arXiv subjects

Ullrich Köthe

Publications and source records attributed to Ullrich Köthe.

At least 19 recordsLinked to original sources

arXiv2026

From Core to Detail: Unsupervised Disentanglement with Entropy-Ordered Flows

The unsupervised discovery of features that are both semantically meaningful and stable across runs remains a central challenge in representation learning. We introduce entropy-ordered flows (EOFlows), a normalizing flow (NF) framework that augments standard maximum likelihood training with an orthogonality regularizer on the decoder Jacobian. The regularizer is rooted in Independent Mechanism Analysis and encourages geometric disentanglement, and a stochastic estimator makes it tractable at image scale (CelebA at $D=2352$ and $12288$). Learned features form near-orthogonal curvilinear coordinates and can be ordered by their $\textit{explained (manifold) entropy}$ after training, analogous to the ranking by explained variance in PCA, which turns EOFlows into a non-linear generalization of PCA. EOFlows identify an order of magnitude more stable features than existing methods, and these features emerge in distinguishable categories (global, local, and generic) and support tentative semantic interpretations. The local features have strikingly sparse support in pixel space, although our method never enforces this. Retaining only the most important, i.e. highest entropy, features turns the bijective flow into an autoencoder with adjustable bottleneck, rivaling the rate-distortion performance of dedicated autoencoders.

cs.LG↗

arXiv2026

Diffusion Models in Simulation-Based Inference: A Tutorial Review

Diffusion models have recently emerged as powerful learners for simulation-based inference (SBI), enabling fast and accurate estimation of latent parameters from simulated and real data. Their score-based formulation offers a flexible way to learn conditional or joint distributions over parameters and observations, thereby providing a versatile solution to various modeling problems. In this tutorial review, we synthesize recent developments on diffusion models for SBI, covering design choices for training, inference, and evaluation. We highlight opportunities created by various concepts such as guidance, score composition, flow matching, consistency models, and joint modeling. Furthermore, we discuss how efficiency and statistical accuracy are affected by noise schedules, parameterizations, and samplers. Finally, we illustrate these concepts with case studies across parameter dimensionalities, simulation budgets, and model types, and outline open questions for future research.

stat.ML↗

arXiv2026

Breaking the Simplification Bottleneck in Amortized Neural Symbolic Regression

Symbolic regression (SR) aims to discover interpretable analytical expressions that accurately describe observed data. Amortized SR promises to be much more efficient than the predominant genetic programming SR methods, but currently struggles to scale to realistic scientific complexity. We find that a key obstacle is the lack of a fast reduction of equivalent expressions to a concise normalized form. Amortized SR has addressed this with general-purpose Computer Algebra Systems (CAS) like SymPy, but the high computational cost severely limits training and inference speed. We propose SimpliPy, a rule-based simplification engine achieving a 100-fold speed-up over SymPy at comparable quality. This enables substantial improvements in amortized SR, including scalability to much larger training sets, more efficient use of the per-expression token budget, and systematic training set decontamination with respect to equivalent test expressions. We demonstrate these advantages in our Flash-ANSR framework, which achieves much better accuracy than amortized baselines (NeSymReS, E2E) on the FastSRB benchmark. Moreover, it performs on par with state-of-the-art direct optimization (PySR) while recovering more concise rather than more complex expressions with increasing inference budget.

cs.LG↗

arXiv2026

Split-Flows: Measure Transport and Information Loss Across Molecular Resolutions

By reducing resolution, coarse-grained models greatly accelerate molecular simulations, unlocking access to long-timescale phenomena, though at the expense of microscopic information. Recovering this fine-grained detail is essential for tasks that depend on atomistic accuracy, making backmapping a central challenge in molecular modeling. We introduce split-flows, a novel flow-based approach that reinterprets backmapping as a continuous-time measure transport across resolutions. Unlike existing generative strategies, split-flows establish a direct probabilistic link between resolutions, enabling expressive conditional sampling of atomistic structures and -- for the first time -- a tractable route to computing mapping entropies, an information-theoretic measure of the irreducible detail lost in coarse-graining. We demonstrate these capabilities on diverse molecular systems, including chignolin, a lipid bilayer, and alanine dipeptide, highlighting split-flows as a principled framework for accurate backmapping and systematic evaluation of coarse-grained models.

physics.chem-ph↗

arXiv2026

Show Me What You Don't Know: Efficient Sampling from Invariant Sets for Model Validation

The performance of machine learning models is determined by the quality of their learned features. They should be invariant under irrelevant data variation but sensitive to task-relevant details. To visualize whether this is the case, we propose a method to analyze feature extractors by sampling from their fibers -- equivalence classes defined by their invariances -- given an arbitrary representative. Unlike existing work where a dedicated generative model is trained for each feature detector, our algorithm is training-free and exploits a pretrained diffusion or flow-matching model as a prior. The fiber loss -- which penalizes mismatch in features -- guides the denoising process toward the desired equivalence class, via non-linear diffusion trajectory matching. This replaces days of training for invariance learning with a single guided generation procedure at comparable fidelity. Experiments on popular datasets (ImageNet, CheXpert) and model types (ResNet, DINO, BiomedClip) demonstrate that our framework can reveal invariances ranging from very desirable to concerning behaviour. For instance, we show how Qwen-2B places patients with situs inversus (heart on the right side) in the same fiber as typical anatomy.

cs.LG↗

arXiv2026

Economical Jet Taggers -- Equivariant, Slim, and Quantized

Modern machine learning is transforming jet tagging at the LHC, but the leading transformer architectures are large, not particularly fast, and training-intensive. We present a slim version of the L-GATr tagger, reduce the number of parameters of jet-tagging transformers, and quantize them. We compare different quantization methods for standard and Lorentz-equivariant transformers and estimate their gains in resource efficiency. We find an order-of-magnitude reduction in energy cost for an moderate performance decrease, down to 1000-parameter taggers. This might be a step towards trigger-level jet tagging with small and quantized versions of the leading equivariant transformer architectures.

hep-ph↗

arXiv2025

Towards Context-Aware Domain Generalization: Understanding the Benefits and Limits of Marginal Transfer Learning

In this work, we analyze the conditions under which information about the context of an input $X$ can improve the predictions of deep learning models in new domains. Following work in marginal transfer learning in Domain Generalization (DG), we formalize the notion of context as a permutation-invariant representation of a set of data points that originate from the same domain as the input itself. We offer a theoretical analysis of the conditions under which this approach can, in principle, yield benefits, and formulate two necessary criteria that can be easily verified in practice. Additionally, we contribute insights into the kind of distribution shifts for which the marginal transfer learning approach promises robustness. Empirical analysis shows that our criteria are effective in discerning both favorable and unfavorable scenarios. Finally, we demonstrate that we can reliably detect scenarios where a model is tasked with unwarranted extrapolation in out-of-distribution (OOD) domains, identifying potential failure cases. Consequently, we showcase a method to select between the most predictive and the most robust model, circumventing the well-known trade-off between predictive performance and robustness.

cs.LG↗

arXiv2025

Beyond Diagonal Covariance: Flexible Posterior VAEs via Free-Form Injective Flows

Variational Autoencoders (VAEs) are powerful generative models widely used for learning interpretable latent spaces, quantifying uncertainty, and compressing data for downstream generative tasks. VAEs typically rely on diagonal Gaussian posteriors due to computational constraints. Using arguments grounded in differential geometry, we demonstrate inherent limitations in the representational capacity of diagonal covariance VAEs, as illustrated by explicit low-dimensional examples. In response, we show that a regularized variant of the recently introduced Free-form Injective Flow (FIF) can be interpreted as a VAE featuring a highly flexible, implicitly defined posterior. Crucially, this regularization yields a posterior equivalent to a full Gaussian covariance distribution, yet maintains computational costs comparable to standard diagonal covariance VAEs. Experiments on image datasets validate our approach, demonstrating that incorporating full covariance substantially improves model likelihood.

cs.LG↗

arXiv2025

Learning Distances from Data with Normalizing Flows and Score Matching

Density-based distances (DBDs) provide a principled approach to metric learning by defining distances in terms of the underlying data distribution. By employing a Riemannian metric that increases in regions of low probability density, shortest paths naturally follow the data manifold. Fermat distances, a specific type of DBD, have attractive properties, but existing estimators based on nearest neighbor graphs suffer from poor convergence due to inaccurate density estimates. Moreover, graph-based methods scale poorly to high dimensions, as the proposed geodesics are often insufficiently smooth. We address these challenges in two key ways. First, we learn densities using normalizing flows. Second, we refine geodesics through relaxation, guided by a learned score model. Additionally, we introduce a dimension-adapted Fermat distance that scales intuitively to high dimensions and improves numerical stability. Our work paves the way for the practical use of density-based distances, especially in high-dimensional spaces.

cs.LG↗

arXiv2025

Analyzing Generative Models by Manifold Entropic Metrics

Good generative models should not only synthesize high quality data, but also utilize interpretable representations that aid human understanding of their behavior. However, it is difficult to measure objectively if and to what degree desirable properties of disentangled representations have been achieved. Inspired by the principle of independent mechanisms, we address this difficulty by introducing a novel set of tractable information-theoretic evaluation metrics. We demonstrate the usefulness of our metrics on illustrative toy examples and conduct an in-depth comparison of various normalizing flow architectures and $β$-VAEs on the EMNIST dataset. Our method allows to sort latent features by importance and assess the amount of residual correlations of the resulting concepts. The most interesting finding of our experiments is a ranking of model architectures and training procedures in terms of their inductive bias to converge to aligned and disentangled representations during training.

cs.LG↗

arXiv2025

TRADE: Transfer of Distributions between External Conditions with Normalizing Flows

Modeling distributions that depend on external control parameters is a common scenario in diverse applications like molecular simulations, where system properties like temperature affect molecular configurations. Despite the relevance of these applications, existing solutions are unsatisfactory as they require severely restricted model architectures or rely on energy-based training, which is prone to instability. We introduce TRADE, which overcomes these limitations by formulating the learning process as a boundary value problem. By initially training the model for a specific condition using either i.i.d.~samples or backward KL training, we establish a boundary distribution. We then propagate this information across other conditions using the gradient of the unnormalized density with respect to the external parameter. This formulation, akin to the principles of physics-informed neural networks, allows us to efficiently learn parameter-dependent distributions without restrictive assumptions. Experimentally, we demonstrate that TRADE achieves excellent results in a wide range of applications, ranging from Bayesian inference and molecular simulations to physical lattice models.

cs.LG↗

arXiv2025

OOD Detection with immature Models

Likelihood-based deep generative models (DGMs) have gained significant attention for their ability to approximate the distributions of high-dimensional data. However, these models lack a performance guarantee in assigning higher likelihood values to in-distribution (ID) inputs, data the models are trained on, compared to out-of-distribution (OOD) inputs. This counter-intuitive behaviour is particularly pronounced when ID inputs are more complex than OOD data points. One potential approach to address this challenge involves leveraging the gradient of a data point with respect to the parameters of the DGMs. A recent OOD detection framework proposed estimating the joint density of layer-wise gradient norms for a given data point as a model-agnostic method, demonstrating superior performance compared to the Typicality Test across likelihood-based DGMs and image dataset pairs. In particular, most existing methods presuppose access to fully converged models, the training of which is both time-intensive and computationally demanding. In this work, we demonstrate that using immature models,stopped at early stages of training, can mostly achieve equivalent or even superior results on this downstream task compared to mature models capable of generating high-quality samples that closely resemble ID data. This novel finding enhances our understanding of how DGMs learn the distribution of ID data and highlights the potential of leveraging partially trained models for downstream tasks. Furthermore, we offer a possible explanation for this unexpected behaviour through the concept of support overlap.

cs.LG↗

arXiv2025

On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows

We present a novel theoretical framework for understanding the expressive power of normalizing flows. Despite their prevalence in scientific applications, a comprehensive understanding of flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. We propose a distributional universality theorem for well-conditioned coupling-based normalizing flows such as RealNVP. In addition, we show that volume-preserving normalizing flows are not universal, what distribution they learn instead, and how to fix their expressivity. Our results support the general wisdom that affine and related couplings are expressive and in general outperform volume-preserving flows, bridging a gap between empirical results and theoretical understanding.

cs.LG↗

arXiv2025

Deciphering the Definition of Adversarial Robustness for post-hoc OOD Detectors

Detecting out-of-distribution (OOD) inputs is critical for safely deploying deep learning models in real-world scenarios. In recent years, many OOD detectors have been developed, and even the benchmarking has been standardized, i.e. OpenOOD. The number of post-hoc detectors is growing fast. They are showing an option to protect a pre-trained classifier against natural distribution shifts and claim to be ready for real-world scenarios. However, its effectiveness in dealing with adversarial examples (AdEx) has been neglected in most studies. In cases where an OOD detector includes AdEx in its experiments, the lack of uniform parameters for AdEx makes it difficult to accurately evaluate the performance of the OOD detector. This paper investigates the adversarial robustness of 16 post-hoc detectors against various evasion attacks. It also discusses a roadmap for adversarial defense in OOD detectors that would help adversarial robustness. We believe that level 1 (AdEx on a unified dataset) should be added to any OOD detector to see the limitations. The last level in the roadmap (defense against adaptive attacks) we added for integrity from an adversarial machine learning (AML) point of view, which we do not believe is the ultimate goal for OOD detectors.

cs.CR↗

arXiv2025

Application-driven Validation of Posteriors in Inverse Problems

Current deep learning-based solutions for image analysis tasks are commonly incapable of handling problems to which multiple different plausible solutions exist. In response, posterior-based methods such as conditional Diffusion Models and Invertible Neural Networks have emerged; however, their translation is hampered by a lack of research on adequate validation. In other words, the way progress is measured often does not reflect the needs of the driving practical application. Closing this gap in the literature, we present the first systematic framework for the application-driven validation of posterior-based methods in inverse problems. As a methodological novelty, it adopts key principles from the field of object detection validation, which has a long history of addressing the question of how to locate and match multiple object instances in an image. Treating modes as instances enables us to perform mode-centric validation, using well-interpretable metrics from the application perspective. We demonstrate the value of our framework through instantiations for a synthetic toy example and two medical vision use cases: pose estimation in surgery and imaging-based quantification of functional tissue parameters for diagnostics. Our framework offers key advantages over common approaches to posterior validation in all three examples and could thus revolutionize performance assessment in inverse problems.

cs.CV↗

arXiv2024

Learning Distributions on Manifolds with Free-Form Flows

We propose Manifold Free-Form Flows (M-FFF), a simple new generative model for data on manifolds. The existing approaches to learning a distribution on arbitrary manifolds are expensive at inference time, since sampling requires solving a differential equation. Our method overcomes this limitation by sampling in a single function evaluation. The key innovation is to optimize a neural network via maximum likelihood on the manifold, possible by adapting the free-form flow framework to Riemannian manifolds. M-FFF is straightforwardly adapted to any manifold with a known projection. It consistently matches or outperforms previous single-step methods specialized to specific manifolds. It is typically two orders of magnitude faster than multi-step methods based on diffusion or flow matching, achieving better likelihoods in several experiments. We provide our code at https://github.com/vislearn/FFF.

cs.LG↗

arXiv2024

Consistency Models for Scalable and Fast Simulation-Based Inference

Simulation-based inference (SBI) is constantly in search of more expressive and efficient algorithms to accurately infer the parameters of complex simulation models. In line with this goal, we present consistency models for posterior estimation (CMPE), a new conditional sampler for SBI that inherits the advantages of recent unconstrained architectures and overcomes their sampling inefficiency at inference time. CMPE essentially distills a continuous probability flow and enables rapid few-shot inference with an unconstrained architecture that can be flexibly tailored to the structure of the estimation problem. We provide hyperparameters and default architectures that support consistency training over a wide range of different dimensions, including low-dimensional ones which are important in SBI workflows but were previously difficult to tackle even with unconditional consistency models. Our empirical evaluation demonstrates that CMPE not only outperforms current state-of-the-art algorithms on hard low-dimensional benchmarks, but also achieves competitive performance with much faster sampling speed on two realistic estimation problems with high data and/or parameter dimensions.

cs.LG↗

arXiv2024

Sensitivity-Aware Amortized Bayesian Inference

Sensitivity analyses reveal the influence of various modeling choices on the outcomes of statistical analyses. While theoretically appealing, they are overwhelmingly inefficient for complex Bayesian models. In this work, we propose sensitivity-aware amortized Bayesian inference (SA-ABI), a multifaceted approach to efficiently integrate sensitivity analyses into simulation-based inference with neural networks. First, we utilize weight sharing to encode the structural similarities between alternative likelihood and prior specifications in the training process with minimal computational overhead. Second, we leverage the rapid inference of neural networks to assess sensitivity to data perturbations and preprocessing steps. In contrast to most other Bayesian approaches, both steps circumvent the costly bottleneck of refitting the model for each choice of likelihood, prior, or data set. Finally, we propose to use deep ensembles to detect sensitivity arising from unreliable approximation (e.g., due to model misspecification). We demonstrate the effectiveness of our method in applied modeling problems, ranging from disease outbreak dynamics and global warming thresholds to human decision-making. Our results support sensitivity-aware inference as a default choice for amortized Bayesian workflows, automatically providing modelers with insights into otherwise hidden dimensions.

stat.ML↗