Frederik Hoppe

Frederik Hoppe, Felix Krahmer, Claudio Mayrink Verdun et al.

One of the most promising solutions for uncertainty quantification in high-dimensional statistics is the debiased LASSO that relies on unconstrained $\ell_1$-minimization. The initial works focused on real Gaussian designs as a toy model for this problem. However, in medical imaging applications, such as compressive sensing for MRI, the measurement system is represented by a (subsampled) complex Fourier matrix. The purpose of this work is to extend the method to the MRI case in order to construct confidence intervals for each pixel of an MR image. We show that a sufficient amount of data is $n \gtrsim \max\{ s_0\log^2 s_0\log p, s_0 \log^2 p \}$.

Frederik Hoppe, Claudio Mayrink Verdun, Felix Krahmer et al.

Model-based deep learning solutions to inverse problems have attracted increasing attention in recent years as they bridge state-of-the-art numerical performance with interpretability. In addition, the incorporated prior domain knowledge can make the training more efficient as the smaller number of parameters allows the training step to be executed with smaller datasets. Algorithm unrolling schemes stand out among these model-based learning techniques. Despite their rapid advancement and their close connection to traditional high-dimensional statistical methods, they lack certainty estimates and a theory for uncertainty quantification is still elusive. This work provides a step towards closing this gap proposing a rigorous way to obtain confidence intervals for the LISTA estimator.

Frederik Hoppe, Claudio Mayrink Verdun, Hannah Laus et al.

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional regression or learning problems to increase the confidence of a given predictor. We develop a new data-driven approach for UQ in regression that applies both to classical regression approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to be neglected, resulting in overly narrow confidence intervals. Our work rigorously addresses this issue and derives a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating uncertainty in critical applications such as MRI diagnosis. Importantly, our analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings bridge the gap between established theory and the practical applicability of such debiased methods.

Frederik Hoppe, Claudio Mayrink Verdun, Felix Krahmer et al.

Over the last few years, debiased estimators have been proposed in order to establish rigorous confidence intervals for high-dimensional problems in machine learning and data science. The core argument is that the error of these estimators with respect to the ground truth can be expressed as a Gaussian variable plus a remainder term that vanishes as long as the dimension of the problem is sufficiently high. Thus, uncertainty quantification (UQ) can be performed exploiting the Gaussian model. Empirically, however, the remainder term cannot be neglected in many realistic situations of moderately-sized dimensions, in particular in certain structured measurement scenarios such as Magnetic Resonance Imaging (MRI). This, in turn, can downgrade the advantage of the UQ methods as compared to non-UQ approaches such as the standard LASSO. In this paper, we present a method to improve the debiased estimator by sampling without replacement. Our approach leverages recent results of ours on the structure of the random nature of certain sampling schemes showing how a transition between sampling with and without replacement can lead to a weighted reconstruction scheme with improved performance for the standard LASSO. In this paper, we illustrate how this reweighted sampling idea can also improve the debiased estimator and, consequently, provide a better method for UQ in Fourier imaging.

Frederik Hoppe, Lars Kleinemeier, Astrid Franz et al.

Recent foundation models for tabular data achieve strong task-specific performance via in-context learning. Nevertheless, they focus on direct prediction by encapsulating both representation learning and task-specific inference inside a single, resource-intensive network. This work specifically focuses on representation learning, i.e., on transferable, task-agnostic embeddings. We systematically evaluate task-agnostic representations from tabular foundation models (TabPFN and TabICL) alongside with classical feature engineering (TableVectorizer) across a variety of application tasks as outlier detection (ADBench) and supervised learning (TabArena Lite). We find that simple TableVectorizer features achieve comparable or superior performance while being up to three orders of magnitude faster than tabular foundation models. The code is available at https://github.com/ContactSoftwareAI/TabEmbedBench.

Astrid Franz, Frederik Hoppe, Marianne Michaelis et al.

Tabular data in relational databases represents a significant portion of industrial data. Hence, analyzing and interpreting tabular data is of utmost importance. Application tasks on tabular data are manifold and are often not specified when setting up an industrial database. To address this, we present a novel framework for generating universal, i.e., task-independent embeddings of tabular data for performing downstream tasks without predefined targets. Our method transforms tabular data into a graph structure, leverages Graph Auto-Encoders to create entity embeddings, which are subsequently aggregated to obtain embeddings for each table row, i.e., each data sample. This two-step approach has the advantage that unseen samples, consisting of similar entities, can be embedded without additional training. Downstream tasks such as regression, classification or outlier detection, can then be performed by applying a distance-based similarity measure in the embedding space. Experiments on real-world datasets demonstrate that our method achieves superior performance compared to existing universal tabular data embedding techniques.

Frederik Hoppe, Astrid Franz, Lars Kleinemeier et al.

Synthetic tabular data generation has received increasing attention in recent years, particularly with the emergence of foundation models for tabular data. The breakthrough success of TabPFN (Hollmann et al.,2025), which leverages vast quantities of synthetic tabular datasets derived from structural causal models (SCMs), demonstrates the critical role synthetic data plays in developing powerful tabular foundation models. However, most real-world tabular data exists in relational formats spanning multiple interconnected tables - a structure not adequately addressed by current generation methods. In this work, we extend the SCM-based approach by developing a novel framework that generates realistic synthetic relational tabular data including causal relationships across tables. Our experiments confirm that this framework is able to construct relational datasets with complex inter-table dependencies mimicking real-world scenarios.