Michael Habeck

Henrik Voigt, Michael Habeck, Joachim Giesen

Large-scale transformers achieve impressive results on program synthesis benchmarks, yet their true generalization capabilities remain obscured by data contamination and opaque training corpora. To rigorously assess whether models are truly generalizing or merely retrieving memorized templates, we introduce a strictly controlled program synthesis environment based on a domain-specific arithmetic grammar. By systematically enumerating and evaluating millions of unique programs, we construct interpretable syntactic and semantic metric spaces. This allows us to precisely map data distributions and sample train and test splits that isolate specific distributional shifts. Our experiments demonstrate that optimizing density generalization -- through diverse sampling over both semantic and syntactic spaces -- induces robust out-of-distribution generalization. Conversely, evaluating support generalization reveals that transformers severely struggle with extrapolation, experiencing a performance drop of over 30% when forced to generate syntactically novel programs. While steadily scaling up compute improves generalization, the gains follow a strictly log-linear relationship. We conclude that robust generalization requires maximizing training diversity across multiple manifolds, and our findings indicate the necessity for novel search-based approaches to break through current log-linear scaling bottlenecks.

Paul Kahlmeyer, Joachim Giesen, Michael Habeck et al.

In a regression task, a function is learned from labeled data to predict the labels at new data points. The goal is to achieve small prediction errors. In symbolic regression, the goal is more ambitious, namely, to learn an interpretable function that makes small prediction errors. This additional goal largely rules out the standard approach used in regression, that is, reducing the learning problem to learning parameters of an expansion of basis functions by optimization. Instead, symbolic regression methods search for a good solution in a space of symbolic expressions. To cope with the typically vast search space, most symbolic regression methods make implicit, or sometimes even explicit, assumptions about its structure. Here, we argue that the only obvious structure of the search space is that it contains small expressions, that is, expressions that can be decomposed into a few subexpressions. We show that systematically searching spaces of small expressions finds solutions that are more accurate and more robust against noise than those obtained by state-of-the-art symbolic regression methods. In particular, systematic search outperforms state-of-the-art symbolic regressors in terms of its ability to recover the true underlying symbolic expressions on established benchmark data sets.

Henrik Voigt, Paul Kahlmeyer, Kai Lawonn et al.

Symbolic regression algorithms search a space of mathematical expressions for formulas that explain given data. Transformer-based models have emerged as a promising, scalable approach shifting the expensive combinatorial search to a large-scale pre-training phase. However, the success of these models is critically dependent on their pre-training data. Their ability to generalize to problems outside of this pre-training distribution remains largely unexplored. In this work, we conduct a systematic empirical study to evaluate the generalization capabilities of pre-trained, transformer-based symbolic regression. We rigorously test performance both within the pre-training distribution and on a series of out-of-distribution challenges for several state of the art approaches. Our findings reveal a significant dichotomy: while pre-trained models perform well in-distribution, the performance consistently degrades in out-of-distribution scenarios. We conclude that this generalization gap is a critical barrier for practitioners, as it severely limits the practical use of pre-trained approaches for real-world applications.

Julian L. Möbius, Michael Habeck

Many inverse problems are ill-posed and need to be complemented by prior information that restricts the class of admissible models. Bayesian approaches encode this information as prior distributions that impose generic properties on the model such as sparsity, non-negativity or smoothness. However, in case of complex structured models such as images, graphs or three-dimensional (3D) objects,generic prior distributions tend to favor models that differ largely from those observed in the real world. Here we explore the use of diffusion models as priors that are combined with experimental data within a Bayesian framework. We use 3D point clouds to represent 3D objects such as household items or biomolecular complexes formed from proteins and nucleic acids. We train diffusion models that generate coarse-grained 3D structures at a medium resolution and integrate these with incomplete and noisy experimental data. To demonstrate the power of our approach, we focus on the reconstruction of biomolecular assemblies from cryo-electron microscopy (cryo-EM) images, which is an important inverse problem in structural biology. We find that posterior sampling with diffusion model priors allows for 3D reconstruction from very sparse, low-resolution and partial observations.

Kevin H. Knuth, Michael Habeck, Nabin K. Malakar et al.

In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. Specific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization.

Mikhail Langovoy, Michael Habeck, Bernhard Schoelkopf

Cryo-electron microscopy (cryo-EM) is an emerging experimental method to characterize the structure of large biomolecular assemblies. Single particle cryo-EM records 2D images (so-called micrographs) of projections of the three-dimensional particle, which need to be processed to obtain the three-dimensional reconstruction. A crucial step in the reconstruction process is particle picking which involves detection of particles in noisy 2D micrographs with low signal-to-noise ratios of typically 1:10 or even lower. Typically, each picture contains a large number of particles, and particles have unknown irregular and nonconvex shapes.

Mikhail Langovoy, Michael Habeck, Bernhard Schölkopf

We develop a novel method for detection of signals and reconstruction of images in the presence of random noise. The method uses results from percolation theory. We specifically address the problem of detection of multiple objects of unknown shapes in the case of nonparametric noise. The noise density is unknown and can be heavy-tailed. The objects of interest have unknown varying intensities. No boundary shape constraints are imposed on the objects, only a set of weak bulk conditions is required. We view the object detection problem as a multiple hypothesis testing for discrete statistical inverse problems. We present an algorithm that allows to detect greyscale objects of various shapes in noisy images. We prove results on consistency and algorithmic complexity of our procedures. Applications to cryo-electron microscopy are presented.