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 are sent directly to PhD students in the group or 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 Semester Projects (SP)

• Acoustic sensing and trajectory estimation of objects flying at supersonic speed (with industry)
• Separability of structured data
• Minimal depth to represent continuous piecewise linear (CPWL) functions by ReLU networks
• Continuous-time recurrent neural networks in Banach spaces
• General signal denoising
• Why transformers cannot learn maths?

List of Master Projects (MA)

• Automatic synopsis generation from amendment proposals for German law
• Learning cellular automaton transition rules with transformers
• Minimal depth to represent continuous piecewise linear (CPWL) functions by ReLU networks
• Finite-precision neural networks
• General signal denoising
• The "logic" behind transformers
• Generating singular distributions through neural networks
• Why transformers cannot learn maths?

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]

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]

Minimal depth to represent continuous piecewise linear (CPWL) functions by ReLU networks (MA/SP)

A core problem in machine learning is to analyze the expressive power of neural networks. So-called universal approximation theorems [1, 2, 3] show that even with a single hidden layer, one can essentially approximate any (continuous/measurable) function to within arbitrary precision. Nevertheless, it can be advantageous to employ deeper networks as this can reduce the overall network size in terms of the number of nodes [3, 4]. To get a better quantitative understanding of this phenomenon, it is important to characterize the classes of functions that are exactly representable by neural networks of a certain architecture. Specifically, we will consider the class of continuous piece-wise linear (CPWL) functions and ReLU neural networks.

In [5], a conjecture was recently formulated regarding this question. The conjecture states that the minimum number of hidden layers needed to represent any 𝑑-dimensional CPWL function is exactly ⌈log₂(𝑑+1)⌉. In this project, you would first write a literature review based on [4, 5, 6, 7] and then make attempts towards solving the conjecture formulated in [5].

Type of project: 100% theory, or 80% theory and 20% programming
Prerequisites: Strong mathematical background
Supervisor: Yang Pan
Professor: Helmut Bölcskei

References:
[1] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals and Systems, vol. 2, no 4, pp. 303–314, 1989. [Link to Document]

[2] K. Hornik, “Approximation capabilities of multilayer feedforward networks,” Neural Networks, vol. 4, no 2, pp. 251–257, 1991. [Link to Document]

[3] 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]

[4] R. Eldan and O. Shamir, "The power of depth for feedforward neural networks," Conference on Learning Theory, pp. 907–940, 2016. [Link to Document]

[5] C. Hertrich, A. Basu, M. Di Summa, and M. Skutella, "Towards lower bounds on the depth of ReLU neural networks," Advances in Neural Information Processing Systems, vol. 34, pp. 3336–3348, 2021. [Link to Document]

[6] C. Haase, C. Hertrich, and G. Loho, "Lower bounds on the depth of integral ReLU neural networks via lattice polytopes," arXiv preprint arXiv:2302.12553, 2023. [Link to Document]

[7] P. Petersen, M. Raslan, and F. Voigtlaender, "Topological properties of the set of functions generated by neural networks of fixed size," Foundations of Computational Mathematics, vol. 21, pp. 375-444, 2021. [Link to Document]

Continuous-time recurrent neural networks in Banach spaces (SP)

Since Hopfield introduced the first model of continuous-time recurrent neural networks (CTRNNs) in [1], there has been sustained interest in studying their properties, both from a theoretical and a practical point of view. Applications of CTRNNs can be found in domains as varied as computational neuroscience [2], analog mathematical optimization [3], and statistical learning [4].

Although there exists a rich literature on approximation-theoretic properties of CTRNNs, there is a lack of explicit constructive results. The standard statements in the literature often implicitly rely on the fundamental theorem of approximation for neural networks [5, 6]. The goal of the present project is to fill this gap by explicitly constructing CTRNNs that approximate functions in Banach spaces, including Besov spaces, Sobolev spaces, spaces of analytic functions, Hölder spaces, and Lipschitz spaces. You will approach this problem by studying how bases in such spaces, such as bases of sinusoids, wavelets, or splines, can be approximated by CTRNNs.

Type of project: 100% theory
Prerequisites: Strong mathematical background
Supervisor: Thomas Allard
Professor: Helmut Bölcskei

References:
[1] J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Nat. Acad. Sci. United States Amer., vol. 79, no. 8, pp. 2554–2558, Apr. 1982. [Link to Document]

[2] P. Dayan and L. F. Abbott, "Theoretical neuroscience: Computational and mathematical modeling of neural systems," MIT press, 2001. [Link to Document]

[3] M. Kennedy and L. Chua, “Neural networks for nonlinear programming,” IEEE Trans. Circuits Syst., vol. 35, no. 5, pp. 554–562, May 1988. [Link to Document]

[4] E. D. Sontag, “A learning result for continuous-time recurrent neural networks,” Systems and Control Letters, vol. 34, no. 3, pp. 151–158, June 1998. [Link to Document]

[5] K.-I. Funahashi and Y. Nakamura, “Approximation of dynamical systems by continuous time recurrent neural networks,” Neural Networks, vol. 6, pp. 801–806, Jan. 1993. [Link to Document]

[6] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals and Systems, vol. 2, no 4, pp. 303-314, 1989. [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]

General signal denoising (MA/SP)

Denoising signals contaminated by Gaussian noise has been a prevailing problem in the fields of statistics and signal processing. The majority of the results available in the literature assume that the true signal comes from certain specific subsets of Euclidean spaces and then design and analyze the denoising algorithm accordingly. Examples include one-dimensional intervals [1], hyperrectangles [2], 𝓁_p-balls [3], and unions of manifolds corresponding to low-rank matrices [4].

This project seeks to explore the signal denoising problem under more general assumptions on the data structures, while imposing low dimensionality in a suitable sense. The first goal of the project is to identify the best possible performance achievable by any denoising algorithm, and the second is to design algorithms that can provably attain this best performance.

Type of project: 80% theory, 20% simulation
Prerequisites: Strong mathematical background
Supervisor: Weigutian Ou
Professor: Helmut Bölcskei

References:
[1] P. J. Bickel, "Minimax estimation of the mean of a normal distribution when the parameter space is restricted," The Annals of Statistics, 9(6): 1301–1309, 1981. [Link to Document]

[2] D. L. Donoho, R. C. Liu, and B. MacGibbon, "Minimax risk over hyperrectangles, and implications," The Annals of Statistics, 18(3): 1416–1437, 1990. [Link to Document]

[3] D. L. Donoho and I. M. Johnstone, "Minimax risk over 𝓁_p-balls for 𝓁_q-error," Probability Theory and Related Fields, 99(2):277–303, 1994. [Link to Document]

[4] D. L. Donoho and M. Gavish, "Minimax risk of matrix denoising by singular value thresholding," The Annals of Statistics, 42(6): 2413–2440, 2014. [Link to Document]

The "logic" behind transformers (MA)

The transformer [1] is a neural network architecture–-underlying software such as ChatGPT–-that can simulate Turing machines [2]. Specifically, this is done through the simulation of a recurrent neural network (RNN) with a transformer, and then using the fact that RNNs can, in turn, simulate Turing machines [3]. This simulation process does, however, not yield a clear vision of which parts of the transformer architecture are connected to the different constituents of the Turing machine.

In this project, you will familiarize yourself with literature on links between Turing machines and neural networks. You would then establish direct connections between the transformer architecture and Turing machines, with the goal of understanding better how transformers simulate Turing machines.

 

Type of project: 100% theory
Prerequisites: Knowledge in neural network theory and theoretical computer science, appetite for theory in general
Supervisor: Valentin Abadie
Professor: Helmut Bölcskei

References:
[1] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, "Attention is all you need," Advances in Neural Information Processing Systems, 2017. [Link to Document]

[2] S. Bhattamishra, A. Patel, and N. Goyal, "On the computational power of transformers and its implications in sequence modeling," arXiv preprint arXiv:2006.09286, 2020. [Link to Document]

[3] 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]

Generating singular distributions through neural networks (MA)

There is a large body of work on neural networks as function approximators, either in the context of regression or classification. In recent years another use for neural networks has emerged, namely for generating high-dimensional probability distributions. It was shown in [1] that the set of probability distributions supported on the 𝑑-dimensional unit cube can be approximated arbitrarily well by push-forwards of the uniform distribution on the unit interval through a set of neural networks of cardinality depending on 𝑑.

The goal of this project is to derive results on the generation of singular, e.g., fractal, distributions through a set of neural networks of cardinality depending on the Hausdorff dimension of the probability distribution.

Type of project: 100% theory
Prerequisites: Strong mathematical background, measure theory
Supervisor: Erwin Riegler
Professor: Helmut Bölcskei

References:
[1] D. Perekrestenko, L. Eberhard, and H. Bölcskei, "High-dimensional distribution generation through deep neural networks," Partial Differential Equations and Applications, Springer, vol. 2, no. 64, pp. 1–44, Sep. 2021. [Link to Document]

[2] Y. Yang, L. Zhen, and Y. Wang, "On the capacity of deep generative networks for approximating distributions," Neural Networks, vol. 145, no. C, pp. 144–154, Jan. 2022. [Link to Document]

[3] K. Falconer, "Fractal geometry: Mathematical foundations and applications," Wiley, 2nd ed., 2003. [Link to Document]

Why transformers cannot learn maths? (MA/SP)

The transformer [1] is a neural network architecture—underlying software such as ChatGPT—which can provably simulate Turing machines [2]. However, even state-of-the-art architectures appear unable to perform simple mathematical operations, such as addition and multiplication. Recently, techniques for improving the performance of transformers on such tasks—by changing the way data is represented—were reported in [3]. Moreover, investigations on which kind of heuristics transformers do learn have been carried out in [4].

In this project, you will study which part of the learning process hinders the realization of simple mathematical operations. Specifically, you will implement the transformer architecture simulating Turing machines in [2], and you will try to reach this architecture through a learning algorithm.

Type of project: 100% simulation
Prerequisites: Knowledge in neural network theory and good programming skills
Supervisor: Valentin Abadie
Professor: Helmut Bölcskei

References:
[1] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, "Attention is all you need," Advances in Neural Information Processing Systems, 2017. [Link to Document]

[2] S. Bhattamishra, A. Patel, and N. Goyal, "On the computational power of transformers and its implications in sequence modeling," arXiv preprint arXiv:2006.09286, 2020. [Link to Document]

[3] R. Nogueira, Z. Jiang, and J. Lin, "Investigating the limitations of transformers with simple arithmetic tasks,” arXiv preprint arXiv:2102.13019, 2021. [Link to Document]

[4] F. Charton, ”Can transformers learn the greatest common divisor?,” arXiv preprint arXiv:2308.15594, 2023. [Link to Document]