arXiv subjects

Philipp Tuchel

Publications and source records attributed to Philipp Tuchel.

8 recordsLinked to original sources

arXiv2026

Realising atomless laws as distance distributions on metric measure spaces

We study which probability measures on $[0,\infty)$ can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every $L>1$, we show that the realising metric may be chosen $L$-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.

math.PR↗

arXiv2026

Mod-phi convergence for random variables with cyclotomic generating functions

We consider a sequence of discrete random variables whose generating functions are polynomials only having roots on the unit circle in the complex plane. For such sequences we prove, under mild assumptions, mod-$\phi$ convergence with explicit rate and limiting function. This allows us to recover, in a unified way, classical central limit theorems as well as finer asymptotic expansions. The general theory is applied to random permutations and random partitions.

math.PR↗

arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

math.HO↗

arXiv2026

Large deviations for sums of multivariate stretched-exponential random variables: the few-big-jumps principle

Large deviations for sums of i.i.d.\ random variables with stretched-exponential tails (also called Weibull or semi-exponential tails) have been well understood since the 60's, going back to Nagaev's seminal work. Many extensions in the $1$-dimensional setting have been developed since then, showing that such deviations are typically governed by a single big jump. In higher dimensions, a corresponding theory has remained largely undeveloped. This work provides such a multivariate extension and establishes large deviation results for sums of i.i.d.\ random vectors in $\mathbb{R}^k$ under fairly general assumptions. Roughly speaking, for some $\alpha\in(0,1)$, the log-probability of one random vector divided by $x$ exceeding a threshold $t$ in all components behaves asymptotically, for large $x$, as $x^\alpha$ times a negative infimum of a function $\mathcal{J}$. We prove large deviation results for sums of i.i.d.\ copies, where the rate function is given by a minimization of at most $k$ summands of $\mathcal{J}$. This establishes a few-big-jumps principle that generalizes the classical $1$-dimensional phenomenon: the deviation is typically realized by \emph{at most} $k$ independent vectors. The results are applied to absolute powers of multivariate Gaussian vectors as well as to various other examples. They also allow us to study random projections of high-dimensional $\ell_p^N$-balls, revealing interesting insights about the appearance of light- and heavy-tailed distributions in high-dimensional geometry.

math.PR↗

arXiv2025

On the Universal Representation Property of Spiking Neural Networks

Inspired by biology, spiking neural networks (SNNs) process information via discrete spikes over time, offering an energy-efficient alternative to the classical computing paradigm and classical artificial neural networks (ANNs). In this work, we analyze the representational power of SNNs by viewing them as sequence-to-sequence processors of spikes, i.e., systems that transform a stream of input spikes into a stream of output spikes. We establish the universal representation property for a natural class of spike train functions. Our results are fully quantitative, constructive, and near-optimal in the number of required weights and neurons. The analysis reveals that SNNs are particularly well-suited to represent functions with few inputs, low temporal complexity, or compositions of such functions. The latter is of particular interest, as it indicates that deep SNNs can efficiently capture composite functions via a modular design. As an application of our results, we discuss spike train classification. Overall, these results contribute to a rigorous foundation for understanding the capabilities and limitations of spike-based neuromorphic systems.

cs.NE↗

arXiv2025

Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold

Let $A_N$ be distributed according to the Haar probability measure on the orthogonal group $\mathscr{O}(N)$ for each $N\in\mathbb{N}$. It is well-known that the upper left $m_N\times k_N$ block of $\sqrt{N}A_N$ with $m_Nk_N = o(N)$ converges in total variation distance to a matrix of same size consisting of i.i.d. standard normal entries as $N\to\infty$. In this work, we characterize this convergence on the scale of large deviations. More precisely, we show that under the same condition $m_Nk_N = o(N)$ the empirical measure of entries of this block satisfies a large deviation principle with speed $m_Nk_N$ and rate function given by the relative entropy with respect to the standard normal distribution. Further, we complement the large deviation principle (LDP) obtained by Kabluchko and Prochno in [Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, Annales de l'Institut Henri Poincar\'e. 60 (2024), 990 -- 1024] for the whole block $A_N$ with a moderate deviation principle (MDP). Concretely, we show an MDP for the sequence of matrices $\beta_N A_N$ in the product topology, where $\beta_N\to\infty$ is a sequence of real numbers such that $\beta_N = o(\sqrt{N})$. Here, in contrast to the LDP, the Gaussian behavior of the entries is reflected in the rate function.

math.PR↗

arXiv2024

Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls

Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$. We also consider geometric quantities other than the volume such as the intrinsic volumes or the dual volumes. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture. In particular, they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719] and another result of Adamczak, Pivovarov, and Simanjuntak [Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls, Ann. Probab. 52 (2024), 93-126].

math.PR↗

arXiv2024

A Continual and Incremental Learning Approach for TinyML On-device Training Using Dataset Distillation and Model Size Adaption

A new algorithm for incremental learning in the context of Tiny Machine learning (TinyML) is presented, which is optimized for low-performance and energy efficient embedded devices. TinyML is an emerging field that deploys machine learning models on resource-constrained devices such as microcontrollers, enabling intelligent applications like voice recognition, anomaly detection, predictive maintenance, and sensor data processing in environments where traditional machine learning models are not feasible. The algorithm solve the challenge of catastrophic forgetting through the use of knowledge distillation to create a small, distilled dataset. The novelty of the method is that the size of the model can be adjusted dynamically, so that the complexity of the model can be adapted to the requirements of the task. This offers a solution for incremental learning in resource-constrained environments, where both model size and computational efficiency are critical factors. Results show that the proposed algorithm offers a promising approach for TinyML incremental learning on embedded devices. The algorithm was tested on five datasets including: CIFAR10, MNIST, CORE50, HAR, Speech Commands. The findings indicated that, despite using only 43% of Floating Point Operations (FLOPs) compared to a larger fixed model, the algorithm experienced a negligible accuracy loss of just 1%. In addition, the presented method is memory efficient. While state-of-the-art incremental learning is usually very memory intensive, the method requires only 1% of the original data set.

cs.LG↗