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Julius von Kügelgen

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24 papers

2 author rows

ICLR Conference 2025 Conference Paper

Interaction Asymmetry: A General Principle for Learning Composable Abstractions

Jack Brady
Julius von Kügelgen
Sébastien Lachapelle
Simon Buchholz
Thomas Kipf
Wieland Brendel

Learning disentangled representations of concepts and re-composing them in unseen ways is crucial for generalizing to out-of-domain situations. However, the underlying properties of concepts that enable such disentanglement and compositional generalization remain poorly understood. In this work, we propose the principle of interaction asymmetry which states: "Parts of the same concept have more complex interactions than parts of different concepts". We formalize this via block diagonality conditions on the $(n+1)$th order derivatives of the generator mapping concepts to observed data, where different orders of "complexity" correspond to different $n$. Using this formalism, we prove that interaction asymmetry enables both disentanglement and compositional generalization. Our results unify recent theoretical results for learning concepts of objects, which we show are recovered as special cases with $n=0$ or $1$. We provide results for up to $n=2$, thus extending these prior works to more flexible generator functions, and conjecture that the same proof strategies generalize to larger $n$. Practically, our theory suggests that, to disentangle concepts, an autoencoder should penalize its latent capacity and the interactions between concepts during decoding. We propose an implementation of these criteria using a flexible Transformer-based VAE, with a novel regularizer on the attention weights of the decoder. On synthetic image datasets consisting of objects, we provide evidence that this model can achieve comparable object disentanglement to existing models that use more explicit object-centric priors.

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NeurIPS Conference 2025 Conference Paper

Transferring Causal Effects using Proxies

Manuel Iglesias-Alonso
Felix Schur
Julius von Kügelgen
Jonas Peters

We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

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ICML Conference 2024 Conference Paper

A Sparsity Principle for Partially Observable Causal Representation Learning

Danru Xu
Dingling Yao
Sébastien Lachapelle
Perouz Taslakian
Julius von Kügelgen
Francesco Locatello
Sara Magliacane

Causal representation learning aims at identifying high-level causal variables from perceptual data. Most methods assume that all latent causal variables are captured in the high-dimensional observations. We instead consider a partially observed setting, in which each measurement only provides information about a subset of the underlying causal state. Prior work has studied this setting with multiple domains or views, each depending on a fixed subset of latents. Here, we focus on learning from unpaired observations from a dataset with an instance-dependent partial observability pattern. Our main contribution is to establish two identifiability results for this setting: one for linear mixing functions without parametric assumptions on the underlying causal model, and one for piecewise linear mixing functions with Gaussian latent causal variables. Based on these insights, we propose two methods for estimating the underlying causal variables by enforcing sparsity in the inferred representation. Experiments on different simulated datasets and established benchmarks highlight the effectiveness of our approach in recovering the ground-truth latents.

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TMLR Journal 2024 Journal Article

Deep Backtracking Counterfactuals for Causally Compliant Explanations

Klaus-Rudolf Kladny
Julius von Kügelgen
Bernhard Schölkopf
Michael Muehlebach

Counterfactuals answer questions of what would have been observed under altered circumstances and can therefore offer valuable insights. Whereas the classical interventional interpretation of counterfactuals has been studied extensively, backtracking constitutes a less studied alternative where all causal laws are kept intact. In the present work, we introduce a practical method called deep backtracking counterfactuals (DeepBC) for computing backtracking counterfactuals in structural causal models that consist of deep generative components. We propose two distinct versions of our method—one utilizing Langevin Monte Carlo sampling and the other employing constrained optimization—to generate counterfactuals for high-dimensional data. As a special case, our formulation reduces to methods in the field of counterfactual explanations. Compared to these, our approach represents a causally compliant, versatile and modular alternative. We demonstrate these properties experimentally on a modified version of MNIST and CelebA.

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ICLR Conference 2024 Conference Paper

Multi-View Causal Representation Learning with Partial Observability

Dingling Yao
Danru Xu
Sébastien Lachapelle
Sara Magliacane
Perouz Taslakian
Georg Martius
Julius von Kügelgen
Francesco Locatello

We present a unified framework for studying the identifiability of representations learned from simultaneously observed views, such as different data modalities. We allow a partially observed setting in which each view constitutes a nonlinear mixture of a subset of underlying latent variables, which can be causally related. We prove that the information shared across all subsets of any number of views can be learned up to a smooth bijection using contrastive learning and a single encoder per view. We also provide graphical criteria indicating which latent variables can be identified through a simple set of rules, which we refer to as identifiability algebra. Our general framework and theoretical results unify and extend several previous work on multi-view nonlinear ICA, disentanglement, and causal representation learning. We experimentally validate our claims on numerical, image, and multi-modal data sets. Further, we demonstrate that the performance of prior methods is recovered in different special cases of our setup. Overall, we find that access to multiple partial views offers unique opportunities for identifiable representation learning, enabling the discovery of latent structures from purely observational data.

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NeurIPS Conference 2023 Conference Paper

Causal Component Analysis

Liang Wendong
Armin Kekić
Julius von Kügelgen
Simon Buchholz
Michel Besserve
Luigi Gresele
Bernhard Schölkopf

Independent Component Analysis (ICA) aims to recover independent latent variables from observed mixtures thereof. Causal Representation Learning (CRL) aims instead to infer causally related (thus often statistically dependent ) latent variables, together with the unknown graph encoding their causal relationships. We introduce an intermediate problem termed Causal Component Analysis (CauCA). CauCA can be viewed as a generalization of ICA, modelling the causal dependence among the latent components, and as a special case of CRL. In contrast to CRL, it presupposes knowledge of the causal graph, focusing solely on learning the unmixing function and the causal mechanisms. Any impossibility results regarding the recovery of the ground truth in CauCA also apply for CRL, while possibility results may serve as a stepping stone for extensions to CRL. We characterize CauCA identifiability from multiple datasets generated through different types of interventions on the latent causal variables. As a corollary, this interventional perspective also leads to new identifiability results for nonlinear ICA—a special case of CauCA with an empty graph—requiring strictly fewer datasets than previous results. We introduce a likelihood-based approach using normalizing flows to estimate both the unmixing function and the causal mechanisms, and demonstrate its effectiveness through extensive synthetic experiments in the CauCA and ICA setting.

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UAI Conference 2023 Conference Paper

Causal effect estimation from observational and interventional data through matrix weighted linear estimators

Klaus-Rudolf Kladny
Julius von Kügelgen
Bernhard Schölkopf
Michael Muehlebach

We study causal effect estimation from a mixture of observational and interventional data in a confounded linear regression model with multivariate treatments. We show that the statistical efficiency in terms of expected squared error can be improved by combining estimators arising from both the observational and interventional setting. To this end, we derive methods based on matrix weighted linear estimators and prove that our methods are asymptotically unbiased in the infinite sample limit. This is an important improvement compared to the pooled estimator using the union of interventional and observational data, for which the bias only vanishes if the ratio of observational to interventional data tends to zero. Studies on synthetic data confirm our theoretical findings. In settings where confounding is substantial and the ratio of observational to interventional data is large, our estimators outperform a Stein-type estimator and various other baselines.

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ICLR Conference 2023 Conference Paper

DCI-ES: An Extended Disentanglement Framework with Connections to Identifiability

Cian Eastwood
Andrei Liviu Nicolicioiu
Julius von Kügelgen
Armin Kekic
Frederik Träuble
Andrea Dittadi
Bernhard Schölkopf

In representation learning, a common approach is to seek representations which disentangle the underlying factors of variation. Eastwood & Williams (2018) proposed three metrics for quantifying the quality of such disentangled representations: disentanglement (D), completeness (C) and informativeness (I). In this work, we first connect this DCI framework to two common notions of linear and nonlinear identifiability, thereby establishing a formal link between disentanglement and the closely-related field of independent component analysis. We then propose an extended DCI-ES framework with two new measures of representation quality—explicitness (E) and size (S)—and point out how D and C can be computed for black-box predictors. Our main idea is that the functional capacity required to use a representation is an important but thus-far neglected aspect of representation quality, which we quantify using explicitness or ease-of-use (E). We illustrate the relevance of our extensions on the MPI3D and Cars3D datasets.

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NeurIPS Conference 2023 Conference Paper

Nonparametric Identifiability of Causal Representations from Unknown Interventions

Julius von Kügelgen
Michel Besserve
Liang Wendong
Luigi Gresele
Armin Kekić
Elias Bareinboim
David Blei
Bernhard Schölkopf

We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional functions (“mixtures”) of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision.

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ICML Conference 2023 Conference Paper

Provably Learning Object-Centric Representations

Jack Brady
Roland S. Zimmermann
Yash Sharma 0001
Bernhard Schölkopf
Julius von Kügelgen
Wieland Brendel

Learning structured representations of the visual world in terms of objects promises to significantly improve the generalization abilities of current machine learning models. While recent efforts to this end have shown promising empirical progress, a theoretical account of when unsupervised object-centric representation learning is possible is still lacking. Consequently, understanding the reasons for the success of existing object-centric methods as well as designing new theoretically grounded methods remains challenging. In the present work, we analyze when object-centric representations can provably be learned without supervision. To this end, we first introduce two assumptions on the generative process for scenes comprised of several objects, which we call compositionality and irreducibility. Under this generative process, we prove that the ground-truth object representations can be identified by an invertible and compositional inference model, even in the presence of dependencies between objects. We empirically validate our results through experiments on synthetic data. Finally, we provide evidence that our theory holds predictive power for existing object-centric models by showing a close correspondence between models’ compositionality and invertibility and their empirical identifiability.

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NeurIPS Conference 2023 Conference Paper

Spuriosity Didn’t Kill the Classifier: Using Invariant Predictions to Harness Spurious Features

Cian Eastwood
Shashank Singh
Andrei L Nicolicioiu
Marin Vlastelica Pogančić
Julius von Kügelgen
Bernhard Schölkopf

To avoid failures on out-of-distribution data, recent works have sought to extract features that have an invariant or stable relationship with the label across domains, discarding "spurious" or unstable features whose relationship with the label changes across domains. However, unstable features often carry complementary information that could boost performance if used correctly in the test domain. In this work, we show how this can be done without test-domain labels. In particular, we prove that pseudo-labels based on stable features provide sufficient guidance for doing so, provided that stable and unstable features are conditionally independent given the label. Based on this theoretical insight, we propose Stable Feature Boosting (SFB), an algorithm for: (i) learning a predictor that separates stable and conditionally-independent unstable features; and (ii) using the stable-feature predictions to adapt the unstable-feature predictions in the test domain. Theoretically, we prove that SFB can learn an asymptotically-optimal predictor without test-domain labels. Empirically, we demonstrate the effectiveness of SFB on real and synthetic data.

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NeurIPS Conference 2022 Conference Paper

Active Bayesian Causal Inference

Christian Toth
Lars Lorch
Christian Knoll
Andreas Krause
Franz Pernkopf
Robert Peharz
Julius von Kügelgen

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e. g. , the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference—quantities that are not of direct interest ought to be marginalized out in this process, thus contributing to our overall uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, i. e. , for jointly inferring a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient nonlinear additive Gaussian noise models, which we model using Gaussian processes. To capture the space of causal graphs, we use a continuous latent graph representation, allowing our approach to scale to practically relevant problem sizes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, update our beliefs, and repeat. Through simulations, we demonstrate that our approach is more data-efficient than existing methods that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples, while providing well-calibrated uncertainty estimates of the quantities of interest.

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NeurIPS Conference 2022 Conference Paper

Causal Discovery in Heterogeneous Environments Under the Sparse Mechanism Shift Hypothesis

Ronan Perry
Julius von Kügelgen
Bernhard Schölkopf

Machine learning approaches commonly rely on the assumption of independent and identically distributed (i. i. d. ) data. In reality, however, this assumption is almost always violated due to distribution shifts between environments. Although valuable learning signals can be provided by heterogeneous data from changing distributions, it is also known that learning under arbitrary (adversarial) changes is impossible. Causality provides a useful framework for modeling distribution shifts, since causal models encode both observational and interventional distributions. In this work, we explore the sparse mechanism shift hypothesis which posits that distribution shifts occur due to a small number of changing causal conditionals. Motivated by this idea, we apply it to learning causal structure from heterogeneous environments, where i. i. d. data only allows for learning an equivalence class of graphs without restrictive assumptions. We propose the Mechanism Shift Score (MSS), a score-based approach amenable to various empirical estimators, which provably identifies the entire causal structure with high probability if the sparse mechanism shifts hypothesis holds. Empirically, we verify behavior predicted by the theory and compare multiple estimators and score functions to identify the best approaches in practice. Compared to other methods, we show how MSS bridges a gap by both being nonparametric as well as explicitly leveraging sparse changes.

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ICML Conference 2022 Conference Paper

Causal Inference Through the Structural Causal Marginal Problem

Luigi Gresele
Julius von Kügelgen
Jonas M. Kübler
Elke Kirschbaum
Bernhard Schölkopf
Dominik Janzing

We introduce an approach to counterfactual inference based on merging information from multiple datasets. We consider a causal reformulation of the statistical marginal problem: given a collection of marginal structural causal models (SCMs) over distinct but overlapping sets of variables, determine the set of joint SCMs that are counterfactually consistent with the marginal ones. We formalise this approach for categorical SCMs using the response function formulation and show that it reduces the space of allowed marginal and joint SCMs. Our work thus highlights a new mode of falsifiability through additional variables, in contrast to the statistical one via additional data.

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NeurIPS Conference 2022 Conference Paper

Embrace the Gap: VAEs Perform Independent Mechanism Analysis

Patrik Reizinger
Luigi Gresele
Jack Brady
Julius von Kügelgen
Dominik Zietlow
Bernhard Schölkopf
Georg Martius
Wieland Brendel

Variational autoencoders (VAEs) are a popular framework for modeling complex data distributions; they can be efficiently trained via variational inference by maximizing the evidence lower bound (ELBO), at the expense of a gap to the exact (log-)marginal likelihood. While VAEs are commonly used for representation learning, it is unclear why ELBO maximization would yield useful representations, since unregularized maximum likelihood estimation cannot invert the data-generating process. Yet, VAEs often succeed at this task. We seek to elucidate this apparent paradox by studying nonlinear VAEs in the limit of near-deterministic decoders. We first prove that, in this regime, the optimal encoder approximately inverts the decoder---a commonly used but unproven conjecture---which we refer to as self-consistency. Leveraging self-consistency, we show that the ELBO converges to a regularized log-likelihood. This allows VAEs to perform what has recently been termed independent mechanism analysis (IMA): it adds an inductive bias towards decoders with column-orthogonal Jacobians, which helps recovering the true latent factors. The gap between ELBO and log-likelihood is therefore welcome, since it bears unanticipated benefits for nonlinear representation learning. In experiments on synthetic and image data, we show that VAEs uncover the true latent factors when the data generating process satisfies the IMA assumption.

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AAAI Conference 2022 Conference Paper

On the Fairness of Causal Algorithmic Recourse

Julius von Kügelgen
Amir-Hossein Karimi
Umang Bhatt
Isabel Valera
Adrian Weller
Bernhard Schölkopf

Algorithmic fairness is typically studied from the perspective of predictions. Instead, here we investigate fairness from the perspective of recourse actions suggested to individuals to remedy an unfavourable classification. We propose two new fairness criteria at the group and individual level, which—unlike prior work on equalising the average group-wise distance from the decision boundary—explicitly account for causal relationships between features, thereby capturing downstream effects of recourse actions performed in the physical world. We explore how our criteria relate to others, such as counterfactual fairness, and show that fairness of recourse is complementary to fairness of prediction. We study theoretically and empirically how to enforce fair causal recourse by altering the classifier and perform a case study on the Adult dataset. Finally, we discuss whether fairness violations in the data generating process revealed by our criteria may be better addressed by societal interventions as opposed to constraints on the classifier.

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NeurIPS Conference 2022 Conference Paper

Probable Domain Generalization via Quantile Risk Minimization

Cian Eastwood
Alexander Robey
Shashank Singh
Julius von Kügelgen
Hamed Hassani
George J. Pappas
Bernhard Schölkopf

Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions or domains. To achieve this, DG is commonly formulated as an average- or worst-case problem over the set of possible domains. However, predictors that perform well on average lack robustness while predictors that perform well in the worst case tend to be overly-conservative. To address this, we propose a new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the $\alpha$-quantile of predictor's risk distribution over domains, QRM seeks predictors that perform well with probability $\alpha$. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) a generalization bound for EQRM; and (ii) the conditions under which EQRM recovers the causal predictor as $\alpha \to 1$. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG, and demonstrate that EQRM outperforms state-of-the-art baselines on datasets from WILDS and DomainBed.

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ICLR Conference 2022 Conference Paper

Visual Representation Learning Does Not Generalize Strongly Within the Same Domain

Lukas Schott
Julius von Kügelgen
Frederik Träuble
Peter V. Gehler
Chris Russell 0001
Matthias Bethge
Bernhard Schölkopf
Francesco Locatello

An important component for generalization in machine learning is to uncover underlying latent factors of variation as well as the mechanism through which each factor acts in the world. In this paper, we test whether 17 unsupervised, weakly supervised, and fully supervised representation learning approaches correctly infer the generative factors of variation in simple datasets (dSprites, Shapes3D, MPI3D) from controlled environments, and on our contributed CelebGlow dataset. In contrast to prior robustness work that introduces novel factors of variation during test time, such as blur or other (un)structured noise, we here recompose, interpolate, or extrapolate only existing factors of variation from the training data set (e.g., small and medium-sized objects during training and large objects during testing). Models that learn the correct mechanism should be able to generalize to this benchmark. In total, we train and test 2000+ models and observe that all of them struggle to learn the underlying mechanism regardless of supervision signal and architectural bias. Moreover, the generalization capabilities of all tested models drop significantly as we move from artificial datasets towards more realistic real-world datasets. Despite their inability to identify the correct mechanism, the models are quite modular as their ability to infer other in-distribution factors remains fairly stable, providing only a single factor is out-of-distribution. These results point to an important yet understudied problem of learning mechanistic models of observations that can facilitate generalization.

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ICLR Conference 2022 Conference Paper

You Mostly Walk Alone: Analyzing Feature Attribution in Trajectory Prediction

Osama Makansi
Julius von Kügelgen
Francesco Locatello
Peter V. Gehler
Dominik Janzing
Thomas Brox
Bernhard Schölkopf

Predicting the future trajectory of a moving agent can be easy when the past trajectory continues smoothly but is challenging when complex interactions with other agents are involved. Recent deep learning approaches for trajectory prediction show promising performance and partially attribute this to successful reasoning about agent-agent interactions. However, it remains unclear which features such black-box models actually learn to use for making predictions. This paper proposes a procedure that quantifies the contributions of different cues to model performance based on a variant of Shapley values. Applying this procedure to state-of-the-art trajectory prediction methods on standard benchmark datasets shows that they are, in fact, unable to reason about interactions. Instead, the past trajectory of the target is the only feature used for predicting its future. For a task with richer social interaction patterns, on the other hand, the tested models do pick up such interactions to a certain extent, as quantified by our feature attribution method. We discuss the limits of the proposed method and its links to causality.

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NeurIPS Conference 2021 Conference Paper

Backward-Compatible Prediction Updates: A Probabilistic Approach

Frederik Träuble
Julius von Kügelgen
Matthäus Kleindessner
Francesco Locatello
Bernhard Schölkopf
Peter Gehler

When machine learning systems meet real world applications, accuracy is only one of several requirements. In this paper, we assay a complementary perspective originating from the increasing availability of pre-trained and regularly improving state-of-the-art models. While new improved models develop at a fast pace, downstream tasks vary more slowly or stay constant. Assume that we have a large unlabelled data set for which we want to maintain accurate predictions. Whenever a new and presumably better ML models becomes available, we encounter two problems: (i) given a limited budget, which data points should be re-evaluated using the new model? ; and (ii) if the new predictions differ from the current ones, should we update? Problem (i) is about compute cost, which matters for very large data sets and models. Problem (ii) is about maintaining consistency of the predictions, which can be highly relevant for downstream applications; our demand is to avoid negative flips, i. e. , changing correct to incorrect predictions. In this paper, we formalize the Prediction Update Problem and present an efficient probabilistic approach as answer to the above questions. In extensive experiments on standard classification benchmark data sets, we show that our method outperforms alternative strategies along key metrics for backward-compatible prediction updates.

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NeurIPS Conference 2021 Conference Paper

Independent mechanism analysis, a new concept?

Luigi Gresele
Julius von Kügelgen
Vincent Stimper
Bernhard Schölkopf
Michel Besserve

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.

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NeurIPS Conference 2021 Conference Paper

Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style

Julius von Kügelgen
Yash Sharma
Luigi Gresele
Wieland Brendel
Bernhard Schölkopf
Michel Besserve
Francesco Locatello

Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice.

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NeurIPS Conference 2020 Conference Paper

Algorithmic recourse under imperfect causal knowledge: a probabilistic approach

Amir-Hossein Karimi
Julius von Kügelgen
Bernhard Schölkopf
Isabel Valera

Recent work has discussed the limitations of counterfactual explanations to recommend actions for algorithmic recourse, and argued for the need of taking causal relationships between features into consideration. Unfortunately, in practice, the true underlying structural causal model is generally unknown. In this work, we first show that it is impossible to guarantee recourse without access to the true structural equations. To address this limitation, we propose two probabilistic approaches to select optimal actions that achieve recourse with high probability given limited causal knowledge (e. g. , only the causal graph). The first captures uncertainty over structural equations under additive Gaussian noise, and uses Bayesian model averaging to estimate the counterfactual distribution. The second removes any assumptions on the structural equations by instead computing the average effect of recourse actions on individuals similar to the person who seeks recourse, leading to a novel subpopulation-based interventional notion of recourse. We then derive a gradient-based procedure for selecting optimal recourse actions, and empirically show that the proposed approaches lead to more reliable recommendations under imperfect causal knowledge than non-probabilistic baselines.

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UAI Conference 2020 Conference Paper

Semi-supervised learning, causality, and the conditional cluster assumption

Julius von Kügelgen
Alexander Mey
Marco Loog
Bernhard Schölkopf

While the success of semi-supervised learning (SSL) is still not fully understood, Schölkopf et al. (2012) have established a link to the principle of independent causal mechanisms. They conclude that SSL should be impossible when predicting a target variable from its causes, but possible when predicting it from its effects. Since both these cases are restrictive, we extend their work by considering classification using cause and effect features at the same time, such as predicting a disease from both risk factors and symptoms. While standard SSL exploits information contained in the marginal distribution of all inputs (to improve the estimate of the conditional distribution of the target given in-puts), we argue that in our more general setting we should use information in the conditional distribution of effect features given causal features. We explore how this insight generalises the previous understanding, and how it relates to and can be exploited algorithmically for SSL.

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