Seminar abstracts: WiSe 2025

The set of germs of differentiable real-valued functions forms a ring with the induced addition and multiplication of functions. Hardy fields are subfields of this ring which are closed under taking derivatives and come with a natural ordering. Moreover, every element of a Hardy field has a limit in the extended real line. To compare different sizes of asymptotic growth of germs we equip Hardy fields with the standard valuation. One can associate to each Hardy field its asymptotic couple, which is a pair consisting of the value group of the Hardy field together with a map induced by the logarithmic derivative. These objects can also be defined purely algebraically, and this raises the question which asymptotic couples come from Hardy fields. Rosenlicht gave a partial explicit answer to a relative question: when does an extension of an asymptotic couple of a Hardy field also come from a corresponding Hardy field extension? He showed that this is possible under the assumption that the value groups are of finite rank. We give a self-contained treatment of this result and construct Hardy fields for asymptotic couples of Hardy type with small derivation which are of countable rank.

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We present a greedy approach for solving time-independent linear PDEs by growing neural networks.

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In this talk, we introduce a framework for Fourier analysis on the group of truncated signatures. After a brief introduction to signatures and their applications, we present some key concepts from representation theory, and show how they can be used to construct the Fourier transform of functions defined on the group of signatures. Further, we show that this transform has desirable properties, including an inversion formula and an analogue of the Plancherel theorem.

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Dispersion of a point set measures the volume of the largest axis-aligned empty box within the unit cube that avoids the given points. Closely related to discrepancy, dispersion quantifies the uniformity of point distributions and has applications in various fields such as numerical integration. We present improved lower bound on dispersion for sufficiently large volumes, which is asymptotically optimal up to a logarithmic factor. A generalization of this method, combined with the superlinear property of dispersion, yields a series of further lower bounds valid in arbitrary dimensions and for arbitrary volumes, establishing new state-of-the-art results.

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In this talk, we explore two complementary approaches for recovering frequency-localized functions from pointwise data. First, we introduce the Slepian basis and establish a recovery theorem for least-squares approximation using uniformly random samples in low dimensions. Second, we present a recovery guarantee for approximating frequency-localized functions using deep learning. This result, formulated as a practical existence theorem, identifies conditions on the network architecture, training procedure, and data acquisition that are sufficient to ensure accurate approximation. To conclude, we provide numerical examples to illustrate and compare the performance of least-squares methods and deep-learning-based approaches.

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We study the full discretization of the elliptic and parabolic p-Laplacian using discontinuous Galerkin (DG) methods. Quasi-norm error estimates are derived for a DG time-stepping scheme combined with conforming finite element discretization in space. In addition, we consider an hp-version DG method for the elliptic p-Laplacian on general polygonal and curved meshes, as well as a space–time DG formulation for the corresponding parabolic problem. By employing novel quasi-norm trace–inverse estimates, we establish quasi-norm error bounds of optimal order with respect to the local mesh sizes and of slightly reduced order with respect to the local polynomial degrees. Numerical experiments are presented to validate and illustrate the theoretical results. This is joint work with Konstantinos Chrysafinos (NTU Athens) and Emmanuil H. Georgoulis (NTU Athens & Heriot-Watt Uni.).

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Low-rank representations have a long history in numerical methods, signal processing, and data analysis. The classical view, through methods such as PCA, SVD, or POD, concerns finite-dimensional data in R^d, with some extensions to infinite-dimensional data in the context of functional data analysis. Low-rank structure also plays an important role in neural network approximations of operators between function spaces. A common rationale in the classical setting as well as in functional data analysis is that the low-rank representation can be chosen as an orthonormal basis. At present, this is still an underexplored topic within the framework of operator learning and typically appears via eigendecompositions of infinite-dimensional operators, realized by algorithms that sequentially learn one eigenfunction after another. In this work, we formulate low-rank representations of infinite-dimensional data as an SVD for operators mapping from finite-dimensional to infinite-dimensional spaces, which leads to the neural functional singular value decomposition. We then explore algorithms that jointly learn a prescribed number of eigenfunctions from irregularly sampled data. Finally, we present approaches to analyze the required model complexity of jointly versus sequentially learning the basis functions.

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In our ongoing work we propose a simple dictionary of truly infinite dimensional (non-linear) operators. Based on this dictionary, we utilize some recent developments in approximation theory for variation spaces and general type-2 Banach spaces to provide approximation rates for what can be considered a class of infinite dimensional Barron spaces. We show that our dictionary indeed fits the required assumptions for the approximation theorem and based on its structure, we can also identify certain classes of operators which are in the variation space that is associated to our dictionary. We furthermore explore how this class of N-term operator approximations can be extended to deep models and present some preliminary experimental results in which we reach state of the art generalization performance.

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As the limits of classical computing approach, Quantum Computing has emerged as a frontier for computational advantage. This talk provides a mathematically grounded introduction to the field, beginning with the foundations of photonic quantum computing. We will explore how quantum states can be manipulated via optical circuits to perform high-dimensional linear algebra - the core of modern machine learning. The second half of the talk motivates the intersection of optical or quantum machine learning and traditional learning theory. We will specifically focus on the Data Re-uploading paradigm, demonstrating how a single-qubit can be treated as a shallow neural network. By analyzing the capacity and generalization bounds of photonic circuits - utilizing classical concepts such as VC dimension - we aim to show that the future of quantum research requires the rigor of learning theory. Reference: Mauser, M.F.X. et al. (2025) ‘Experimental data re-uploading with provable enhanced learning capabilities’. arXiv. Available at: doi.org/10.48550/arXiv.2507.05120.

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We derive lower bounds on the minimax estimation error for binary classifiers under a geometric margin condition. Our approach is based on constructing a special finite subset within an infinite class of classifiers, where the size of this subset depends on the sample size available to the learning algorithm. We then introduce a family of probability densities associated with this finite set of classifiers and apply Assouad’s lemma, which provides a lower bound on the minimax error corresponding to this construction. In our setting, we consider horizon-type classifiers with regular decision boundaries defined by functions belonging to Hölder spaces, Barron spaces, and convex Lipschitz function classes. This is a joint work with Philipp Petersen.

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We address the floating-point computation of compositionally-rich functions, concentrating on LLM inference. Based on the rounding error analysis of a composition, we provide an adaptive strategy to select components of the inner function that need to be recomputed more accurately to improve the numerical stability. We explain how this strategy can be applied to different compositions within a transformer neural network and illustrate its overall effect on LLM inference.

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