arXiv subjects

Philipp Petersen

Publications and source records attributed to Philipp Petersen.

At least 19 recordsLinked to original sources

arXiv2026

The frame set of the first Hermite function

We determine the frame set of the first Hermite function, proving a conjecture of Lyubarskii and Nes. We use a characterization of semi-regular Gabor frames due to Gr\"ochenig, Romero, and St\"ockler to transform the problem into a question about the existence of a nonzero Gaussian shift-invariant entire function $F$, with bounded coefficients, whose derivative vanishes on a lattice with spacing $\delta=ab$. By taking the Wronskian of $N$ translates of $F$, we amplify its critical points to zeros of multiplicity at least $N-1$. After rescaling, this Wronskian is again a Gaussian shift-invariant function. A Gaussian zero-density theorem then gives $(N-1)/(N\delta)\leq1$. Varying $N$ over the range allowed by the construction forces every subcritical lattice product $\delta=ab$ for which the system is not a frame to equal $(q-1)/q$ for some integer $q\geq2$.

math.FA↗

arXiv2026

Linear dependence of time-frequency shifts of a Schwartz function

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

math.FA↗

arXiv2026

Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.

math.LO↗

arXiv2026

Compositional Approximation Can Strictly Outperform Superpositional Approximation

Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.

math.NA↗

arXiv2026

HEPA: A Self-Supervised Horizon-Conditioned Event Predictive Architecture for Time Series

Critical events in multivariate time series, from turbine failures to cardiac arrhythmias, demand accurate prediction, yet labeled data is scarce because such events are rare and costly to annotate. We introduce HEPA (Horizon-conditioned Event Predictive Architecture), built on two key principles. First, a causal Transformer encoder is pretrained via a Joint-Embedding Predictive Architecture (JEPA): a horizon-conditioned predictor learns to forecast future representations rather than future values, forcing the encoder to capture predictable temporal dynamics from unlabeled data alone. Second, we freeze the encoder and finetune only the predictor toward the target event, producing a monotonic survival cumulative distribution function (CDF) over horizons. With fixed architecture and optimiser hyperparameters across all benchmarks, HEPA handles water contamination, cyberattack detection, volatility regimes, and eight further event types across 11 domains, exceeding leading time-series architectures including PatchTST, iTransformer, MAE, and Chronos-2 on at least 10 of 14 benchmarks, with an order of magnitude fewer tuned parameters and, on lifecycle datasets, an order of magnitude less labeled data.

cs.LG↗

arXiv2026

FactoryNet: A Large-Scale Dataset toward Industrial Time-Series Foundation Models

We introduce the first universal pretraining corpus for industrial time-series data: FactoryNet. 51M datapoints across 23k end-to-end task executions (13.3k real, 9.8k synthetic) on six embodiments, unified by a shared schema that enables robust zero-shot cross-embodiment transfer and highly parameter-efficient anomaly detection. We introduce a novel schema: Setpoint, Effort, Feedback, Context (S-E-F-C) underlying the whole pipeline that maps any actuated system into a common representational frame. The corpus spans 27 annotated anomaly types alongside healthy baselines and counterfactual pairs across robotic manipulation and machining domains. Cross-embodiment transfer experiments yield positive results: under bias-aware metrics our model demonstrates fair cross-embodiment transfer capabilities on the evaluated source-target pair, while 24 schema-aligned signals achieves competitive anomaly detection performance compared to high-dimensional baselines. We release FactoryNet as a growing, multi-embodiment dataset to drive progress toward industrial foundation models.

cs.LG↗

arXiv2026

FactoryBench: Evaluating Industrial Machine Understanding

We introduce FactoryBench, a benchmark for evaluating time-series models and LLMs on machine understanding over industrial robotic telemetry. Q&A pairs are organized along four causal levels (state, intervention, counterfactual, decision) instantiating Pearl's ladder of causation, and span five answer formats: four structured formats are scored deterministically and free-form answers are scored by an LLM-as-judge voting protocol. We propose a scalable Q&A generation framework built around structured question templates, present FactoryWave (a dense, multitask, multivariate sensor dataset collected from a UR3 cobot and a KUKA KR10 industrial arm), and construct FactoryBench as a large-scale benchmark of over 70k Q&A items grounded in roughly 15k normalized episodes from FactoryWave, AURSAD, and voraus-AD. Zero-shot evaluation of six frontier LLMs shows that no model exceeds 50% on structured levels or 18% on decision-making, revealing a wide gap between current models and operational machine understanding.

cs.AI↗

arXiv2026

Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning

We compare in-context learning with fixed queries and agentic learning with adaptive queries for uniform approximation of task families. We consider two settings: an unrestricted regime, where querying and approximation are arbitrary functions, and a realizable regime, where we require these operations to be implemented by ReLU neural networks. In both settings, adaptivity never hinders approximation performance. However, this advantage can change when one passes from the unrestricted regime to the realizable regime. We identify four distinct approximation scenarios, each witnessed by an explicit task family: (a) no advantage of adaptivity; (b) an advantage in the unrestricted regime that persists under ReLU realizability; (c) an advantage that arises only under realizability; and (d) an advantage that disappears under realizability. This demonstrates that representational constraints interact profoundly with the effect of adaptivity.

cs.LG↗

arXiv2026

Certified and accurate computation of function space norms of deep neural networks

Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do not provide enough information to derive tight deterministic and guaranteed bounds on function space norms. In this work, we move beyond a purely black-box setting and exploit the neural network structure directly. We present a framework for the certified and accurate computation of integral quantities of neural networks, including Lebesgue and Sobolev norms, by combining interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement and quadrature-based aggregation. On each box, we compute guaranteed lower and upper bounds for function values and derivatives, and propagate these local certificates to global lower and upper bounds for the target integrals. Our analysis provides a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, yielding certified computation of $L^p$, $W^{1,p}$, and $W^{2,p}$ norms. We further show how these ingredients lead to practical certified bounds for PINN interior residuals. Numerical experiments illustrate the accuracy and practical behavior of the proposed methods.

math.NA↗

arXiv2026

LAMP: Look-Ahead Mixed-Precision Inference of Large Language Models

Mixed-precision computations are a hallmark of the current stage of AI, driving the progress in large language models towards efficient, locally deployable solutions. This article addresses the floating-point computation of compositionally-rich functions, concentrating on transformer inference. Based on the rounding error analysis of a composition $f(g(\mathrm{x}))$, we provide an adaptive strategy that selects a small subset of components of $g(\mathrm{x})$ to be computed more accurately while all other computations can be carried out with lower accuracy. We then explain how this strategy can be applied to different compositions within a transformer and illustrate its overall effect on transformer inference. We study the effectiveness of this algorithm numerically on GPT-2 models and demonstrate that already very low recomputation rates allow for improvements of up to two orders of magnitude in accuracy.

cs.LG↗

arXiv2025

Experimental data re-uploading with provable enhanced learning capabilities

The last decades have seen the development of quantum machine learning, stemming from the intersection of quantum computing and machine learning. This field is particularly promising for the design of alternative quantum (or quantum inspired) computation paradigms that could require fewer resources with respect to standard ones, e.g. in terms of energy consumption. In this context, we present the implementation of a data re-uploading scheme on a photonic integrated processor, achieving high accuracies in several image classification tasks. We thoroughly investigate the capabilities of this apparently simple model, which relies on the evolution of one-qubit states, by providing an analytical proof that our implementation is a universal classifier and an effective learner, capable of generalizing to new, unknown data. Hence, our results not only demonstrate data re-uploading in a potentially resource-efficient optical implementation but also provide new theoretical insight into this algorithm, its trainability, and generalizability properties. This lays the groundwork for developing more resource-efficient machine learning algorithms, leveraging our scheme as a subroutine.

quant-ph↗

arXiv2025

Minimax learning rates for estimating binary classifiers under margin conditions

We study classification problems using binary estimators where the decision boundary is described by horizon functions and where the data distribution satisfies a geometric margin condition. A key novelty of our work is the derivation of lower bounds for the worst-case learning rates over broad classes of functions, under a geometric margin condition -- a setting that is almost universally satisfied in practice, but remains theoretically challenging. Moreover, we work in the noiseless setting, where lower bounds are particularly hard to establish. Our general results cover, in particular, classification problems with decision boundaries belonging to several classes of functions: for Barron-regular functions, H\"older-continuous functions, and convex-Lipschitz functions with strong margins, we identify optimal rates close to the fast learning rates of $\mathcal{O}(n^{-1})$ for $n \in \mathbb{N}$ samples.

stat.ML↗

arXiv2025

Numerical stability analysis of large language models

Transformers are the state-of-the-art architecture for large language models, and a key to their scalability is the strategic usage of low-precision arithmetic. We develop a mixed-precision analysis of transformer inference, deriving bounds for the condition numbers and forward error of the architecture's constituent parts. Notably, we compare the numerical stability of LayerNorm and RMSNorm in the massive-outlier regime, tighten the error bound of softmax in the presence of attention sinks, and quantify the impact of its shifted evaluation on the sensitivity to perturbations. Furthermore, we derive novel sequence-length-independent bounds on the local Lipschitz constant of self-attention. Our worst-case error bound for transformer inference suggests that its numerical stability is determined by the interplay between weight magnitude and the growth of the residual stream. Crucially, and as validated by experiments with GPT-2, our analysis establishes that the scaling of residual-projection weights preserves the propagation of the relative rounding error unless it forces a qualitative transition in the dynamics of the residual stream.

math.NA↗

arXiv2024

High-dimensional classification problems with Barron regular boundaries under margin conditions

We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.

cs.LG↗

arXiv2024

The sampling complexity of learning invertible residual neural networks

In recent work it has been shown that determining a feedforward ReLU neural network to within high uniform accuracy from point samples suffers from the curse of dimensionality in terms of the number of samples needed. As a consequence, feedforward ReLU neural networks are of limited use for applications where guaranteed high uniform accuracy is required. We consider the question of whether the sampling complexity can be improved by restricting the specific neural network architecture. To this end, we investigate invertible residual neural networks which are foundational architectures in deep learning and are widely employed in models that power modern generative methods. Our main result shows that the residual neural network architecture and invertibility do not help overcome the complexity barriers encountered with simpler feedforward architectures. Specifically, we demonstrate that the computational complexity of approximating invertible residual neural networks from point samples in the uniform norm suffers from the curse of dimensionality. Similar results are established for invertible convolutional Residual neural networks.

stat.ML↗

arXiv2024

Dimension-independent learning rates for high-dimensional classification problems

We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every $RBV^2$ function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. And we leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.

cs.LG↗

arXiv2024

Mathematical theory of deep learning

This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural network theory. Serving as a guide for students and researchers in mathematics and related fields, the book aims to equip readers with foundational knowledge on the topic. It prioritizes simplicity over generality, and presents rigorous yet accessible results to help build an understanding of the essential mathematical concepts underpinning deep learning.

cs.LG↗

arXiv2023

Large Language Models for Mathematicians

Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code. For many professions, LLMs represent an invaluable tool that can speed up and improve the quality of work. In this note, we discuss to what extent they can aid professional mathematicians. We first provide a mathematical description of the transformer model used in all modern language models. Based on recent studies, we then outline best practices and potential issues and report on the mathematical abilities of language models. Finally, we shed light on the potential of LLMs to change how mathematicians work.

cs.CL↗