Proceedings of the Geometry, Topology, and Machine Learning Workshop Held in Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany on 10-14 November 2025 Published as Volume 325 by the Proceedings of Machine Learning Research on 29 June 2026. Volume Edited by: Michael Bleher Freya Jensen Levin Maier Diaaeldin Taha Anna Wienhard Series Editors: Neil D. Lawrence https://proceedings.mlr.press/v325/ Tue, 30 Jun 2026 09:22:45 +0000 Tue, 30 Jun 2026 09:22:45 +0000 Jekyll v3.10.0 Maximizing the spectral gap through graph rewiring has been proposed to enhance the performance of message-passing graph neural networks (GNNs) by addressing over-squashing. However, as we show, minimizing the spectral gap can also improve generalization. To explain this, we analyze how rewiring can benefit GNNs within the context of stochastic block models. Since spectral gap optimization primarily influences community strength, it improves performance when the community structure aligns with node labels. Building on this insight, we propose three distinct rewiring strategies that explicitly target community structure, node labels, and their alignment: (a) community structure-based rewiring (ComMa), a more computationally efficient alternative to spectral gap optimization that achieves similar goals; (b) feature similarity-based rewiring (FeaSt), which focuses on maximizing global homophily; and (c) a hybrid approach (ComFy), which enhances local feature similarity while preserving community structure to optimize label-community alignment. Extensive experiments confirm the effectiveness of these strategies and support our theoretical insights.\footnote[2]{This work is an extended abstract which was presented as a lightning talk at GTML 2025. It is based on a previously published work at ICLR 2025 \citep{rubio-madrigal2025gnns}. The appendix reproduces relevant material from the full paper for completeness.} Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/rubio-madrigal26a.html https://proceedings.mlr.press/v325/rubio-madrigal26a.html Chromatic alpha complexes serve as a generalization of alpha complexes for chromatic point sets and were developed beyond two colors by \citet{Cult25}. Instead of only one as in the case in standard persistent homology, six different persistence diagrams result from this construction. Here we present the findings of \citet{Rose25}, in which we study the expected number and total length of persistence pairs for each diagram, assuming uniformly distributed $2$-colored points in the unit square. Additionally, we highlight deeper connections to the research area of Euclidean minimum spanning trees. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/rosenmeier26a.html https://proceedings.mlr.press/v325/rosenmeier26a.html After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of \emph{geometric deep learning}, and paradigms that were once considered to be firmly in the realm of the abstract—like sheaves—have been “tamed” to serve as novel inductive biases for model architectures in \emph{topological deep learning}. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to \emph{lifting} existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/rieck26a.html https://proceedings.mlr.press/v325/rieck26a.html In this article, we give a short survey on the embedded homology of hypergraphs, random hypergraphs, and hypergraphs on manifolds by S. Bressan, J. Li, S. Ren, C. Wu, J. Wu, M. Zhang et al. from 2019 to 2025. We review the motivating problems as well as the meanings for the embedded homology of hypergraphs, the map algebra of hypergraphs, the relations between random hypergraphs and random simplicial complexes, and the double complexes for hypergraphs on manifolds. Besides, we give some further discussions about potential applications of hypergraphs on manifolds in the $r$-ball packing problems and the $r$-ball covering problems. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/ren26a.html https://proceedings.mlr.press/v325/ren26a.html Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as histopathology, topology is an important descriptor that can be used to distinguish between disease-indicating tissue by analyzing the shape characteristics of cells. Current literature suggests that reintroducing topological information using persistent homology can improve medical diagnostics; however, previous methods utilize global topological summaries which do not contain information about the locality of topological features. To address this gap, we present a novel method that generates local persistent homology-based data using a modified version of the convolution operator called \textit{Persistent Homology Convolutions}. This method captures information about the locality and translation equivariance of topological features. We perform a comparative study using various representations of histopathology slides and find that models trained with persistent homology convolutions outperform conventionally trained models and are less sensitive to hyperparameters. These results indicate that persistent homology convolutions extract meaningful geometric information from the histopathology slides. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/pothagoni26a.html https://proceedings.mlr.press/v325/pothagoni26a.html In this extended abstract, we present ongoing work on a novel functional representation of persistence diagrams (PDs). Building on the approach of \citet{gotovac2025topological}, we model PDs as scalar fields on the sphere using the lift zonoid representation of finite integrable measures. Unlike their method, however, our construction yields a bi-continuous operator that is stable with respect to the 1-Wasserstein distance. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/pegoraro26a.html https://proceedings.mlr.press/v325/pegoraro26a.html This work studies the asymptotic behavior of discrete graph Laplacians constructed from random samples on Riemannian manifolds whose boundaries may exhibit geometric irregularities. We introduce the class of \emph{manifolds with kinks} (MFK)—a broad generalization of smooth manifolds with boundaries and corners—and establish convergence results of the graph Laplacian at interior, border, and cusp points. The results unify earlier analyses on smooth domains (Belkin–Niyogi, Hein–Luxburg, Peoples–Harlim) and extend them to non-smooth geometries that frequently occur in data analysis. We also discuss applications to edge detection in image processing and possible extensions to curvature-dependent asymptotics. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/pal26a.html https://proceedings.mlr.press/v325/pal26a.html Motivated by the prevalence of equivariant machine learning models and the success of the framework of linear equivariant convolutional neural networks, we present in this work an extended framework that also includes non-linear equivariant models. More specifically, we represent these models as integral operators and derive conditions on the integrand for the operator to be equivariant. Further, we prove the generality of the proposed framework and show explicitly how common equivariant models, linear as well as non-linear, fit into the proposed formulation. This extended abstract summarises the central points of the preprint \citet{nyholm2025equivariantnonlinearmapsneural}, which is joint work together with Oscar Carlsson, Maurice Weiler and Daniel Persson. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/nyholm26a.html https://proceedings.mlr.press/v325/nyholm26a.html Reward maximization, safe exploration, and intrinsic motivation are often studied as separate objectives in reinforcement learning (RL). We present a unified geometric framework, that views these goals as instances of a single optimization problem on the space of achievable long-term behavior in an environment. Within this framework, classical methods such as policy mirror descent, natural policy gradient, and trust-region algorithms naturally generalize to nonlinear utilities and convex constraints. We illustrate how this perspective captures robustness, safety, exploration, and diversity objectives, and outline open challenges at the interface of geometry and deep RL. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/milosevic26a.html https://proceedings.mlr.press/v325/milosevic26a.html Extract-Transform-Load (ETL) processes are core components of modern data processing infrastructures. The throughput of processed data records can be adjusted by changing the amount of allocated resources, i.e. the number of parallel processing threads for each of the three ETL phases, but also depends on stochastic variations in the per-record processing times. In chains of multiple consecutive ETL processes, the relation between allocated resources and overall throughput is further complicated, for example by the occurrence of bottlenecks affecting all subsequent ETL processes. We develop a mathematical model of ETL process chains that is accurate at the level of time-aggregated throughput and suitable for efficient simulation. The process chain is represented as a controlled discrete-time Markov process on a directed acyclic graph whose edges are individual ETL processes. We model the mean throughput as a bounded, monotone function of the number of parallel threads, to capture the diminishing benefit of allocating more threads. We furthermore introduce a Flow Balance postulate linking number of threads, mean throughput, and mean processing time. The stochastic processing times are then modeled by non-negative heavy-tailed distributions around the mean processing time. This framework provides a principled simulator for ETL networks and a foundation for learning- and control-based resource allocation. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/maier26a.html https://proceedings.mlr.press/v325/maier26a.html We investigate the question: “How can we efficiently describe equivalence classes of finite sets of (colored) points in $\BR^3$, where (colored) point sets are equivalent if they can be transformed into each other by a rotation?” It sounds very simple, but we will see it leads to some interesting mathematical structures. However, they only become a part of the picture when we have to take into account some application specific constraints: We want to characterize these configurations by features that do not depend on the number of points in the set, and that are fast to evaluate. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/maennel26a.html https://proceedings.mlr.press/v325/maennel26a.html We study neural forecasters for dynamical systems through the lens of representational alignment. We introduce anchor-based, geometry-agnostic \emph{relative embeddings} that remove rotational and scaling ambiguities, enabling robust cross-seed and cross-architecture comparison. Across diverse periodic, quasi-periodic, and chaotic systems, we observe consistent family-level patterns: MLPs align with MLPs, RNNs with RNNs, and ESNs show reduced alignment on chaotic dynamics, while transformers often align weakly but still perform well. Alignment generally correlates with forecasting accuracy, yet high accuracy can coexist with low alignment. Relative embeddings thus offer a simple, reproducible basis for comparing learned dynamics. \footnote{This workshop paper is a condensed companion to the extended version published in Transactions on Machine Learning Research (TMLR); \citep{kucukahmetler2026relative}. We thank the Max Planck Computing and Data Facility (MPCDF) for providing GPU resources. D.K. is supported by BMFTR in DAAD project 57616814 (SECAI). N.S. is supported by BMFTR (Federal Ministry of Research, Technology and Space) through ACONITE (16IS22065) and the Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI.) Leipzig and by the European Union and the Free State of Saxony through BIOWIN.} Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/kucukahmetler26a.html https://proceedings.mlr.press/v325/kucukahmetler26a.html We present here, informally, ideas on ongoing work for analyzing chromatic (labeled) point clouds, with the goal of understanding spatial interactions between cell types in biological settings. We describe our starting framework and state some open questions that we are currently addressing. Two specific contexts attract our interest: (1) tumor microenvironment and (2) cell differentiation process. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/jimenez26a.html https://proceedings.mlr.press/v325/jimenez26a.html Implicit bias plays an important role in explaining how overparameterized models generalize well. Explicit regularization like weight decay is often employed in addition to prevent overfitting. While both concepts have been studied separately, in practice, they often act in tandem. Understanding their interplay is key to controlling the shape and strength of implicit bias, as it can be modified by explicit regularization. To this end, we incorporate explicit regularization into the mirror flow framework and analyze its lasting effects on the geometry of the training dynamics, covering three distinct effects: positional bias, type of bias, and range shrinking. Our analytical approach encompasses a broad class of problems, including sparse coding, matrix sensing, single-layer attention, and LoRA, for which we demonstrate the utility of our insights. To exploit the lasting effect of regularization and highlight the potential benefit of dynamic weight decay schedules, we propose to switch off weight decay during training, which can improve generalization, as we demonstrate in experiments. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/jacobs26a.html https://proceedings.mlr.press/v325/jacobs26a.html Effective analysis in neuroscience benefits significantly from robust conceptual frameworks. Traditional metrics of interbrain synchrony in social neuroscience typically depend on fixed, correlation-based approaches, restricting their explanatory capacity to descriptive observations. Inspired by the successful integration of geometric insights in network science, we propose leveraging discrete geometry to examine the dynamic reconfigurations in neural interactions during social exchanges. Unlike conventional synchrony approaches, our method interprets inter-brain connectivity changes through the evolving geometric structures of neural networks. This geometric framework is realized through a pipeline that identifies critical transitions in network connectivity using entropy metrics derived from curvature distributions. By doing so, we significantly enhance the capacity of hyperscanning methodologies to uncover underlying neural mechanisms in interactive social behavior. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/hinrichs26a.html https://proceedings.mlr.press/v325/hinrichs26a.html We use topological data analysis to study neural population activity in the Sensorium 2023 dataset, which records responses from thousands of mouse visual cortex neurons to diverse video stimuli. For each video, we build frame-by-frame cubical complexes from neuronal activity and apply zigzag persistent homology to capture how topological structure evolves over time. These dynamics are summarized with persistence landscapes, providing a compact vectorized representation of temporal features. We focus on one-dimensional topological features—loops in the data—that reflect coordinated, cyclical patterns of neural co-activation. To test their informativeness, we compare repeated trials of different videos by clustering their resulting topological neural representations. Our results show that these topological descriptors reliably distinguish neural responses to distinct stimuli. This work highlights a connection between evolving neuronal activity and interpretable topological signatures, advancing the use of topological data analysis for uncovering neural coding in complex dynamical systems. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/gardinazzi26b.html https://proceedings.mlr.press/v325/gardinazzi26b.html We analyze internal representations of large language models with zigzag persistent homology, treating depth as a discrete time axis for point clouds of last-token embeddings. At each layer we build a k-nearest-neighbors clique complex, connect adjacent layers via intersections, and summarize the resulting diagrams with effective persistence images. From these we derive two descriptors: Births’ Relative Frequency (at what rate new p-dimensional features appear) and Inter-Layer Persistence (how long they survive across depth). On the SST movie reviews dataset and three open-source models (Llama-3.1, OSS-20B, Phi-4), we consistently observe three evolving phases: early rapid changes, a middle regime of stable organization, and a final reorganization before output. Using the stability signal (inter-layer persistence) to guide where to remove contiguous blocks of layers, we find that pruning within high-persistence regions maintains 5-shot MMLU performance (with the same trend visible even for the more pruning-sensitive OSS-20B). This suggests that zigzag-based summaries capture meaningful, system-level dynamics and can inform lightweight pruning. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/gardinazzi26a.html https://proceedings.mlr.press/v325/gardinazzi26a.html Equivariant and invariant machine learning models exploit symmetries and structural patterns in data to improve sample efficiency. While empirical studies suggest that data-driven methods such as regularization and data augmentation can perform comparably to explicitly invariant models, theoretical insights remain scarce. In this paper, we provide a theoretical comparison of three approaches for achieving invariance: data augmentation, regularization, and hard-wiring. We focus on mean squared error regression with deep linear networks, which parametrize rank-bounded linear maps and can be hard-wired to be invariant to specific group actions. We show that the critical points of the optimization problems for hard-wiring and data augmentation are identical, consisting solely of saddles and the global optimum. By contrast, regularization introduces additional critical points, though they remain saddles except for the global optimum. Moreover, we demonstrate that the regularization path is continuous and converges to the hard-wired solution. Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/duan26a.html https://proceedings.mlr.press/v325/duan26a.html Mon, 29 Jun 2026 00:00:00 +0000 https://proceedings.mlr.press/v325/bleher26a.html https://proceedings.mlr.press/v325/bleher26a.html