This workshop explores the interplay between the approximation of functions and computational harmonic analysis. It focuses on the mathematical foundations of cutting-edge algorithms in areas such as image and signal processing, numerical analysis, and data science.
This workshop intersects most directly with the Foundations of Data Science and Machine Learning, Information-Based Complexity, and Foundations of Numerical PDEs workshops.
Organizers
Simon Fraser University
Sorbonne Université
University of Vienna
Speakers
Semi-plenary speakers
University of Warwick
University of California, Davis
Invited speakers
RWTH Aachen
INRIA
University of South California
University of Oregon
Concordia University
Mohammed VI Polytechnic University
CNRS Besancon
CUNEF University
University Ottawa
KU Leuven
Michigan State University
Georgia Institute of Technology
University of Vienna
University of California, San Diego
Texas A&M University
University of Vienna
University of Genoa
Texas A&M University
Princeton University
Monday, 13 July
![[HS 04 – Ground Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS04.png){kind=link}
14:00-15:00 semi-plenary talk
Thomas Strohmer (University of California, Davis)
As AI models are trained on ever-expanding datasets, the ability to remove the influence of specific data from trained models has become essential for privacy protection and regulatory compliance. Unlearning addresses this challenge by selectively removing parametric knowledge from the trained models without retraining from scratch, which is critical for resource-intensive models such as Large Language Models (LLMs). However, existing unlearning methods often severely degrade model performance by removing more information than necessary when attempting to “forget” specific data. We introduce a mathematical framework based on information-theoretic regularization that can accommodate different types of machine unlearning, such as feature unlearning and data point unlearning. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. For LLMs, we propose Forgetting-MarI, an unlearning framework that provably removes only the additional (marginal) information contributed by the data to be unlearned, while preserving the information supported by the data to be retained. Extensive experiments confirm that our approach outperforms current state-of-the-art unlearning methods, delivering reliable forgetting and better preserved general model performance across diverse benchmarks. This advancement represents an important step toward making AI systems more controllable and compliant with privacy and copyright regulations without compromising their effectiveness. We will also discuss applications in machine learning driven scientific discovery. This is joint work with Shizhou Xu, Yuan Ni, and Stefan Broecker.
15:00-15:30
Wenjing Liao (Georgia Institute of Technology)
When training deep neural networks, a model’s generalization error is often observed to follow a power scaling law dependent on the model size and the data size. A prominent example is transformer-based large language models (LLMs), where networks with billions of parameters are trained on trillions of tokens. A theoretical interest in LLMs is to understand why transformer scaling laws emerge. In this talk, we exploit low-dimensional structures in language datasets by estimating its intrinsic dimension and establish statistical estimation and mathematical approximation theories for transformers to predict the scaling laws. This perspective shows that transformer scaling laws can be explained in a manner consistent with the underlying data geometry. We further validate our theory with empirical observations of LLMs and find strong agreement between the observed empirical scaling laws and our theoretical predictions. Finally, we turn to in-context learning, analyzing its scaling behavior by uncovering a connection between the attention mechanism in transformers and classical kernel methods in machine learning.
15:30-16:00
Matteo Santacesaria (University of Genoa)
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional, up to logarithmic factors, to the sparsity of the unknown signal. The classical theory mostly considers either random linear measurements or subsampled
isometries. In particular, the case with the subsampled Fourier transform finds applications to undersampled magnetic resonance imaging. In this talk, I will show how the theory of compressed sensing can also be rigorously applied to the sparse Radon transform, in which only a finite number of angles are considered. One of the main novelties consists in the fact that the Radon transform is associated to an ill-posed inverse problem, and the result follows from a new theory of compressed sensing for abstract inverse problems. This is a joint work with G.S. Alberti, A. Felisi and S.I. Trapasso.
16:00-16:30 Coffee Break
16:30-17:00
Ujué Etayo (CUNEF Universidad Madrid)
Determinantal point processes provide a structured random model for generating repulsive point sets on manifolds, and they are natural candidates for producing low-discrepancy configurations. In this talk I will discuss recent discrepancy bounds for homogeneous determinantal processes on compact, connected two-point homogeneous spaces, including spheres and projective spaces. The key point is a general principle: for homogeneous determinantal processes, discrepancy with respect to metric balls can be bounded in terms of concentration estimates and the variance of the number of points inside balls. This reduces the problem to estimating a variance integral adapted to the kernel. For the harmonic ensemble, this yields a high-probability bound of order (N^{\frac12(1-1/D)}\log N), where (D) is the real dimension of the manifold. For the projective ensemble on (\mathbb{CP}^d), one obtains the sharper order ((N^{1-1/D}\log N)^{1/2}). These results extend earlier sphere-based estimates to all compact, connected two-point homogeneous spaces and illustrate how repulsion can improve geometric uniformity.
17:00-17:30
Markus Bachmayr (RWTH Aachen)
The approximation of high-dimensional antisymmetric functions is a central problem in methods for fermionic quantum systems, in particular in quantum chemistry. Such functions can be represented in terms of occupation numbers with respect to a suitable one-particle basis of orbitals. In this picture, antisymmetry is encoded in operator representations rather than functions, and thus low-rank tensor approximations based on tensor networks become applicable for representing high-dimensional occupation number tensors. In this talk, I give an overview of some first mathematical results on such low-rank approximations, of the role of symmetries and constraints, and of recent results on eigensolvers. Based on joint works with Michael Götte, Sebastian Krämer, and Max Pfeffer
17:30-18:00
Simone Brugiapaglia (Concordia University)
Motivated by applications such as sparse PCA, we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The resulting compressive-sensing-based approaches can approximate the leading eigenvectors of huge approximately low-rank matrices that are too large to store in memory based on a single pass over its entries while utilizing a total memory footprint on the order of the much smaller desired sparse eigenvector approximations. Moreover, the compressive sensing recovery algorithm itself can also be formulated to run in a time which principally depends on the size of the sought sparse approximations, making its runtime sublinear in the size of the large matrix whose eigenvectors one aims to approximate. Preliminary experiments on huge matrices having approximately 10^{16} entries illustrate the developed theory and demonstrate the practical potential of the proposed approach.
This presentation is based on joint work with Edem Boahen, Hung-Hsu Chou, Mark Iwen, and Felix Krahmer.
Tuesday, 14 July
14:00-14:30
Mark Iwen (Michigan State University)
In this talk we discuss sparse spectral methods capable of rapidly and automatically determining a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given PDE on a (potentially very high-dimensional) periodic domain. This small, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE’s data compressibility properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations will be discussed in the elliptic setting, as well as initial attempts to extend the methods to related parabolic PDE. Numerical experiments will demonstrate good empirical performance on several multiscale and high-dimensional example problems, showcasing the promise of the proposed methods in practice.
14:30-15:00
Laurent Baratchart (INRIA)
Based on joint work with: A. Borichev, S. Chevillard, R. Zarouf and C. Coiffard-Marre. We derive lower bounds in best rational approximation of given degree to Blaschke products, in the Hardy space H2 of the unit disk. It is well-known that Blaschke products, finite or infinite, cannot be approximated by rational functions of lower degree in the sup norm on the circle (or H∞-norm on the disk) : the best approximant is zero. The situation is of course different for the L2 norm (or H2 norm on the disk), but one may surmise that the rate of approximation when the degree increases is poor. After recalling results on the rate of approximation to infinite Blaschke products in terms of the geometry of their zeros, we turn to lower bounds on the L2-approximation error to finite Blaschke products by rational functions of lower degree. We first consider approximation to zN, which can be interpreted as model reduction of pure delays in the sense of minimum variance, using for instance linear neural nets; we prove in particular that a lower bound in rational approximation of degree n is (1 − n/N)1/2 in this case. Then we move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.
15:00-15:30
Amit Singer (Princeton University)
Cryo-EM is a Nobel Prize winning technology for determining 3-D biological molecular structures at high resolution. In cryo-EM, the 3-D structure needs to be determined from many 2-D noisy tomographic projection images of the molecule taken at unknown viewing directions and positions. The maximum likelihood approach has been very successful in determining large molecular structures, but struggles with small molecules for which the signal-to-noise ratio of the images is too low. Motivated by the challenge of reconstructing small molecules by cryo-EM, we have been investigating the method of moments as an alternative statistical and computational framework. The method of moments draws parallels with the phase retrieval problem, and provides theoretical insight about the sample complexity, that is, the number of noisy images required for reconstruction. No prior knowledge of cryo-EM or the method of moments is necessary for this talk.
15:30-16:00
Daan Huybrechs (KU Leuven)
Modern neural networks are typically trained using large amounts of data and large-scale numerical optimization, in noisy settings where loss is acceptable. They are also able to represent members of classical function spaces to high accuracy, but that requires solving a highly nonlinear approximation problem. We show how to construct such networks, for a broad class of functions, with exponential accuracy in the depth of the network. The approach applies to networks with the ReLU activation function, in which case deep networks correspond to piecewise linear approximations with exponentially many pieces. This talk is joint work with Matteo Margini.
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Clarice Poon (University of Warwick)
In this talk, I will discuss a particular inverse problem arising from Optimal transport (OT). OT is now a central modeling tool to compare and couple probability distributions, with applications spanning economics, imaging, generative modeling, and computational biology. In many modern pipelines, however, the transport cost is not known a priori and must be inferred from data. This leads to inverse optimal transport (iOT): recover the ground cost (or metric parameters) from an observed optimal coupling. On the other hand, modern computational pipelines typically exploit an entropic regularization variant (eOT) of OT. I will discuss iOT in a regime that is both mathematically delicate and practically unavoidable: the entropic regularization level is small (approaching the unregularized OT model), while the coupling is observed through a finite number of samples.
17:30-18:00
Genevieve Dusson (CNRS Besancon)
Constructing N variables functions that are both permutation-invariant (PI) and group-equivariant (GE) under some Lie group action is required in many scientific fields ranging from molecular modeling to machine learning on set-based models. In general, the cost of generating such functions suffers from an exponential scaling with respect to the number of variables N, as existing constructions symmetrise explicitly over permutations, limiting their applicability for large systems.
In this talk, we present a general, scalable construction of such functions. Starting from a given finite-dimensional space stable with respect to the group action, we leverage the Lie algebra to build a matrix whose kernel spans the space with desired equivariance and invariance properties. For SO(3) and SU(2), we simplify the form of the linear system and prove explicit dimensionalities. Thus our method scales linearly with respect to the dimension of the considered spaces, largely outperforming traditional approaches. Finally we analyze the dimensionalities depending on the considered symmetries (PI or GE or GE-PI) and show that asymptotically the number of PI and GE-PI functions are of same order, while preasymptotically, a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.
This is a joint work with Eloïse Barthélémy, Camille Hernandez, Liwei Zhang
18:00-18:30
Moulay Abdellah Chkifa (Mohammed VI Polytechnic University)
We study the approximation of functions from point evaluations in general approximation spaces and reproducing kernel Hilbert spaces (RKHS), with emphasis on scalable algorithms for large-scale problems. Building on randomized least-squares and spectral sparsification techniques, we develop constructive sampling and discretization procedures achieving near-optimal guarantees with minimal oversampling. We extend these guarantees to infinite-dimensional RKHS settings via a generalized two-sided sparsification framework, deriving discretization inequalities in both L2 and uniform norms, constructive recovery procedures, and quantitative error bounds. These results strengthen several previously known bounds, improve oversampling factors, and yield practical algorithms for high-dimensional and structured approximation spaces, replacing non-constructive existence arguments by implementable sampling schemes with reduced sample complexity.
A central contribution is the design of fast computational procedures based on resolvent evaluations of low-rank matrices. Exploiting Woodbury-type identities, we efficiently compute inverses and resolvents arising in weighted least-squares and kernel discretization systems, reducing key linear-algebraic operations from full-dimensional solves to updates involving small matrices. This makes the overall methods practical in high-dimensional and large-sample regimes.
[1] M. A. Chkifa and M. Dolbeault, Randomized least-squares with minimal oversampling and interpolation in general spaces, SIAM Journal on Numerical Analysis, 62(4), 1515–1538, 2024.
[2] M. A. Chkifa, M. Dolbeault, D. Krieg, and M. Ullrich, Constructive discretization and approximation in reproducing kernel Hilbert spaces, arXiv preprint arXiv:2602.18719, 2026.
Wednesday, 15 July
14:00-14:30
Jonathan Siegel (Texas A&M University)
We consider two problems related to recovery of functions from point samples. For the first, we suppose that an unknown function is convex and Lipschitz on the unit cube in $d$ dimensions. We wish to recover this function from $n$ point samples, i.e., the values of the function at $n$ chosen points, with error measured in the $L_p$-norm. We also consider the same problem when the target function is only convex and bounded. For the second problem, suppose that an unknown convex subset $C$ of the unit cube is given. We wish to recover this subset, with error in the symmetric difference metric, from the following information. We choose $n$ points in the unit cube and observe which of these points lie in the set $C$.
For both of these problems, the natural questions are: where should the sample points be chosen? Given the observed data, how should the function or set be recovered? and what error rate can be achieved?
We will provide nearly tight (up to logarithmic factors) answers to these questions in both of these settings, and discuss the techniques and ideas involved in proving them.
14:30-15:00
Peter Binev (University of South Carolina)
We consider adaptive approximation of data points with unknown distribution in high dimensions. The case of lower intrinsic dimension of the point clouds is of particular interest. The domain can be partitioned into hyper rectangles or simplices and the adaptive approximation uses the idea of sparse occupancy tries to avoid some curse-of-dimensionality issues. While the piece-wise constant approximation is well understood, the use of local approximation by higher order polynomials is problematic due to the possible unbalanced distribution of the point cloud within the cells of the partition. We propose to use a distribution-based local approximation scheme that provides higher order approximation with high probability.
15:00-15:30
Diane Guignard (University of Ottawa)
We consider the problem of recovering the velocity and pressure fields of a Stokes system when the boundary conditions are unknown. Assuming access to finitely many velocity–pressure measurements together with a regularity model class for the solution, the problem can be formulated within the framework of optimal recovery.
Extending previous work for scalar elliptic equations, we develop a recovery algorithm based on numerical approximations of the Riesz representers of the measurement functionals. This requires, in particular, the solution of fractional problems posed on the boundary of the domain. We prove that the method is near-optimal not only for recovering the velocity–pressure pair itself, but also for approximating arbitrary linear quantities of interest, including drag and lift coefficients.
Compared with the Poisson setting, the Stokes system introduces several additional challenges: the Riesz representers are themselves velocity–pressure pairs, nontrivial compatibility conditions must be enforced, and the mean value of the pressure must also be recovered. Numerical experiments illustrate the accuracy, stability, and robustness of the proposed approach.
The talk is based on joint work with Andrea Bonito.
15:30-16:00
José Luis Romero (University of Vienna)
We address the problem of computing the zero set of a complex-valued function on the complex plane. Assuming access to the function only via a finite grid of evaluation points, we aim to approximate the zero set from this limited data. In practice, zeros of such functions often serve as landmarks for signal processing, and the grid density is dictated by the constraints of the acquisition device. We develop algorithms based on local changes in complex phase and magnitude, and analyze them under a stochastic model that describes practical performance. We demonstrate that ‘failure’ instances are fragile in the sense that they are regularized by additive noise (smoothed analysis). Accuracy is measured by estimating the cost of transporting the true zero set into the computed set (Wasserstein metric). Based on joint work with Luis Alberto Escudero, Naomi Feldheim, Antti Haimi and Günther Koliander.
16:00-16:30 Coffee Break
16:30-17:30 (talk Popov prize)
Rahul Parhi (University of California, San Diego)
This talk investigates the fundamental differences between low-norm and flat solutions of shallow ReLU networks training problems, particularly in high-dimensional settings. We sharply characterize the regularity of the functions learned by neural networks in these two regimes. This enables us to show that global minima with small weight norms exhibit strong generalization guarantees that are dimension-independent. In contrast, local minima that are “flat” can generalize poorly as the input dimension increases. We attribute this gap to a phenomenon we call neural shattering, where neurons specialize to extremely sparse input regions, resulting in activations that are nearly disjoint across data points. This forces the network to rely on large weight magnitudes, leading to poor generalization. Our analysis establishes an exponential separation between flat and low-norm minima. In particular, while flatness does imply some degree of generalization, we show that the corresponding convergence rates necessarily deteriorate exponentially with input dimension. These findings suggest that flatness alone does not fully explain the generalization performance of neural networks.
17:30-18:00
Guergana Petrova (Texas A&M University)
Classical Kolmogorov widths were introduced in the 1930’s as a theoretical benchmark for how well a compact class K in a Banach space X can be approximated by linear methods. In recent years, in the pursuit of numerically more effective methods of approximation, new types of widths have been introduced. In this talk, we will focus on two of these new widths: the Lipschitz widths and the constrained widths. We motivate their definition, discuss some of their properties, and determine their asymptotic rate of decay for some natural compact classes.
18:00-18:30
Marcin Bownik (University of Oregon)
In this talk we discuss the relationship between the frame measure function in certain reproducing kernel Hilbert spaces on metric measure spaces and the Beurling density. We show that a frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. As an application, we settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for non-localized Gabor frames. The techniques used to show these results combine a selector form of Weaver’s conjecture and various methods for quantifying the overcompleteness of frames. This talk is based on a joint work with Jordy van Velthoven.
Posters
![[PZ01 & PZ04 – Ground Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_erdgeschoss.png){kind=link}
Thomas Allard
(Stanford University)
Entropy numbers of operators are a fundamental tool in approximation theory, harmonic analysis, and statistical learning. Standard approaches for estimating them usually rely on spectral methods, leveraging information about the operator’s eigenvalues. In many situations, however, such information is not directly available, and only (partial) information about the time-frequency representation of the operator is accessible. We introduce a new method for this setting. Specifically, our approach combines techniques from microlocal analysis with new sharp entropy estimates for diagonal operators, and leads to a simple, ready-to-use, formula. We illustrate how this formula can be used with two applications. First, we apply it to characterize the metric entropy of unit balls in Sobolev spaces on arbitrary multi-dimensional domains. Second, we obtain sharp asymptotic characterizations on the minimax risk in non-parametric estimation from the time-frequency information of the functions being estimated. This poster presentation is based on [1] and is joint work with H. Bölcskei.
[1] Allard, T., & Bölcskei, H. (2026). Entropy and Minimax Risk of Hypoelliptic Pseudodifferential Operators. arXiv preprint arXiv:2603.23744.
Jinhui Bai
(Fudan University Shanghai)
Inspired by the structure of spherical harmonics, we propose the truncated kernel stochastic gradient descent (T-kernel SGD) algorithm with a least-square loss for spherical data fitting. T-kernel SGD introduces a novel regularization strategy by implementing stochastic gradient descent through a closed-form solution of the projection of the stochastic gradient in a low-dimensional subspace. In contrast to traditional kernel SGD, the regularization strategy implemented by T-kernel SGD is more effective in balancing bias and variance by dynamically adjusting the hypothesis space during iterations. The most significant advantage of the proposed algorithm is achieving theoretically optimal convergence rates using a constant step size (independent of the sample size) while overcoming the inherent saturation problem of kernel SGD. Additionally, we leverage the structure of spherical polynomials to derive an equivalent T-kernel SGD, significantly reducing storage and computational costs compared to kernel SGD. Our main results quantitatively characterize how this prior information influences the convergence of T-kernel SGD. Numerical experiments demonstrate that T-kernel SGD effectively fits spherical data.
Peter Balazs
(ARI Vienna)
Frame multipliers are an abstract version of Toeplitz operators in frame theory and consist of a composition of a multiplication operator with the symbol, between the analysis and synthesis operators of thje frame. We consider Banach spaces associated to a frame, so-called coorbit spaces, which are defined by the coefficents being in a weighted sequence space.
Whereas the boundedness properties of frame multipliers are well understood, their invertibility is much more difficult. We show that frame multipliers with a positive symbol are Banach space isomorphisms between coorbit spaces. To offer a flavor of our result: Under suitable conditions on the frame and the weights, the frame multiplier is a Banach space isomorphism between the co-orbit spaces with weights related by the symbol.
The main techniques are the theory of localized frames and existence of inverse-closed matrix algebras. The results resemble the lifting theorems in the theory of Besov spaces and modulation spaces. Indeed, the application of the abstract lifting theorem to Gabor frames yields a new lifting theorem between modulation spaces. A second application to Fock spaces yields isomorphisms between weighted Fock spaces.
This is joint work with K. Gröchenig
Leandro Bentancur
(Universidad de la República)
A basic problem in optimization and statistics is the recovery of a measure from a given collection of moments. The Christoffel-Darboux (CD) function (and its reproducing kernel) is a classical tool from approximation theory and orthogonal polynomials for this task. The CD function allows us to obtain an approximation of the density of the measure as an explicit rational function involving the inverse of the given truncated moment matrix.
We introduce Mollified Christoffel-Darboux kernels on algebraic varieties, a systematic regularization of the classical CD kernel. The main motivations are twofold: first, sharpening the classical dichotomy between the behavior of the CD function on and off the support; second, obtaining consistent and quantitatively controlled recovery of densities from moment data, without the need to know the equilibrium measure of the underlying domain.
Our main contributions are as follows: (i) We introduce families of mollifiers on algebraic varieties. For each measure and degree d on such a variety, we define a mollified CD kernel, which can be computed from the moments of the underlying measure by linear algebra. (ii) We prove that an improved dichotomy property holds: on the interior of the support, the mollified CD polynomial is uniformly bounded in d, while outside the support it grows exponentially fast in d. (iii) Assuming Sobolev regularity of the density with respect to a reference measure, we derive explicit convergence rates for density recovery in R^n via mollified CD kernels. (iv) On the unit sphere, we show that suitably chosen algebraic mollifiers, constructed from zonal polynomials, lead to kernels with improved rates, building on classical constructive approximation results. We illustrate the theory with numerical experiments on the sphere.
This is joint work with Didier Henrion (LAAS, CNRS, France; FEL, CTU Prague, Czechia) and Mauricio Velasco (Udelar, Uruguay).
Pablo M. Berná
(CUNEF Universidad)
This work investigates the relationship between classical non-linear approximation spaces, greedy classes and weighted Lorentz sequence spaces. While previous characterizations often required strong conditions like unconditionality or almost greediness, we extend these results to a broader class of bases by utilizing the property of being truncation quasi-greedy. We establish optimal embeddings by proving Bernstein-type inequalities for truncation quasi-greedy bases and Jackson-type inequalities for bases satisfying the Approximability by Projections Property (APP).
The main contribution of this research is a complete characterization theorem: for any truncation quasi-greedy basis with the APP, the classical approximation spaces, greedy classes, and Chebyshev-greedy classes coincide if and only if the basis is democratic or superdemocratic. This provides a unified framework that relaxes traditional constraints and clarifies the role of democracy in the efficiency of the Thresholding Greedy Algorithm (TGA).
Peter Binev
(University of South Carolina)
We consider adaptive approximation of data points with unknown distribution in high dimensions. The case of lower intrinsic dimension of the point clouds is of particular interest. The domain can be partitioned into hyper rectangles or simplices and the adaptive approximation uses the idea of sparse occupancy tries to avoid some curse-of-dimensionality issues. While the piece-wise constant approximation is well understood, the use of local approximation by higher order polynomials is problematic due to the possible unbalanced distribution of the point cloud within the cells of the partition. We propose to use a distribution-based local approximation scheme that provides higher order approximation with high probability.
Biraj Dahal
(Georgia Institute of Technology)
Chenguang Duan
(RWTH Aachen)
In the context of training neural network-based approximations of solutions of parameter-dependent PDEs, we investigate the effect of preconditioning via well-conditioned frame representations of operators and demonstrate a significant improvement on the performance of standard training methods. We also observe that standard representations of preconditioned matrices are insufficient for obtaining numerical stability and propose a generally applicable form of stable representations that enables computations with single- and half-precision floating point numbers without loss of precision. Joint work with Markus Bachmayr, Wolfgang Dahmen, and Mathias Oster.
Markus Faulhuber
(University of Vienna)
We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.
Christian Fiedler
(TU Munich)
Large-scale multiagent systems play an important role in many areas of engineering, mathematics, and natural sciences. One way to handle systems with a very large number of agents is the mean field limit from kinetic theory, which provides a principled way to transition from a microscopic to a mesoscopic modelling level. Motivated by learning problems for large-scale interacting particle systems, we introduce and investigate the mean field limit of kernels and their reproducing kernel Hilbert spaces. In turn, this is applied to kernel-based statistical learning, where we establish appropriate mean field limit notions of statistical learning problems, and prove mean field convergence of support vector machines for these problems. On the one hand, this provides a solid theoretical foundation for learning problems for functions of many inputs, as appearing in the context of learning on large-scale interacting particle systems. On the other hand, it constitutes a new kind of large-scale limit in the theory of machine learning, complementing existing settings like infinite-width neural networks and the classic asymptotic high-dimensional setting of statistics.
References
Fiedler, C., Herty, M., & Trimpe, S. (2023). On kernel-based statistical learning theory in the mean field limit. NeurIPS 2023
Fiedler, C., Herty, M., Rom, M., Segala, C., & Trimpe, S. (2023). Reproducing kernel Hilbert spaces in the mean field limit. Kinetic and Related Models, 16(6), 850-870.
Fiedler, C., Herty, M., Segala, C., & Trimpe, S. (2025). Recent kernel methods for interacting particle systems: first numerical results. European Journal of Applied Mathematics, 36(2), 464-489.
Luca Finotti
(University of Genoa)
Reconstructing an infinite-dimensional signal from a finite set of measurements is a pivotal problem in sampling theory and signal processing. The generalized sampling framework provides a reliable methodology for recovering elements of arbitrary separable Hilbert spaces from a sufficient (yet finite) amount of its samples. However, deterministic approaches suffer from severe basis-dependent dimensionality constraints, often requiring a quadratic sampling rate $m\gtrsim n^2$ to avoid numerical instability. In this paper, we develop a stochastic framework to generalized sampling that overcomes such deterministic barriers. By picking the samples at our disposal according to a suitable probability distribution, we demonstrate that stable reconstruction can be achieved with high probability at a near-linear sampling rate of $m\gtrsim n\log(n)$. This result is independent of the chosen bases and is proven to hold even when both the sampling and the reconstruction systems are frames. Furthermore, we provide a new concentration inequality to rigorously study the behavior of the empirical cross-term. We showcase the effectiveness of this method on the classic problem of recovering analytic functions from Fourier measurements, where we achieve near-exponential convergence rates.
Max Getter
(RWTH Aachen)
We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Our main result establishes Cheeger-type bounds that relate reconstruction stability to a connectivity quantity of the measurement structure, relative to kernel localization. The resulting inequalities give interpretable necessary conditions for local stability and, in the real case, also sufficient ones. They provide a diagnostic for when phaseless recovery becomes ill-conditioned.
Andrea García
(Universidad CEU San Pablo, CEU Universities)
Greedy algorithms are essential tools for nonlinear approximation in Hilbert spaces using redundant dictionaries. A landmark in this field is the Relaxed Greedy Algorithm (RGA), introduced by DeVore and Temlyakov, which utilizes a relaxation step—a convex combination of the previous approximation and the new dictionary element—to achieve optimal convergence rates for functions with bounded atomic norm.
In this poster, we present and analyze new algorithms that generalize the RGA framework. First, we explore the Power-Relaxed Greedy Algorithm (PRGA), which replaces the standard 1/m relaxation parameter with 1/m^alpha. We provide a negative answer to the open question of its convergence for alpha > 1, demonstrating both theoretically and numerically that the algorithm may fail to converge in this range. Furthermore, we introduce the Convex-Relaxed Greedy Algorithm (CRGA), which incorporates an adaptive relaxation step calculated via an exact line search. Our results establish that the CRGA restores and maintains optimal convergence rates, offering a more robust and efficient generalization of the classical RGA.
Max Getter
(RWTH Aachen University)
We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Our main result establishes Cheeger-type bounds that relate reconstruction stability to a connectivity quantity of the measurement structure, relative to kernel localization. The resulting inequalities give interpretable necessary conditions for local stability and, in the real case, also sufficient ones. They provide a diagnostic for when phaseless recovery becomes ill-conditioned.
Avi Gupta
(Simon Fraser University)
A central problem in approximation theory is the recovery of high-dimensional functions from limited data. In many cases, the functions of interest exhibit anisotropic smoothness. Moreover, in many practical settings, the nature of this anisotropy may be unknown a priori. Therefore, an important question involves the development of `universal algorithms’, namely, algorithms that simultaneously achieve optimal or near-optimal rates of convergence across a range of different anisotropic smoothness classes. In this work, we consider universal approximation of periodic functions that belong to anisotropic Sobolev spaces and anisotropic dominating mixed smoothness Sobolev spaces. Our first result is the construction of a universal algorithm. This algorithm recasts function recovery as a sparse recovery problem for Fourier coefficients of such a function. It then exploits compressed sensing to yield the desired approximation rates. Next, we demonstrate optimality of this universal algorithm up to a dimension-independent polylogarithmic factor. We do this by deriving a lower bound for the adaptive m-width for the unit balls of such function classes. Finally, we demonstrate the necessity of nonlinear algorithms. We show that universal linear algorithms can achieve rates that are suboptimal by a dimensional dependent polylogarithmic factor. In other words, they suffer from a curse of dimensionality in the rate – a phenomenon which justifies the necessity of nonlinear algorithms for universal recovery of anisotropic functions.
This is joint work with Ben Adcock (SFU).
Andrew Horning
(Rensselaer Polytechnic Institute)
Linear operators with a continuous spectrum often lurk behind complex physical phenomena in nature, from wave attractors in geometrically confined fluids to topological bifurcations in dynamical systems. However, they are notoriously tricky to learn from data. For example, finite-dimensional approximations of the operator must “discretize” the continuous spectrum into finitely many points and cannot converge uniformly.
This poster explores two principled approaches to data-driven approximation of unitary operators with continuous spectrum. The first approach synthesizes recent advances in computational spectral theory with stable algorithms for Gauss-Szego quadrature rules on the unit circle. The second approach leverages new algorithms for quadrature rules that adaptively deform and discretize the contour inside the unit circle, resembling classical resonance expansions for wave operators. We present strong convergence rates for the approximate operators, data-driven aspects of the approximation, and highlight the unifying power of quadrature rules derived from rational approximations.
Gregor Maier
(University of Bonn)
Operator approximation has gained increasing research attention in recent years and gave rise to a new field called “Operator Learning”. Approximate operators, learned from data, hold promise to serve as efficient surrogate models for problems in computational science and engineering, complementing traditional numerical methods. Previous research efforts in understanding the underlying mathematical theory mainly focused on holomorphic operators and revealed that these can be approximated efficiently: The worst-case error scales algebraically or even exponentially with the reciprocal of the number of training samples.
We show that this data efficiency deteriorates if we consider less regular operators. More specifically, approximating operators which are merely Sobolev regular with respect to a Gaussian measure exhibits an inherent curse of sample complexity: No method (polynomials, neural networks, kernel methods, etc.) to approximate such operators on infinite-dimensional domains based on m (possibly adaptively chosen) linear samples can achieve algebraic convergence rates in m. This includes the important class of Lipschitz operators, which arise, e.g., as data-to-solution mappings in parametric variational inequalities.
To assess the consequences of these asymptotic theoretical limitations for practical applications, we present a data-driven spectral algorithm for the recovery of Gaussian Sobolev operators from finitely many noisy point samples, which has near-optimal sample complexity. It is based on weighted least-squares approximation, does not require knowledge of the underlying Gaussian measure, and achieves higher convergence rates the smoother the operators are. We complement our theoretical analysis with numerical experiments.
This is joint work with Ben Adcock (Simon Fraser University), Michael Griebel (University of Bonn), and Rahul Parhi (University of California, San Diego). It is partially based on the following papers: arXiv:2410.23440, arXiv:2512.17805.
Nicolas Mil-Homens Cavaco
(UCLouvain)
Sinusoidal neural networks, such as SIREN, has recently emerged as a particularly convenient architecture for Implicit neural representations (INRs); a powerful tool allowing efficient encoding of complex signals.
These INRs have indeed achieved very promizing results in a large range of applications, e.g., in computer vision, physics-informed machine learning and inverse problems, namely thanks to their ability to mitigate the so-called spectral bias. Despite recent advances (see, e.g., Yüce et al., A Structured Dictionary Perspective on Implicit Neural Representations (2022)), INRs still remain poorly understood from a theoretical point of view.
In this work, we rely on frame theory from signal processing and reproducing kernel Hilbert space, a core component of kernel methods in machine learning, to characterize the expressive power of INRs under the lazy regime, where only the weights of the last layer are trained. This regime has become of great interest since both the so-called neural network gaussian process (NNGP) and neural tangent kernel (NTK) have been discovered.
Through asymptotic and non-asymptotic analyses, we show that a lazy sinusoidal neural network also shares the same spectrum expansion property as the fully trained neural network,
and in particular is able to boost the bandwidth of the positional encoding provided at the first layer of the network.
More precisely, the kernel induced by a sinusoidal NNGP has been derived in closed form and a non-asymptotic analysis shows that
the properties derived from NNGP also hold for a finite lazy sinusoidal neural network.
Our results are supported by several numerical experiments.
Lukas Odelius
(University of Vienna)
In recent work with Antti Haimi and José Luis Romero we have studied the the local behavior of the zeros and critical points of the standard Gaussian Entire Function (GEF), in particular we have considered the asymptotics of the second order correlations of the corresponding number statistics on small observation disks and interpreted the results in terms of (local) repulsion.
It was known that the zeros of the GEF are repulsive of order 6 and that the critical point are (weakly) repulsive of order 4, in our work we use a method due to Safa Ladgham and Raphaël Lachièze-Rey to show the following:
1) The critical points of the same index are repulsive of order 7.
2) The overall (weak) repulsion of critical points is due to interactions between critical points of different indices.
3) The repulsion between zeros and critical points of positive index is of order 6.
4) The repulsion between zeros and critical points of negative index is much stronger of order 20.
5) The overall repulsion between zeros and critical points is of order 6 and is achieved by the subclass of critical points of positive index.
This result can be reformulated in terms of the weighted amplitude S of the GEF, where now critical points of positive index are saddle points of S and critical points of negative index are local maxima of S, as the STFT (with Gaussian window) of standard complex (Gaussian) white noise can be identified with the square of the weighted amplitude of the GEF our result has implications for signal processing and supports heuristics from the signal processing literature. For example spectrogram reassignment is a popular method used to sharpen spectrograms based on a vector field whose attractors are the spectrogram local maxima and repellers are the spectrogram zeros, the very strong repulsion between zeros and maxima of S is strongly consistent with the success of this sharpening method.
Maximilian Penka
(TU Munich)
We investigate nonlinear reduced-order modeling techniques for parametrized eigenvalue problems, motivated by applications in electronic structure calculations. In this setting, traditional linear reduced-order models (ROMs), such as those based on Galerkin projections, can perform poorly due to the highly nonlinear and often non-smooth dependence of the solution on atomic positions.
Building on recent work on nonlinear reduced basis methods using mixture Wasserstein barycenters [1], we consider reduced approximations constructed not in a linear trial space, but on a nonlinear manifold generated by barycenters of precomputed snapshots. This changes the underlying approximation geometry: instead of combining solutions by linear superposition, one interpolates them in a transport-informed metric space.
This point of view is particularly well suited for problems where the solutions exhibit localization, symmetry, or other structures that are poorly captured by linear spaces. A central objective of the ongoing work is to develop a rigorous mathematical analysis of such nonlinear reduced models, including approximation properties, stability, and computable error indicators.
[1] Dalery, M., Dusson, G., Ehrlacher, V. et al. Nonlinear Reduced Basis using Mixture Wasserstein Barycenters: Application to an Eigenvalue Problem Inspired from Quantum Chemistry. Constr Approx (2026).
Polina Sachsenmaier
(RWTH Aachen)
Standard numerical methods for solving PDEs typically suffer from the curse of dimensionality: their computational cost scales exponentially with the dimension of the underlying domain, making them impractical even at low resolution. In many cases of interest, however, such limitations can be overcome by appropriately compressed representations of approximate solutions, for example, by low-rank tensor representations.
For time-dependent PDEs, several approaches to low-rank approximation exist that control ranks in different ways. These range from methods that keep ranks fixed, such as dynamical low-rank approximations, which may lead to uncontrolled errors, to methods that approximate standard time-stepping schemes to any desired accuracy but can produce unnecessarily large ranks.
We develop time integration methods in low‑rank tensor representations that adaptively adjust the approximation ranks to meet a prescribed accuracy while simultaneously maintaining control over the ranks of the computed approximations and all intermediate quantities. These ranks are a main determining factor in the computational costs of such methods. Our approach combines an iterative time-stepping scheme with soft thresholding of the iterates. In the matrix case, the proposed strategy yields iterates whose ranks remain comparable to natural benchmark quantities — namely, the best approximation ranks of the sought solutions at the achieved accuracy. In the higher-dimensional tensor case the algorithm can be adapted appropriately, leading to global error and rank bounds that depend only polynomially on the dimension. Numerical experiments illustrate the theory for linear time-dependent Schrödinger equations.
This poster is based on joint works with Markus Bachmayr, Matthieu Dolbeault, Tianyu Jin and Federico Vismara.
Simone Sanna
(University of Genoa)
The Calderón problem consists in recovering an unknown coefficient of a partial differential equation from boundary measurements of its solution. These measurements give rise to a highly nonlinear forward operator. As a consequence, the development of reconstruction methods for this inverse problem is challenging, as they usually suffer from the problem of local convergence. To circumvent this issue, we propose an alternative approach based on lifting and convex relaxation techniques, that have been successfully developed for solving finite-dimensional quadratic inverse problems. This leads to a convex optimization problem whose solution coincides with the sought-after coefficient, provided that a non- degenerate source condition holds. We demonstrate the validity of our approach on a toy model where the solution of the partial differential equation is known everywhere in the domain. In this simplified setting, we verify that the non-degenerate source condition holds under certain assumptions on the unknown coefficient. We leave the investigation of its validity in the Calderón setting for future works.
Alexander Sietsema
(University of California, Los Angeles)
Certain machine learning models exhibit benign overfitting, where a model is capable of both perfectly interpolating a training dataset as well as generalizing well to unseen data. Recent work in overparameterized machine learning has studied benign overfitting in a variety of settings. We study the generalization behavior of functions in Sobolev spaces that perfectly fit a noisy training data set. Under assumptions of label noise and sufficient regularity in the data distribution, we show that approximately norm-minimizing interpolators, which are canonical solutions selected by smoothness bias, exhibit harmful overfitting: even as the training sample size grows to infinity, the generalization error remains bounded below by a positive constant with high probability. Our results generalize prior results studying the Hilbert space case using kernel methods. Our proof uses a geometric argument which identifies harmful neighborhoods of the training data using Sobolev inequalities.
Tizian Wenzel
(LMU Munich)
This poster presents sharp direct and inverse as well as saturation statements for kernel-based approximation using finitely smooth Sobolev kernels on bounded Lipschitz regions. The analysis focuses on the superconvergence regime, for which direct statements have only recently been obtained.
The resulting theory yields a one-to-one correspondence between the smooth-ness of a target function – quantified in terms of power spaces – and the achievable approximation rates by kernel-based approximation. In this way, we extend existing results beyond the escaping-the-native-space regime and provide a unified characterization covering the full scale of admissible smooth-ness spaces.
Liane Xu
(Princeton University)
Laplacian-based methods are popular for the dimensionality reduction of data lying in a high-dimensional Euclidean space. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian. This is joint work with Amit Singer.
Jiaqi Yang
(Fudan University Shanghai)
Operator learning aims to approximate mappings between infinite-dimensional function spaces, while practical implementations rely on finite-dimensional encoder–decoder representations, which often appear to induce resolution-dependent hypothesis classes and obscure the underlying operator being learned.
We resolve this ambiguity through a kernel-based framework that identifies a resolution-independent limiting operator class associated with encoder–decoder architectures. Starting from matrix-valued kernel learning at a fixed finite resolution, we show that the encoder–decoder structure admits an isometric lifting to an operator-valued kernel formulation on function spaces. As the input and output resolutions increase, the encoder-induced kernels converge to a resolution-independent limiting kernel that characterizes the intrinsic operator class.
Building on this limiting-kernel perspective, we analyze stochastic gradient descent (SGD) for operator learning under encoder–decoder representations. Under a source condition formulated with respect to the integral operator induced by the limiting kernel, we derive a resolution-invariant error decomposition for SGD, in which the prediction error splits into an encoder-induced bias controlled by the kernel discrepancy, an output-encoding error determined by the output discretization scheme, and an optimization error decaying polynomially in the iteration index. The optimization rate can be chosen arbitrarily close to $t^{-1}$, independently of the encoder–decoder resolution.
When the source regularity parameter satisfies r<1, the encoder bias and the output-encoding error form unavoidable approximation barriers for the operator class induced by the limiting kernel. The framework applies to broad classes of kernels and encoder–decoder constructions, including radial and dot-product kernels, spectral truncation–based encoders such as Fourier, polynomial, and wavelet representations, empirical principal component encoders, and kernel interpolation–based encoding schemes. It further extends to infinite-width neural networks through neural tangent kernel (NTK) limits, yielding analogous resolution-invariant error decompositions together with finite-width correction terms and stability guarantees along the SGD trajectory.
Ahmad Zorkot
(University of Vienna)
We address the approximation of high-dimensional Fokker–Planck equations using Gaussian Mixture Models (GMM). Our approach is based on a splitting strategy: the transport step is approximated by a GMM representation, while the diffusion step is handled analytically. This framework leads to an efficient and scalable numerical method suitable for high-dimensional problems. We provide a convergence analysis of the proposed scheme. The performance and reliability of the method are further validated through a series of numerical experiments in various dimensions.
This is a joint work with Markus Bachmayr and Tianyu Jin (RWTH Aachen), Damiano Lombardi (INRIA Paris), and Olga Mula (University of Vienna).