volumetric arguments to investigate the number of linear regions realized by a DNN.

Type of project: 40%-100% algorithms/programming/simulations, 0%-60% mathematical theory; depending on student's interests

Prerequisites: Programming and visualization skills, mathematical background in geometry

Supervisor: Clemens Hutter

Professor:
Helmut Bölcskei

References:

[1] D. Elbrächter, D. Perekrestenko, P. Grohs, and H. Bölcskei, “Deep neural network approximation theory,” *IEEE Transactions on Information Theory*, vol. 67, no. 5, pp. 2581–2623, May 2021, doi: 10.1109/TIT.2021.3062161. [Link to Document]

[2] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals, “Understanding deep learning requires rethinking generalization,” in *The Fifth International Conference on Learning Representations*, Feb. 2017. [Link to Document]

[3] W. R. Huang et al., “Understanding generalization through visualizations,” in *I Can’t Believe It’s Not Better! Workshop at the 34th International Conference on Neural Information Processing Systems*, 2020. [Link to Document]

[4] P. Chiang et al., “Loss landscapes are all you need: Neural network generalization can be explained without the implicit bias of gradient descent,” in *The Eleventh International Conference on Learning Representations*, 2023. [Link to Document]

[5] H. Shah, K. Tamuly, A. Raghunathan, P. Jain, and P. Netrapalli, “The pitfalls of simplicity bias in neural networks,” in *Proceedings of the 34th International Conference on Neural Information Processing Systems*, 2020, pp. 9573–9585. [Link to Document]

[6] A. G. Wilson and P. Izmailov, “Bayesian deep learning and a probabilistic perspective of generalization,” in *Proceedings of the 34th International Conference on Neural Information Processing Systems*, 2020, pp. 4697–4708. [Link to Document]

[7] B. Hanin and D. Rolnick, “Deep ReLU networks have surprisingly few activation patterns,” in *Proceedings of the 33rd International Conference on Neural Information Processing Systems*, Oct. 2019, pp. 361–370. [Link to Document]

### Approximation rates of transformer networks (SP/MA)

Recently, the transformer neural network architecture [1] has seen tremendous success in natural language processing [2, 3], enabling applications such as ChatGPT. Nonetheless, a comprehensive theoretical understanding of its approximation capabilities –- similar to what is known about deep neural networks [4] –- remains elusive.

The goal of this project is to derive a mathematical theory for transformer neural networks. Concretely, you would try to understand the fundamental limits of their sequence-to-sequence approximation properties in the spirit of Kolmogorov exponents à la [4], starting from the universal approximation results in [5].

Type of project: 100% theory

Prerequisites: Strong mathematical background

Supervisor: Clemens Hutter

Professor:
Helmut Bölcskei

References:

[1] A. Vaswani et al., “Attention is all you need,” in *Proceedings of the 31st International Conference on Neural Information Processing Systems*, 2017, pp. 6000–6010. [Link to Document]

[2] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of deep bidirectional transformers for language understanding,” in *Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)*, Jun. 2019, pp. 4171–4186. [Link to Document]

[3] T. B. Brown et al., “Language models are few-shot learners,” in *Proceedings of the 34th International Conference on Neural Information Processing Systems*, Jul. 2020, pp. 1877–1901. [Link to Document]

[4] D. Elbrächter, D. Perekrestenko, P. Grohs, and H. Bölcskei, “Deep neural network approximation theory,” *IEEE Transactions on Information Theory*, vol. 67, no. 5, pp. 2581–2623, May 2021, doi: 10.1109/TIT.2021.3062161. [Link to Document]

[5] C. Yun, S. Bhojanapalli, A. S. Rawat, S. J. Reddi, and S. Kumar, “Are transformers universal approximators of sequence-to-sequence functions?,” in *The Eighth International Conference on Learning Representations*, Feb. 2020. [Link to Document]

### Acoustic sensing and trajectory estimation of objects flying at supersonic speed (with industry) (SP)

In shooting sports, hunting, and law-enforcement applications measuring the speed and trajectory of projectile flight at high precision and reliability constitutes an important technical challenge. For supersonic projectiles these quantities are estimated from signals acquired by microphones placed at different locations. Recently, more powerful microprocessors have made it possible to employ more sophisticated algorithms.

The goal of this project is to investigate new techniques for the task at hand, such as linearization of non-linear systems of equations, least squares fitting, and neural network driven machine learning. Existing hardware and algorithms provide a starting point for the project, which will be carried out in collaboration with an industry partner called SIUS (located in Effretikon, Zurich). SIUS offers close supervision and the possibility to use hardware and a test laboratory.

About the industry partner: SIUS is the world’s leading manufacturer of electronic scoring systems in shooting sports. The company is specialized in producing high speed and high precision measurement equipment capable of measuring projectile position and trajectory and has been equipping the most important international competitions including the Olympic Games for decades.

Type of project: 20% literature research, 20% theory, 50% implementation/programming, 10% experiments

Prerequisites: Solid mathematical background, knowledge of SciPy, Matlab or a similar toolset, ideally knowledge on (deep) neural networks

Supervisor: Michael Lerjen, Steven Müllener

Professor:
Helmut Bölcskei

References:

[1] SIUS Homepage [Link to Document]

### Separability of structured data (SP)

Deep neural networks have demonstrated remarkable success in learning from datasets which exhibit complex structures, such as images or natural language. In the pursuit of developing an understanding of the reasons behind this success, Cover's function counting theory [1] stands as a significant milestone, addressing the question of how many binary label assignments can be realized. This approach, however, does not take into account potential structural properties of the input data.

The goal of this project is to extend Cover's framework [1] to structured data, such as, e.g., data points that are grouped in tuples [2], as illustrated in the figure, or that exhibit sparsity in the sense of compressed sensing theory [3, 4].

Type of project: 100% theory

Prerequisites: Strong mathematical background

Supervisor: Konstantin Häberle

Professor:
Helmut Bölcskei

References:

[1]
T. Cover, “Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition,” *IEEE Transactions on Electronic Computers*, vol. 14, no. 3, pp. 326-334, 1965.
[Link to Document]

[2]
P. Rotondo, M. Lagomarsino, and M. Gherardi, “Counting the learnable functions of geometrically structured data,” *Physical Review Research*, vol. 2, no. 2, p. 023169, 2020.
[Link to Document]

[3]
E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” *IEEE Transactions on Information Theory*, vol. 52, no. 2, pp. 489–509, 2006.
[Link to Document]

[4]
D. Donoho, “Compressed sensing,” *IEEE Transactions on Information Theory*, vol. 52,
no. 4, pp. 1289–1306, 2006.
[Link to Document]

### Learning cellular automaton transition rules with transformers (MA)

A cellular automaton (CA) is a discrete dynamical system consisting of a regular lattice in one or more dimensions with cell values taken from a finite set. The cells change their states at synchronous discrete time steps based on a transition rule [1]. Despite the simplicity of the CA model, it can exhibit complex global behavior. With suitably chosen transition rules, cellular automata can simulate a plethora of dynamical behaviors [2, 3]. The inverse problem of deducing the transition rule from a given global behavior is extremely difficult [4]. In this project, you will investigate the possibility of training transformers [5] to learn CA transition rules.

Type of project: 30% theory, 70% implementation

Prerequisites: Good programming skills, knowledge in machine learning

Supervisor: Yani Zhang

Professor:
Helmut Bölcskei

References:

[1]
J. Kari, “Theory of cellular automata: A survey,” *Theoretical computer science*, 334(1-3):3–33, 2005.
[Link to Document]

[2] T. Toffoli and N. Margolus, “Cellular automata machines: A new environment for modeling,” MIT press, 1987. [Link to Document]

[3] A. Adamatzky, “Game of life cellular automata,” vol. 1, Springer, 2010. [Link to Document]

[4] N. Ganguly, B. K. Sikdar, A. Deutsch, G. Canright, and P. P. Chaudhuri, “A survey on cellular automata,” 2003. [Link to Document]

[5]
A. Vaswani, et al., “Attention is all you need,” *Advances in Neural Information Processing Systems 30*, 2017.
[Link to Document]

### On the metric entropy of dynamical systems (MA/SP)

The aim of this project is to explore the metric entropy of dynamical systems, i.e., to identify how much information about a system's input-output behavior is needed to be able to describe the system dynamics to within a prescribed accuracy. In particular, you will study the asymptotics of ε-entropy in the Kolmogorov sense [1, 2] of a certain class of causal linear systems [3]. Based on these results you will try to develop a general theory that encompasses more general classes of dynamical systems, including time-varying systems [4] and nonlinear systems [5].

Type of project: 100% theory

Prerequisites: Strong mathematical background

Supervisor: Thomas Allard

Professor:
Helmut Bölcskei

References:

[1]
A. N. Kolmogorov, "On certain asymptotic characteristics of completely bounded metric spaces," *Doklady Akademii Nauk SSSR*, vol. 108, no. 3, pp. 385–389, 1956.

[2]
A. N. Kolmogorov and V. M. Tikhomirov, "ε-entropy and ε-capacity of sets in functional spaces," in *Uspekhi Matematicheskikh Nauk*, vol. 14, no. 2, pp. 3–86, 1959.

[3]
G. Zames, "On the metric complexity of causal linear systems: ε-entropy and ε-dimension for continuous time," *IEEE Transactions on Automatic Control*, vol. 24, no. 2, pp. 222–230, 1979.
[Link to Document]

[4]
G. Matz, H. Bölcskei, and F. Hlawatsch, "Time-frequency foundations of communications," *IEEE Signal Processing Magazine*, vol. 30, no. 6, pp. 87–96, 2013.
[Link to Document]

[5]
M. Schetzen, "Nonlinear system modeling based on the Wiener theory," * Proceedings of the IEEE*, vol. 69, no. 12, pp. 1557–1573, 1981.
[Link to Document]

### Finite-precision neural networks (MA)

Deep feedforward neural networks with quantized real-valued weights can approximate a wide class of functions in a Kolmogorov-optimal manner [1, 2]. These results do, however, not fully explain the success of neural networks in practical applications, where besides network weights also the signals in all layers of the network have to be stored in computer memories and hence have finite precision only.

The first step of the project is to generalize the theory developed in [1, 2] to neural networks with both weights and signals in all layers of finite precision and to establish fundamental limits on function approximation through such networks. Specifically, the new theory should be able to answer the question of how a given overall bit budget for operating the neural network should be distributed across the weights and signals in the network so as to minimize the end-to-end approximation error for a given task. The second major goal of the project is to identify function classes for which approximation through finite-precision neural networks achieves the fundamental limits identified in the first part.

The project is carried out in collaboration with Dr. Van Minh Nguyen in the form of an internship at Huawei Labs in Paris.

Type of project: 70% theory, 30% simulation

Prerequisites: Strong mathematical background and good programming skills

Supervisor: Weigutian Ou

Professor:
Helmut Bölcskei

References:

[1]
H. Bölcskei, P. Grohs, G. Kutyniok, and P. Petersen, "Optimal approximation with sparsely connected deep neural networks," *SIAM Journal on Mathematics of Data Science*, vol. 1, no. 1, pp. 8–45, 2019.
[Link to Document]

[2]
D. Elbrächter, D. Perekrestenko, P. Grohs, and H. Bölcskei, "Deep neural network approximation theory," *IEEE Transactions on Information Theory*, vol. 67, no. 5, pp. 2581–2623, May 2021.
[Link to Document]

### Learning with general scattering networks (SP)

Deep learning methods have dramatically improved state-of-the-art results on image classification, speech recognition, and other perceptual tasks in recent years.

However, common architectures typically require a large number of trainable parameters and therefore a lot of training data and compute power.

The goal of this project is to design and implement general scattering networks for feature extraction [1] with drastically fewer trainable parameters than conventional neural networks (such as e.g. ResNet or Inception) and to assess the performance of the resulting networks relative to fully trained networks. In particular, you will evaluate the performance on specific widely used datasets such as MNIST and ImageNet.

Type of project: 75% simulation, 25% theory

Prerequisites: Affinity for signal processing and functional analysis, programming skills

Supervisor: Ines Haymann

Professor:
Helmut Bölcskei

References:

[1]
T. Wiatowski and H. Bölcskei, "A mathematical theory of deep convolutional neural networks for feature extraction," *IEEE Transactions on Information Theory*, vol. 64, no. 3, pp. 1845–1866, Mar. 2018.
[Link to Document]

[2]
Y. LeCun, Y. Bengio, and G. Hinton, "Deep learning", *Nature*, vol. 521, pp. 436–444, May 2015.
[Link to Document]

### The "logic" behind recurrent neural networks (MA/SP)

Recurrent Neural Networks (RNNs) can simulate Turing machines [1]. Specifically, this is done through a clever mapping of the Turing machine’s logic to the architecture and the weights of the RNN. Conversely, it would be interesting to train RNNs in order to find the logical architecture of the underlying training data, thereby learning the “logical properties” of the mechanism generating the data. In essence this amounts to learning Boolean functions via RNNs. Following this philosophy, you will study the realization of certain classes of Boolean functions through RNNs [2], and you will devise RNN training algorithms to identify these functions. This will also entail studying literature on binary neural networks [4].

Type of project: 60% theory, 40% simulation

Prerequisites: Good programming skills, knowledge in machine learning and appetite for functional analysis

Supervisor: Valentin Abadie

Professor:
Helmut Bölcskei

References:

[1]
H. T. Siegelmann and E. D. Sontag, "On the computational power of neural nets," *Journal of Computer and System Sciences*, Vol. 50(1), pp. 132–150, 1995.
[Link to Document]

[2]
R. O'Donnell, "Analysis of Boolean functions," *Cambridge University Press*, 2014.
[Link to Document]

[3]
Y. LeCun, Y. Bengio, and G. Hinton, "Deep learning," *Nature*, Vol. 521(7553), pp. 436–444, 2015.
[Link to Document]

[4]
H. Qin, R. Gong, X. Liu, X. Bai, J. Song, and N. Sebe, "Binary neural networks: A survey," *Pattern Recognition*, Vol. 105, 107281, 2020.
[Link to Document]