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Simon Buchholz

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14 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

Reparameterized LLM Training via Orthogonal Equivalence Transformation

Zeju Qiu
Simon Buchholz
Tim Xiao
Maximilian Dax
Bernhard Schölkopf
Weiyang Liu

While large language models (LLMs) are driving the rapid advancement of artificial intelligence, effectively and reliably training these large models remains one of the field's most significant challenges. To address this challenge, we propose POET, a novel reParameterized training algorithm that uses Orthogonal Equivalence Transformation to optimize neurons. Specifically, POET reparameterizes each neuron with two learnable orthogonal matrices and a fixed random weight matrix. Because of its provable preservation of spectral properties of weight matrices, POET can stably optimize the objective function with improved generalization. We further develop efficient approximations that make POET flexible and scalable for training large-scale neural networks. Extensive experiments validate the effectiveness and scalability of POET in training LLMs.

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NeurIPS Conference 2024 Conference Paper

From Causal to Concept-Based Representation Learning

Goutham Rajendran
Simon Buchholz
Bryon Aragam
Bernhard Schölkopf
Pradeep Ravikumar

To build intelligent machine learning systems, modern representation learning attempts to recover latent generative factors from data, such as in causal representation learning. A key question in this growing field is to provide rigorous conditions under which latent factors can be identified and thus, potentially learned. Motivated by extensive empirical literature on linear representations and concept learning, we propose to relax causal notions with a geometric notion of concepts. We formally define a notion of concepts and show rigorously that they can be provably recovered from diverse data. Instead of imposing assumptions on the "true" generative latent space, we assume that concepts can be represented linearly in this latent space. The tradeoff is that instead of identifying the "true" generative factors, we identify a subset of desired human-interpretable concepts that are relevant for a given application. Experiments on synthetic data, multimodal CLIP models and large language models supplement our results and show the utility of our approach. In this way, we provide a foundation for moving from causal representations to interpretable, concept-based representations by bringing together ideas from these two neighboring disciplines.

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NeurIPS Conference 2024 Conference Paper

Learning Partitions from Context

Simon Buchholz

In this paper, we study the problem of learning the structure of a discrete set of $N$ tokens based on their interactions with other tokens. We focus on a setting where the tokens can be partitioned into a small number of classes, and there exists a real-valued function $f$ defined on certain sets of tokens. This function, which captures the interactions between tokens, depends only on the class memberships of its arguments. The goal is to recover the class memberships of all tokens from a finite number of samples of $f$. We begin by analyzing this problem from both complexity-theoretic and information-theoretic viewpoints. We prove that it is NP-complete in general, and for random instances, we show that on the order of $N\ln(N)$ samples, implying very sparse interactions, suffice to identify the partition. We then investigate the conditions under which gradient flow dynamics of token embeddings can reveal the class structure, finding that this is achievable in certain settings when given on the order of $N^2\ln^2(N)$ samples.

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UAI Conference 2024 Conference Paper

Products, Abstractions and Inclusions of Causal Spaces

Simon Buchholz
Junhyung Park
Bernhard Schölkopf

Causal spaces have recently been introduced as a measure-theoretic framework to encode the notion of causality. While it has some advantages over established frameworks, such as structural causal models, the theory is so far only developed for single causal spaces. In many mathematical theories, not least the theory of probability spaces of which causal spaces are a direct extension, combinations of objects and maps between objects form a central part. In this paper, taking inspiration from such objects in probability theory, we propose the definitions of products of causal spaces, as well as (stochastic) transformations between causal spaces. In the context of causality, these quantities can be given direct semantic interpretations as causally independent components, abstractions and extensions.

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ICML Conference 2024 Conference Paper

Robustness of Nonlinear Representation Learning

Simon Buchholz
Bernhard Schölkopf

We study the problem of unsupervised representation learning in slightly misspecified settings, and thus formalize the study of robustness of nonlinear representation learning. We focus on the case where the mixing is close to a local isometry in a suitable distance and show based on existing rigidity results that the mixing can be identified up to linear transformations and small errors. In a second step, we investigate Independent Component Analysis (ICA) with observations generated according to $x=f(s)=As+h(s)$ where $A$ is an invertible mixing matrix and $h$ a small perturbation. We show that we can approximately recover the matrix $A$ and the independent components. Together, these two results show approximate identifiability of nonlinear ICA with almost isometric mixing functions. Those results are a step towards identifiability results for unsupervised representation learning for real-world data that do not follow restrictive model classes.

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NeurIPS Conference 2023 Conference Paper

A Measure-Theoretic Axiomatisation of Causality

Junhyung Park
Simon Buchholz
Bernhard Schölkopf
Krikamol Muandet

Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of what happens when one intervenes on a system, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a causal space, consisting of a probability space along with a collection of transition probability kernels, called causal kernels, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.

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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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NeurIPS Conference 2023 Conference Paper

Flow Matching for Scalable Simulation-Based Inference

Jonas Wildberger
Maximilian Dax
Simon Buchholz
Stephen Green
Jakob H Macke
Bernhard Schölkopf

Neural posterior estimation methods based on discrete normalizing flows have become established tools for simulation-based inference (SBI), but scaling them to high-dimensional problems can be challenging. Building on recent advances in generative modeling, we here present flow matching posterior estimation (FMPE), a technique for SBI using continuous normalizing flows. Like diffusion models, and in contrast to discrete flows, flow matching allows for unconstrained architectures, providing enhanced flexibility for complex data modalities. Flow matching, therefore, enables exact density evaluation, fast training, and seamless scalability to large architectures---making it ideal for SBI. We show that FMPE achieves competitive performance on an established SBI benchmark, and then demonstrate its improved scalability on a challenging scientific problem: for gravitational-wave inference, FMPE outperforms methods based on comparable discrete flows, reducing training time by 30\% with substantially improved accuracy. Our work underscores the potential of FMPE to enhance performance in challenging inference scenarios, thereby paving the way for more advanced applications to scientific problems.

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NeurIPS Conference 2023 Conference Paper

Learning Linear Causal Representations from Interventions under General Nonlinear Mixing

Simon Buchholz
Goutham Rajendran
Elan Rosenfeld
Bryon Aragam
Bernhard Schölkopf
Pradeep Ravikumar

We study the problem of learning causal representations from unknown, latent interventions in a general setting, where the latent distribution is Gaussian but the mixing function is completely general. We prove strong identifiability results given unknown single-node interventions, i. e. , without having access to the intervention targets. This generalizes prior works which have focused on weaker classes, such as linear maps or paired counterfactual data. This is also the first instance of identifiability from non-paired interventions for deep neural network embeddings and general causal structures. Our proof relies on carefully uncovering the high-dimensional geometric structure present in the data distribution after a non-linear density transformation, which we capture by analyzing quadratic forms of precision matrices of the latent distributions. Finally, we propose a contrastive algorithm to identify the latent variables in practice and evaluate its performance on various tasks.

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TMLR Journal 2023 Journal Article

Some Remarks on Identifiability of Independent Component Analysis in Restricted Function Classes

Simon Buchholz

In this short note, we comment on recent results on identifiability of independent component analysis. We point out an error in earlier works and clarify that this error cannot be fixed as the chosen approach is not sufficiently powerful to prove identifiability results. In addition, we explain the necessary ingredients to prove stronger identifiability results. Finally, we discuss and extend the flow-based technique to construct spurious solutions for independent component analysis problems and provide a counterexample to an earlier identifiability result.

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NeurIPS Conference 2022 Conference Paper

AutoML Two-Sample Test

Jonas M. Kübler
Vincent Stimper
Simon Buchholz
Krikamol Muandet
Bernhard Schölkopf

Two-sample tests are important in statistics and machine learning, both as tools for scientific discovery as well as to detect distribution shifts. This led to the development of many sophisticated test procedures going beyond the standard supervised learning frameworks, whose usage can require specialized knowledge about two-sample testing. We use a simple test that takes the mean discrepancy of a witness function as the test statistic and prove that minimizing a squared loss leads to a witness with optimal testing power. This allows us to leverage recent advancements in AutoML. Without any user input about the problems at hand, and using the same method for all our experiments, our AutoML two-sample test achieves competitive performance on a diverse distribution shift benchmark as well as on challenging two-sample testing problems.

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NeurIPS Conference 2022 Conference Paper

Function Classes for Identifiable Nonlinear Independent Component Analysis

Simon Buchholz
Michel Besserve
Bernhard Schölkopf

Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning. When such model reflects the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that such models allow generalisation in downstream tasks. It is however well known that such identifiability guaranties are typically not achievable without putting constraints on the model class. This is notably the case for nonlinear Independent Component Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function. Several families of spurious solutions fitting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings. However, recent work suggests that constraining the function class of such models may promote identifiability. Specifically, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns. In the present work, we prove that a subclass of these transformations, conformal maps, is identifiable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identifiability in a generic setting.

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NeurIPS Conference 2021 Conference Paper

The Inductive Bias of Quantum Kernels

Jonas Kübler
Simon Buchholz
Bernhard Schölkopf

It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.

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