Ullrich Köthe

Malte Tölle, Ullrich Köthe, Florian André et al.

Differential privacy (DP) has arisen as the gold standard in protecting an individual's privacy in datasets by adding calibrated noise to each data sample. While the application to categorical data is straightforward, its usability in the context of images has been limited. Contrary to categorical data the meaning of an image is inherent in the spatial correlation of neighboring pixels making the simple application of noise infeasible. Invertible Neural Networks (INN) have shown excellent generative performance while still providing the ability to quantify the exact likelihood. Their principle is based on transforming a complicated distribution into a simple one e.g. an image into a spherical Gaussian. We hypothesize that adding noise to the latent space of an INN can enable differentially private image modification. Manipulation of the latent space leads to a modified image while preserving important details. Further, by conditioning the INN on meta-data provided with the dataset we aim at leaving dimensions important for downstream tasks like classification untouched while altering other parts that potentially contain identifying information. We term our method content-aware differential privacy (CADP). We conduct experiments on publicly available benchmarking datasets as well as dedicated medical ones. In addition, we show the generalizability of our method to categorical data. The source code is publicly available at https://github.com/Cardio-AI/CADP.

Jens Müller, Stefan T. Radev, Robert Schmier et al.

We investigate a "learning to reject" framework to address the problem of silent failures in Domain Generalization (DG), where the test distribution differs from the training distribution. Assuming a mild distribution shift, we wish to accept out-of-distribution (OOD) data from a new domain whenever a model's estimated competence foresees trustworthy responses, instead of rejecting OOD data outright. Trustworthiness is then predicted via a proxy incompetence score that is tightly linked to the performance of a classifier. We present a comprehensive experimental evaluation of existing proxy scores as incompetence scores for classification and highlight the resulting trade-offs between rejection rate and accuracy gain. For comparability with prior work, we focus on standard DG benchmarks and consider the effect of measuring incompetence via different learned representations in a closed versus an open world setting. Our results suggest that increasing incompetence scores are indeed predictive of reduced accuracy, leading to significant improvements of the average accuracy below a suitable incompetence threshold. However, the scores are not yet good enough to allow for a favorable accuracy/rejection trade-off in all tested domains. Surprisingly, our results also indicate that classifiers optimized for DG robustness do not outperform a naive Empirical Risk Minimization (ERM) baseline in the competence region, that is, where test samples elicit low incompetence scores.

Kris K. Dreher, Leonardo Ayala, Melanie Schellenberg et al.

Synthetic medical image generation has evolved as a key technique for neural network training and validation. A core challenge, however, remains in the domain gap between simulations and real data. While deep learning-based domain transfer using Cycle Generative Adversarial Networks and similar architectures has led to substantial progress in the field, there are use cases in which state-of-the-art approaches still fail to generate training images that produce convincing results on relevant downstream tasks. Here, we address this issue with a domain transfer approach based on conditional invertible neural networks (cINNs). As a particular advantage, our method inherently guarantees cycle consistency through its invertible architecture, and network training can efficiently be conducted with maximum likelihood training. To showcase our method's generic applicability, we apply it to two spectral imaging modalities at different scales, namely hyperspectral imaging (pixel-level) and photoacoustic tomography (image-level). According to comprehensive experiments, our method enables the generation of realistic spectral data and outperforms the state of the art on two downstream classification tasks (binary and multi-class). cINN-based domain transfer could thus evolve as an important method for realistic synthetic data generation in the field of spectral imaging and beyond.

Ullrich Köthe

Change-of-variables (CoV) formulas allow to reduce complicated probability densities to simpler ones by a learned transformation with tractable Jacobian determinant. They are thus powerful tools for maximum-likelihood learning, Bayesian inference, outlier detection, model selection, etc. CoV formulas have been derived for a large variety of model types, but this information is scattered over many separate works. We present a systematic treatment from the unifying perspective of encoder/decoder architectures, which collects 28 CoV formulas in a single place, reveals interesting relationships between seemingly diverse methods, emphasizes important distinctions that are not always clear in the literature, and identifies surprising gaps for future research.

Peter Sorrenson, Felix Draxler, Armand Rousselot et al.

Normalizing Flows explicitly maximize a full-dimensional likelihood on the training data. However, real data is typically only supported on a lower-dimensional manifold leading the model to expend significant compute on modeling noise. Injective Flows fix this by jointly learning a manifold and the distribution on it. So far, they have been limited by restrictive architectures and/or high computational cost. We lift both constraints by a new efficient estimator for the maximum likelihood loss, compatible with free-form bottleneck architectures. We further show that naively learning both the data manifold and the distribution on it can lead to divergent solutions, and use this insight to motivate a stable maximum likelihood training objective. We perform extensive experiments on toy, tabular and image data, demonstrating the competitive performance of the resulting model.

Felix Draxler, Christoph Schnörr, Ullrich Köthe

Coupling-based normalizing flows (e.g. RealNVP) are a popular family of normalizing flow architectures that work surprisingly well in practice. This calls for theoretical understanding. Existing work shows that such flows weakly converge to arbitrary data distributions. However, they make no statement about the stricter convergence criterion used in practice, the maximum likelihood loss. For the first time, we make a quantitative statement about this kind of convergence: We prove that all coupling-based normalizing flows perform whitening of the data distribution (i.e. diagonalize the covariance matrix) and derive corresponding convergence bounds that show a linear convergence rate in the depth of the flow. Numerical experiments demonstrate the implications of our theory and point at open questions.

Lasse Elsemüller, Hans Olischläger, Marvin Schmitt et al.

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.

The-Gia Leo Nguyen, Lynton Ardizzone, Ullrich Köthe

Autoencoders are able to learn useful data representations in an unsupervised matter and have been widely used in various machine learning and computer vision tasks. In this work, we present methods to train Invertible Neural Networks (INNs) as (variational) autoencoders which we call INN (variational) autoencoders. Our experiments on MNIST, CIFAR and CelebA show that for low bottleneck sizes our INN autoencoder achieves results similar to the classical autoencoder. However, for large bottleneck sizes our INN autoencoder outperforms its classical counterpart. Based on the empirical results, we hypothesize that INN autoencoders might not have any intrinsic information loss and thereby are not bounded to a maximal number of layers (depth) after which only suboptimal results can be achieved.

Felix Draxler, Lars Kühmichel, Armand Rousselot et al.

Gaussianization is a simple generative model that can be trained without backpropagation. It has shown compelling performance on low dimensional data. As the dimension increases, however, it has been observed that the convergence speed slows down. We show analytically that the number of required layers scales linearly with the dimension for Gaussian input. We argue that this is because the model is unable to capture dependencies between dimensions. Empirically, we find the same linear increase in cost for arbitrary input $p(x)$, but observe favorable scaling for some distributions. We explore potential speed-ups and formulate challenges for further research.

Stefan T Radev, Marvin Schmitt, Lukas Schumacher et al.

Modern Bayesian inference involves a mixture of computational techniques for estimating, validating, and drawing conclusions from probabilistic models as part of principled workflows for data analysis. Typical problems in Bayesian workflows are the approximation of intractable posterior distributions for diverse model types and the comparison of competing models of the same process in terms of their complexity and predictive performance. This manuscript introduces the Python library BayesFlow for simulation-based training of established neural network architectures for amortized data compression and inference. Amortized Bayesian inference, as implemented in BayesFlow, enables users to train custom neural networks on model simulations and re-use these networks for any subsequent application of the models. Since the trained networks can perform inference almost instantaneously, the upfront neural network training is quickly amortized.

Robert Schmier, Ullrich Köthe, Christoph-Nikolas Straehle

Detecting test data deviating from training data is a central problem for safe and robust machine learning. Likelihoods learned by a generative model, e.g., a normalizing flow via standard log-likelihood training, perform poorly as an outlier score. We propose to use an unlabelled auxiliary dataset and a probabilistic outlier score for outlier detection. We use a self-supervised feature extractor trained on the auxiliary dataset and train a normalizing flow on the extracted features by maximizing the likelihood on in-distribution data and minimizing the likelihood on the contrastive dataset. We show that this is equivalent to learning the normalized positive difference between the in-distribution and the contrastive feature density. We conduct experiments on benchmark datasets and compare to the likelihood, the likelihood ratio and state-of-the-art anomaly detection methods.

Stefan T. Radev, Marvin Schmitt, Valentin Pratz et al.

This work proposes ``jointly amortized neural approximation'' (JANA) of intractable likelihood functions and posterior densities arising in Bayesian surrogate modeling and simulation-based inference. We train three complementary networks in an end-to-end fashion: 1) a summary network to compress individual data points, sets, or time series into informative embedding vectors; 2) a posterior network to learn an amortized approximate posterior; and 3) a likelihood network to learn an amortized approximate likelihood. Their interaction opens a new route to amortized marginal likelihood and posterior predictive estimation -- two important ingredients of Bayesian workflows that are often too expensive for standard methods. We benchmark the fidelity of JANA on a variety of simulation models against state-of-the-art Bayesian methods and propose a powerful and interpretable diagnostic for joint calibration. In addition, we investigate the ability of recurrent likelihood networks to emulate complex time series models without resorting to hand-crafted summary statistics.

Jonas Arruda, Niels Bracher, Ullrich Köthe et al.

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.

Tim J. Adler, Jan-Hinrich Nölke, Annika Reinke et al.

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.

Marvin Schmitt, Desi R. Ivanova, Daniel Habermann et al.

We propose a method to improve the efficiency and accuracy of amortized Bayesian inference by leveraging universal symmetries in the joint probabilistic model of parameters and data. In a nutshell, we invert Bayes' theorem and estimate the marginal likelihood based on approximate representations of the joint model. Upon perfect approximation, the marginal likelihood is constant across all parameter values by definition. However, errors in approximate inference lead to undesirable variance in the marginal likelihood estimates across different parameter values. We penalize violations of this symmetry with a \textit{self-consistency loss} which significantly improves the quality of approximate inference in low data regimes and can be used to augment the training of popular neural density estimators. We apply our method to a number of synthetic problems and realistic scientific models, discovering notable advantages in the context of both neural posterior and likelihood approximation.

Jonathan Wider, Jakob Kruse, Nils Weitzel et al.

Simulating abundances of stable water isotopologues, i.e. molecules differing in their isotopic composition, within climate models allows for comparisons with proxy data and, thus, for testing hypotheses about past climate and validating climate models under varying climatic conditions. However, many models are run without explicitly simulating water isotopologues. We investigate the possibility to replace the explicit physics-based simulation of oxygen isotopic composition in precipitation using machine learning methods. These methods estimate isotopic composition at each time step for given fields of surface temperature and precipitation amount. We implement convolutional neural networks (CNNs) based on the successful UNet architecture and test whether a spherical network architecture outperforms the naive approach of treating Earth's latitude-longitude grid as a flat image. Conducting a case study on a last millennium run with the iHadCM3 climate model, we find that roughly 40\% of the temporal variance in the isotopic composition is explained by the emulations on interannual and monthly timescale, with spatially varying emulation quality. A modified version of the standard UNet architecture for flat images yields results that are equally good as the predictions by the spherical CNN. We test generalization to last millennium runs of other climate models and find that while the tested deep learning methods yield the best results on iHadCM3 data, the performance drops when predicting on other models and is comparable to simple pixel-wise linear regression. An extended choice of predictor variables and improving the robustness of learned climate--oxygen isotope relationships should be explored in future work.

Felix Draxler, Peter Sorrenson, Lea Zimmermann et al.

Normalizing Flows are generative models that directly maximize the likelihood. Previously, the design of normalizing flows was largely constrained by the need for analytical invertibility. We overcome this constraint by a training procedure that uses an efficient estimator for the gradient of the change of variables formula. This enables any dimension-preserving neural network to serve as a generative model through maximum likelihood training. Our approach allows placing the emphasis on tailoring inductive biases precisely to the task at hand. Specifically, we achieve excellent results in molecule generation benchmarks utilizing $E(n)$-equivariant networks. Moreover, our method is competitive in an inverse problem benchmark, while employing off-the-shelf ResNet architectures.

Peter Sorrenson, Daniel Behrend-Uriarte, Christoph Schnörr et al.

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.

Peter Sorrenson, Felix Draxler, Armand Rousselot et al.

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.

Sander Hummerich, Tristan Bereau, Ullrich Köthe

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.

Marvin Schmitt, Valentin Pratz, Ullrich Köthe et al.

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.

Felix Draxler, Stefan Wahl, Christoph Schnörr et al.

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.

Peter Sorrenson, Lukas Lührs, Hans Olischläger et al.

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.

Daniel Galperin, Ullrich Köthe

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.

Peter Lorenz, Mario Fernandez, Jens Müller et al.

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.

Marvin Schmitt, Paul-Christian Bürkner, Ullrich Köthe et al.

Recent advances in probabilistic deep learning enable efficient amortized Bayesian inference in settings where the likelihood function is only implicitly defined by a simulation program (simulation-based inference; SBI). But how faithful is such inference if the simulation represents reality somewhat inaccurately, that is, if the true system behavior at test time deviates from the one seen during training? We conceptualize the types of such model misspecification arising in SBI and systematically investigate how the performance of neural posterior approximators gradually deteriorates as a consequence, making inference results less and less trustworthy. To notify users about this problem, we propose a new misspecification measure that can be trained in an unsupervised fashion (i.e., without training data from the true distribution) and reliably detects model misspecification at test time. Our experiments clearly demonstrate the utility of our new measure both on toy examples with an analytical ground-truth and on representative scientific tasks in cell biology, cognitive decision making, disease outbreak dynamics, and computer vision. We show how the proposed misspecification test warns users about suspicious outputs, raises an alarm when predictions are not trustworthy, and guides model designers in their search for better simulators.

Jens Müller, Lars Kühmichel, Martin Rohbeck et al.

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.

Titus Leistner, Radek Mackowiak, Lynton Ardizzone et al.

Light field applications, especially light field rendering and depth estimation, developed rapidly in recent years. While state-of-the-art light field rendering methods handle semi-transparent and reflective objects well, depth estimation methods either ignore these cases altogether or only deliver a weak performance. We argue that this is due current methods only considering a single "true" depth, even when multiple objects at different depths contributed to the color of a single pixel. Based on the simple idea of outputting a posterior depth distribution instead of only a single estimate, we develop and explore several different deep-learning-based approaches to the problem. Additionally, we contribute the first "multimodal light field depth dataset" that contains the depths of all objects which contribute to the color of a pixel. This allows us to supervise the multimodal depth prediction and also validate all methods by measuring the KL divergence of the predicted posteriors. With our thorough analysis and novel dataset, we aim to start a new line of depth estimation research that overcomes some of the long-standing limitations of this field.

Marvin Schmitt, Paul-Christian Bürkner, Ullrich Köthe et al.

Neural density estimators have proven remarkably powerful in performing efficient simulation-based Bayesian inference in various research domains. In particular, the BayesFlow framework uses a two-step approach to enable amortized parameter estimation in settings where the likelihood function is implicitly defined by a simulation program. But how faithful is such inference when simulations are poor representations of reality? In this paper, we conceptualize the types of model misspecification arising in simulation-based inference and systematically investigate the performance of the BayesFlow framework under these misspecifications. We propose an augmented optimization objective which imposes a probabilistic structure on the latent data space and utilize maximum mean discrepancy (MMD) to detect potentially catastrophic misspecifications during inference undermining the validity of the obtained results. We verify our detection criterion on a number of artificial and realistic misspecifications, ranging from toy conjugate models to complex models of decision making and disease outbreak dynamics applied to real data. Further, we show that posterior inference errors increase as a function of the distance between the true data-generating distribution and the typical set of simulations in the latent summary space. Thus, we demonstrate the dual utility of MMD as a method for detecting model misspecification and as a proxy for verifying the faithfulness of amortized Bayesian inference.

Lynton Ardizzone, Jakob Kruse, Carsten Lüth et al.

We introduce a new architecture called a conditional invertible neural network (cINN), and use it to address the task of diverse image-to-image translation for natural images. This is not easily possible with existing INN models due to some fundamental limitations. The cINN combines the purely generative INN model with an unconstrained feed-forward network, which efficiently preprocesses the conditioning image into maximally informative features. All parameters of a cINN are jointly optimized with a stable, maximum likelihood-based training procedure. Even though INN-based models have received far less attention in the literature than GANs, they have been shown to have some remarkable properties absent in GANs, e.g. apparent immunity to mode collapse. We find that our cINNs leverage these properties for image-to-image translation, demonstrated on day to night translation and image colorization. Furthermore, we take advantage of our bidirectional cINN architecture to explore and manipulate emergent properties of the latent space, such as changing the image style in an intuitive way.

Jakob Kruse, Lynton Ardizzone, Carsten Rother et al.

Recent work demonstrated that flow-based invertible neural networks are promising tools for solving ambiguous inverse problems. Following up on this, we investigate how ten invertible architectures and related models fare on two intuitive, low-dimensional benchmark problems, obtaining the best results with coupling layers and simple autoencoders. We hope that our initial efforts inspire other researchers to evaluate their invertible architectures in the same setting and put forth additional benchmarks, so our evaluation may eventually grow into an official community challenge.

Jan-Hinrich Nölke, Tim Adler, Janek Gröhl et al.

Multispectral photoacoustic imaging (PAI) is an emerging imaging modality which enables the recovery of functional tissue parameters such as blood oxygenation. However, the underlying inverse problems are potentially ill-posed, meaning that radically different tissue properties may - in theory - yield comparable measurements. In this work, we present a new approach for handling this specific type of uncertainty by leveraging the concept of conditional invertible neural networks (cINNs). Specifically, we propose going beyond commonly used point estimates for tissue oxygenation and converting single-pixel initial pressure spectra to the full posterior probability density. This way, the inherent ambiguity of a problem can be encoded with multiple modes in the output. Based on the presented architecture, we demonstrate two use cases which leverage this information to not only detect and quantify but also to compensate for uncertainties: (1) photoacoustic device design and (2) optimization of photoacoustic image acquisition. Our in silico studies demonstrate the potential of the proposed methodology to become an important building block for uncertainty-aware reconstruction of physiological parameters with PAI.

Jens Müller, Robert Schmier, Lynton Ardizzone et al.

Standard supervised learning breaks down under data distribution shift. However, the principle of independent causal mechanisms (ICM, Peters et al. (2017)) can turn this weakness into an opportunity: one can take advantage of distribution shift between different environments during training in order to obtain more robust models. We propose a new gradient-based learning framework whose objective function is derived from the ICM principle. We show theoretically and experimentally that neural networks trained in this framework focus on relations remaining invariant across environments and ignore unstable ones. Moreover, we prove that the recovered stable relations correspond to the true causal mechanisms under certain conditions. In both regression and classification, the resulting models generalize well to unseen scenarios where traditionally trained models fail.

Stefan T. Radev, Frederik Graw, Simiao Chen et al.

Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and intervention measures during an ongoing outbreak. However, reliably inferring the dynamics of ongoing outbreaks by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address this problem with a novel combination of epidemiological modeling with specialized neural networks. Our approach entails two computational phases: In an initial training phase, a mathematical model describing the epidemic is used as a coach for a neural network, which acquires global knowledge about the full range of possible disease dynamics. In the subsequent inference phase, the trained neural network processes the observed data of an actual outbreak and infers the parameters of the model in order to realistically reproduce the observed dynamics and reliably predict future progression. With its flexible framework, our simulation-based approach is applicable to a variety of epidemiological models. Moreover, since our method is fully Bayesian, it is designed to incorporate all available prior knowledge about plausible parameter values and returns complete joint posterior distributions over these parameters. Application of our method to the early Covid-19 outbreak phase in Germany demonstrates that we are able to obtain reliable probabilistic estimates for important disease characteristics, such as generation time, fraction of undetected infections, likelihood of transmission before symptom onset, and reporting delays using a very moderate amount of real-world observations.

Radek Mackowiak, Lynton Ardizzone, Ullrich Köthe et al.

With the maturing of deep learning systems, trustworthiness is becoming increasingly important for model assessment. We understand trustworthiness as the combination of explainability and robustness. Generative classifiers (GCs) are a promising class of models that are said to naturally accomplish these qualities. However, this has mostly been demonstrated on simple datasets such as MNIST and CIFAR in the past. In this work, we firstly develop an architecture and training scheme that allows GCs to operate on a more relevant level of complexity for practical computer vision, namely the ImageNet challenge. Secondly, we demonstrate the immense potential of GCs for trustworthy image classification. Explainability and some aspects of robustness are vastly improved compared to feed-forward models, even when the GCs are just applied naively. While not all trustworthiness problems are solved completely, we observe that GCs are a highly promising basis for further algorithms and modifications. We release our trained model for download in the hope that it serves as a starting point for other generative classification tasks, in much the same way as pretrained ResNet architectures do for discriminative classification.

Stefan T. Radev, Marco D'Alessandro, Ulf K. Mertens et al.

Comparing competing mathematical models of complex natural processes is a shared goal among many branches of science. The Bayesian probabilistic framework offers a principled way to perform model comparison and extract useful metrics for guiding decisions. However, many interesting models are intractable with standard Bayesian methods, as they lack a closed-form likelihood function or the likelihood is computationally too expensive to evaluate. With this work, we propose a novel method for performing Bayesian model comparison using specialized deep learning architectures. Our method is purely simulation-based and circumvents the step of explicitly fitting all alternative models under consideration to each observed dataset. Moreover, it requires no hand-crafted summary statistics of the data and is designed to amortize the cost of simulation over multiple models and observable datasets. This makes the method particularly effective in scenarios where model fit needs to be assessed for a large number of datasets, so that per-dataset inference is practically infeasible.Finally, we propose a novel way to measure epistemic uncertainty in model comparison problems. We demonstrate the utility of our method on toy examples and simulated data from non-trivial models from cognitive science and single-cell neuroscience. We show that our method achieves excellent results in terms of accuracy, calibration, and efficiency across the examples considered in this work. We argue that our framework can enhance and enrich model-based analysis and inference in many fields dealing with computational models of natural processes. We further argue that the proposed measure of epistemic uncertainty provides a unique proxy to quantify absolute evidence even in a framework which assumes that the true data-generating model is within a finite set of candidate models.

Stefan T. Radev, Ulf K. Mertens, Andreas Voss et al.

Estimating the parameters of mathematical models is a common problem in almost all branches of science. However, this problem can prove notably difficult when processes and model descriptions become increasingly complex and an explicit likelihood function is not available. With this work, we propose a novel method for globally amortized Bayesian inference based on invertible neural networks which we call BayesFlow. The method uses simulation to learn a global estimator for the probabilistic mapping from observed data to underlying model parameters. A neural network pre-trained in this way can then, without additional training or optimization, infer full posteriors on arbitrary many real datasets involving the same model family. In addition, our method incorporates a summary network trained to embed the observed data into maximally informative summary statistics. Learning summary statistics from data makes the method applicable to modeling scenarios where standard inference techniques with hand-crafted summary statistics fail. We demonstrate the utility of BayesFlow on challenging intractable models from population dynamics, epidemiology, cognitive science and ecology. We argue that BayesFlow provides a general framework for building amortized Bayesian parameter estimation machines for any forward model from which data can be simulated.

Lynton Ardizzone, Radek Mackowiak, Carsten Rother et al.

The Information Bottleneck (IB) objective uses information theory to formulate a task-performance versus robustness trade-off. It has been successfully applied in the standard discriminative classification setting. We pose the question whether the IB can also be used to train generative likelihood models such as normalizing flows. Since normalizing flows use invertible network architectures (INNs), they are information-preserving by construction. This seems contradictory to the idea of a bottleneck. In this work, firstly, we develop the theory and methodology of IB-INNs, a class of conditional normalizing flows where INNs are trained using the IB objective: Introducing a small amount of {\em controlled} information loss allows for an asymptotically exact formulation of the IB, while keeping the INN's generative capabilities intact. Secondly, we investigate the properties of these models experimentally, specifically used as generative classifiers. This model class offers advantages such as improved uncertainty quantification and out-of-distribution detection, but traditional generative classifier solutions suffer considerably in classification accuracy. We find the trade-off parameter in the IB controls a mix of generative capabilities and accuracy close to standard classifiers. Empirically, our uncertainty estimates in this mixed regime compare favourably to conventional generative and discriminative classifiers.

Peter Sorrenson, Carsten Rother, Ullrich Köthe

A central question of representation learning asks under which conditions it is possible to reconstruct the true latent variables of an arbitrarily complex generative process. Recent breakthrough work by Khemakhem et al. (2019) on nonlinear ICA has answered this question for a broad class of conditional generative processes. We extend this important result in a direction relevant for application to real-world data. First, we generalize the theory to the case of unknown intrinsic problem dimension and prove that in some special (but not very restrictive) cases, informative latent variables will be automatically separated from noise by an estimating model. Furthermore, the recovered informative latent variables will be in one-to-one correspondence with the true latent variables of the generating process, up to a trivial component-wise transformation. Second, we introduce a modification of the RealNVP invertible neural network architecture (Dinh et al. (2016)) which is particularly suitable for this type of problem: the General Incompressible-flow Network (GIN). Experiments on artificial data and EMNIST demonstrate that theoretical predictions are indeed verified in practice. In particular, we provide a detailed set of exactly 22 informative latent variables extracted from EMNIST.

Ricard Durall, Franz-Josef Pfreundt, Ullrich Köthe et al.

Recent deep learning based approaches have shown remarkable success on object segmentation tasks. However, there is still room for further improvement. Inspired by generative adversarial networks, we present a generic end-to-end adversarial approach, which can be combined with a wide range of existing semantic segmentation networks to improve their segmentation performance. The key element of our method is to replace the commonly used binary adversarial loss with a high resolution pixel-wise loss. In addition, we train our generator employing stochastic weight averaging fashion, which further enhances the predicted output label maps leading to state-of-the-art results. We show, that this combination of pixel-wise adversarial training and weight averaging leads to significant and consistent gains in segmentation performance, compared to the baseline models.

Lynton Ardizzone, Carsten Lüth, Jakob Kruse et al.

In this work, we address the task of natural image generation guided by a conditioning input. We introduce a new architecture called conditional invertible neural network (cINN). The cINN combines the purely generative INN model with an unconstrained feed-forward network, which efficiently preprocesses the conditioning input into useful features. All parameters of the cINN are jointly optimized with a stable, maximum likelihood-based training procedure. By construction, the cINN does not experience mode collapse and generates diverse samples, in contrast to e.g. cGANs. At the same time our model produces sharp images since no reconstruction loss is required, in contrast to e.g. VAEs. We demonstrate these properties for the tasks of MNIST digit generation and image colorization. Furthermore, we take advantage of our bi-directional cINN architecture to explore and manipulate emergent properties of the latent space, such as changing the image style in an intuitive way.

Jakob Kruse, Gianluca Detommaso, Ullrich Köthe et al.

Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical architecture, HINT allows sampling from a joint distribution p(y,x) and the corresponding posterior p(x|y) using a single invertible network. We evaluate our method on some standard data sets and benchmark its full power for density estimation and Bayesian inference on a novel data set of 2D shapes in Fourier parameterization, which enables consistent visualization of samples for different dimensionalities.

Steffen Wolf, Alberto Bailoni, Constantin Pape et al.

Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments, or equivalently to detect closed contours. Most prior work either requires seeds, one per segment; or a threshold; or formulates the task as multicut / correlation clustering, an NP-hard problem. Here, we propose an efficient algorithm for graph partitioning, the "Mutex Watershed''. Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need for explicit seeds or a tunable threshold. We also prove that this simple algorithm solves to global optimality an objective function that is intimately related to the multicut / correlation clustering integer linear programming formulation. The algorithm is deterministic, very simple to implement, and has empirically linearithmic complexity. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the Mutex Watershed gives the best results currently known for the competitive ISBI 2012 EM segmentation benchmark.

Tim J. Adler, Lynton Ardizzone, Anant Vemuri et al.

Purpose: Optical imaging is evolving as a key technique for advanced sensing in the operating room. Recent research has shown that machine learning algorithms can be used to address the inverse problem of converting pixel-wise multispectral reflectance measurements to underlying tissue parameters, such as oxygenation. Assessment of the specific hardware used in conjunction with such algorithms, however, has not properly addressed the possibility that the problem may be ill-posed. Methods: We present a novel approach to the assessment of optical imaging modalities, which is sensitive to the different types of uncertainties that may occur when inferring tissue parameters. Based on the concept of invertible neural networks, our framework goes beyond point estimates and maps each multispectral measurement to a full posterior probability distribution which is capable of representing ambiguity in the solution via multiple modes. Performance metrics for a hardware setup can then be computed from the characteristics of the posteriors. Results: Application of the assessment framework to the specific use case of camera selection for physiological parameter estimation yields the following insights: (1) Estimation of tissue oxygenation from multispectral images is a well-posed problem, while (2) blood volume fraction may not be recovered without ambiguity. (3) In general, ambiguity may be reduced by increasing the number of spectral bands in the camera. Conclusion: Our method could help to optimize optical camera design in an application-specific manner.

Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert et al.

In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution, conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task -- so-called Invertible Neural Networks (INNs). Although INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.

Steffen Wolf, Lukas Schott, Ullrich Köthe et al.

Learned boundary maps are known to outperform hand- crafted ones as a basis for the watershed algorithm. We show, for the first time, how to train watershed computation jointly with boundary map prediction. The estimator for the merging priorities is cast as a neural network that is con- volutional (over space) and recurrent (over iterations). The latter allows learning of complex shape priors. The method gives the best known seeded segmentation results on the CREMI segmentation challenge.