Author name cluster

Max Zimmer

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9 papers

2 author rows

ICML Conference 2025 Conference Paper

Approximating Latent Manifolds in Neural Networks via Vanishing Ideals

Nico Pelleriti
Max Zimmer
Elias Samuel Wirth
Sebastian Pokutta

Deep neural networks have reshaped modern machine learning by learning powerful latent representations that often align with the manifold hypothesis: high-dimensional data lie on lower-dimensional manifolds. In this paper, we establish a connection between manifold learning and computational algebra by demonstrating how vanishing ideals can characterize the latent manifolds of deep networks. To that end, we propose a new neural architecture that (i) truncates a pretrained network at an intermediate layer, (ii) approximates each class manifold via polynomial generators of the vanishing ideal, and (iii) transforms the resulting latent space into linearly separable features through a single polynomial layer. The resulting models have significantly fewer layers than their pretrained baselines, while maintaining comparable accuracy, achieving higher throughput, and utilizing fewer parameters. Furthermore, drawing on spectral complexity analysis, we derive sharper theoretical guarantees for generalization, showing that our approach can in principle offer tighter bounds than standard deep networks. Numerical experiments confirm the effectiveness and efficiency of the proposed approach.

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ICML Conference 2025 Conference Paper

Capturing Temporal Dynamics in Large-Scale Canopy Tree Height Estimation

Jan Pauls
Max Zimmer
Berkant Turan
Sassan Saatchi
Philippe Ciais
Sebastian Pokutta
Fabian Gieseke

With the rise in global greenhouse gas emissions, accurate large-scale tree canopy height maps are essential for understanding forest structure, estimating above-ground biomass, and monitoring ecological disruptions. To this end, we present a novel approach to generate large-scale, high-resolution canopy height maps over time. Our model accurately predicts canopy height over multiple years given Sentinel 2 time series satellite data. Using GEDI LiDAR data as the ground truth for training the model, we present the first 10 m resolution temporal canopy height map of the European continent for the period 2019–2022. As part of this product, we also offer a detailed canopy height map for 2020, providing more precise estimates than previous studies. Our pipeline and the resulting temporal height map are publicly available, enabling comprehensive large-scale monitoring of forests and, hence, facilitating future research and ecological analyses. For an interactive viewer, see https: //europetreemap. projects. earthengine. app/view/europeheight.

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NeurIPS Conference 2025 Conference Paper

Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms

Hiroshi Kera
Nico Pelleriti
Yuki Ishihara
Max Zimmer
Sebastian Pokutta

Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gröbner and Border bases are fundamental but suffer from high computational costs, which have motivated recent Deep Learning approaches to improve efficiency, albeit at the expense of output correctness. In this work, we introduce the Oracle Border Basis Algorithm, the first Deep Learning approach that accelerates Border basis computation while maintaining output guarantees. To this end, we design and train a Transformer-based oracle that identifies and eliminates computationally expensive reduction steps, which we find to dominate the algorithm's runtime. By selectively invoking this oracle during critical phases of computation, we achieve substantial speedup factors of up to 3. 5x compared to the base algorithm, without compromising the correctness of results. To generate the training data, we develop a sampling method and provide the first sampling theorem for border bases. We construct a tokenization and embedding scheme tailored to monomial-centered algebraic computations, resulting in a compact and expressive input representation, which reduces the number of tokens to encode an $n$-variate polynomial by a factor of $O(n)$. Our learning approach is data efficient, stable, and a practical enhancement to traditional computer algebra algorithms and symbolic computation.

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ICML Conference 2025 Conference Paper

DUNIA: Pixel-Sized Embeddings via Cross-Modal Alignment for Earth Observation Applications

Ibrahim Fayad
Max Zimmer
Martin Schwartz
Fabian Gieseke
Philippe Ciais
Gabriel Belouze
Sarah Brood
Aurélien de Truchis

Significant efforts have been directed towards adapting self-supervised multimodal learning for Earth observation applications. However, most current methods produce coarse patch-sized embeddings, limiting their effectiveness and integration with other modalities like LiDAR. To close this gap, we present DUNIA, an approach to learn pixel-sized embeddings through cross-modal alignment between images and full-waveform LiDAR data. As the model is trained in a contrastive manner, the embeddings can be directly leveraged in the context of a variety of environmental monitoring tasks in a zero-shot setting. In our experiments, we demonstrate the effectiveness of the embeddings for seven such tasks: canopy height mapping, fractional canopy cover, land cover mapping, tree species identification, plant area index, crop type classification, and per-pixel waveform-based vertical structure mapping. The results show that the embeddings, along with zero-shot classifiers, often outperform specialized supervised models, even in low-data regimes. In the fine-tuning setting, we show strong performances near or better than the state-of-the-art on five out of six tasks.

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ICML Conference 2025 Conference Paper

Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?

Konrad Mundinger
Max Zimmer
Aldo Kiem
Christoph Spiegel 0002
Sebastian Pokutta

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al. , 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

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ICLR Conference 2025 Conference Paper

On the Byzantine-Resilience of Distillation-Based Federated Learning

Christophe Roux
Max Zimmer
Sebastian Pokutta

Federated Learning (FL) algorithms using Knowledge Distillation (KD) have received increasing attention due to their favorable properties with respect to privacy, non-i.i.d. data and communication cost. These methods depart from transmitting model parameters and instead communicate information about a learning task by sharing predictions on a public dataset. In this work, we study the performance of such approaches in the byzantine setting, where a subset of the clients act in an adversarial manner aiming to disrupt the learning process. We show that KD-based FL algorithms are remarkably resilient and analyze how byzantine clients can influence the learning process. Based on these insights, we introduce two new byzantine attacks and demonstrate their ability to break existing byzantine-resilient methods. Additionally, we propose a novel defence method which enhances the byzantine resilience of KD-based FL algorithms. Finally, we provide a general framework to obfuscate attacks, making them significantly harder to detect, thereby improving their effectiveness.

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ICML Conference 2024 Conference Paper

Estimating Canopy Height at Scale

Jan Pauls
Max Zimmer
Una M. Kelly
Martin Schwartz
Sassan Saatchi
Philippe Ciais
Sebastian Pokutta
Martin Brandt

We propose a framework for global-scale canopy height estimation based on satellite data. Our model leverages advanced data preprocessing techniques, resorts to a novel loss function designed to counter geolocation inaccuracies inherent in the ground-truth height measurements, and employs data from the Shuttle Radar Topography Mission to effectively filter out erroneous labels in mountainous regions, enhancing the reliability of our predictions in those areas. A comparison between predictions and ground-truth labels yields an MAE/RMSE of 2. 43 / 4. 73 (meters) overall and 4. 45 / 6. 72 (meters) for trees taller than five meters, which depicts a substantial improvement compared to existing global-scale products. The resulting height map as well as the underlying framework will facilitate and enhance ecological analyses at a global scale, including, but not limited to, large-scale forest and biomass monitoring.

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ICLR Conference 2024 Conference Paper

Sparse Model Soups: A Recipe for Improved Pruning via Model Averaging

Max Zimmer
Christoph Spiegel 0002
Sebastian Pokutta

Neural networks can be significantly compressed by pruning, yielding sparse models with reduced storage and computational demands while preserving predictive performance. Model soups (Wortsman et al., 2022) enhance generalization and out-of-distribution (OOD) performance by averaging the parameters of multiple models into a single one, without increasing inference time. However, achieving both sparsity and parameter averaging is challenging as averaging arbitrary sparse models reduces the overall sparsity due to differing sparse connectivities. This work addresses these challenges by demonstrating that exploring a single retraining phase of Iterative Magnitude Pruning (IMP) with varied hyperparameter configurations such as batch ordering or weight decay yields models suitable for averaging, sharing identical sparse connectivity by design. Averaging these models significantly enhances generalization and OOD performance over their individual counterparts. Building on this, we introduce Sparse Model Soups (SMS), a novel method for merging sparse models by initiating each prune-retrain cycle with the averaged model from the previous phase. SMS preserves sparsity, exploits sparse network benefits, is modular and fully parallelizable, and substantially improves IMP's performance. We further demonstrate that SMS can be adapted to enhance state-of-the-art pruning-during-training approaches.

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ICLR Conference 2023 Conference Paper

How I Learned to Stop Worrying and Love Retraining

Max Zimmer
Christoph Spiegel 0002
Sebastian Pokutta

Many Neural Network Pruning approaches consist of several iterative training and pruning steps, seemingly losing a significant amount of their performance after pruning and then recovering it in the subsequent retraining phase. Recent works of Renda et al. (2020) and Le & Hua (2021) demonstrate the significance of the learning rate schedule during the retraining phase and propose specific heuristics for choosing such a schedule for IMP (Han et al., 2015). We place these findings in the context of the results of Li et al. (2020) regarding the training of models within a fixed training budget and demonstrate that, consequently, the retraining phase can be massively shortened using a simple linear learning rate schedule. Improving on existing retraining approaches, we additionally propose a method to adaptively select the initial value of the linear schedule. Going a step further, we propose similarly imposing a budget on the initial dense training phase and show that the resulting simple and efficient method is capable of outperforming significantly more complex or heavily parameterized state-of-the-art approaches that attempt to sparsify the network during training. These findings not only advance our understanding of the retraining phase, but more broadly question the belief that one should aim to avoid the need for retraining and reduce the negative effects of ‘hard’ pruning by incorporating the sparsification process into the standard training.

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