Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

We study a linear mixing model with dependent latent components, assuming multiple data domains. Most existing models assume that the components are independent or at least uncorrelated, in line with independent component analysis (ICA). Some recent work allows for dependent components, but then makes specific assumptions such as parametric forms of dependencies, multi-view settings, or interventions. In contrast, we consider a multi-domain setting in which domains differ through domain-specific scalings of the components, while the distribution of the underlying latent components is the same across domains. This approach can model data collected, for example, from different sensors measuring the same process, different laboratories conducting the same experiment, different experimental conditions, or different subjects that might differ in biological or physiological factors. We show that, under sufficient domain variability, latent variables and mixing functions can be identified from second-order statistics alone. We propose the lti-main variance atching (MuDo-CoM) algorithm that generalizes previous methods of joint diagonalization. MuDo-CoM is validated on simulated data and a real-world fMRI dataset.

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Despite decades of research, discovering instantaneous causal relationships from observational brain imaging data, such as spontaneous MEG energies or fMRI, remains a difficult problem. Popular methods, such as Granger Causality and Non-Gaussian Structural Equation Models (SEM), either are unable to handle instantaneous effects or do not work because the data are not non-Gaussian enough. Here, we propose a model with instantaneous causality for temporally dependent variables; these are both very common properties in neuroimaging data. Then, we propose a method to estimate the causal directions based on likelihood ratios, which are related to mutual information between the residual and data variables. We thus construct a simple decision criterion that allows for instantaneous causal discovery in time-dependent data. The proposed method is computationally and conceptually very simple, and we show with simulated data that it performs well even in the case of limited sample sizes, presumably due to the general optimality properties of likelihood. We further apply it to an MEG dataset from the Cam-CAN repository, for which the method gives consistent causal directionalities of energies both intra-subject and inter-subject, as measured by split-half tests. It also gives better performance than Granger causality and non-Gaussian SEM methods in a brain age prediction task. The results also demonstrate that our method might be useful in analyzing causal brain connectomes in functional brain-imaging data.

Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

Molecules are essential building blocks of life and their different conformations (i.e., shapes) crucially determine the functional role that they play in living organisms. Cryogenic Electron Microscopy (cryo-EM) allows for acquisition of large image datasets of individual molecules. Recent advances in computational cryo-EM have made it possible to learn latent variable models of conformation landscapes. However, interpreting these latent spaces remains a challenge as their individual dimensions are often arbitrary. The key message of our work is that this interpretation challenge can be viewed as an Independent Component Analysis (ICA) problem where we seek models that have the property of identifiability. That means, they have an essentially unique solution, representing a conformational latent space that separates the different degrees of freedom a molecule is equipped with in nature. Thus, we aim to advance the computational field of cryo-EM beyond visualizations as we connect it with the theoretical framework of (nonlinear) ICA and discuss the need for identifiable models, improved metrics, and benchmarks. Moving forward, we propose future directions for enhancing the disentanglement of latent spaces in cryo-EM, refining evaluation metrics and exploring techniques that leverage physics-based decoders of biomolecular systems. Moreover, we discuss how future technological developments in time-resolved single particle imaging may enable the application of nonlinear ICA models that can discover the true conformation changes of molecules in nature. The pursuit of interpretable conformational latent spaces will empower researchers to unravel complex biological processes and facilitate targeted interventions. This has significant implications for drug discovery and structural biology more broadly. More generally, latent variable models are deployed widely across many scientific disciplines. Thus, the argument we present in this work has much broader applications in AI for science if we want to move from impressive nonlinear neural network models to mathematically grounded methods that can help us learn something new about nature.

Recently, nonlinear ICA has surfaced as a popular alternative to the many heuristic models used in deep representation learning and disentanglement. An advantage of nonlinear ICA is that a sophisticated identifiability theory has been developed; in particular, it has been proven that the original components can be recovered under sufficiently strong latent dependencies. Despite this general theory, practical nonlinear ICA algorithms have so far been mainly limited to data with one-dimensional latent dependencies, especially time-series data. In this paper, we introduce a new nonlinear ICA framework that employs $t$-process (TP) latent components which apply naturally to data with higher-dimensional dependency structures, such as spatial and spatio-temporal data. In particular, we develop a new learning and inference algorithm that extends variational inference methods to handle the combination of a deep neural network mixing function with the TP prior, and employs the method of inducing points for computational efficacy. On the theoretical side, we show that such TP independent components are identifiable under very general conditions. Further, Gaussian Process (GP) nonlinear ICA is established as a limit of the TP Nonlinear ICA model, and we prove that the identifiability of the latent components at this GP limit is more restricted. Namely, those components are identifiable if and only if they have distinctly different covariance kernels. Our algorithm and identifiability theorems are explored on simulated spatial data and real world spatio-temporal data.

The Langevin algorithm is a classic method for sampling from a given pdf in a real space. In its basic version, it only requires knowledge of the gradient of the log-density, also called the score function. However, in deep learning, it is often easier to learn the so-called "noisy-data score function", i.e. the gradient of the log-density of noisy data, more precisely when Gaussian noise is added to the data. Such an estimate is biased and complicates the use of the Langevin method. Here, we propose a noise-corrected version of the Langevin algorithm, where the bias due to noisy data is removed, at least regarding first-order terms. Unlike diffusion models, our algorithm only needs to know the noisy score function for one single noise level. We further propose a simple special case which has an interesting intuitive interpretation of iteratively adding noise the data and then attempting to remove half of that noise.

A topic of great current interest is Causal Representation Learning (CRL), whose goal is to learn a causal model for hidden features in a data-driven manner. Unfortunately, CRL is severely ill-posed since it is a combination of the two notoriously ill-posed problems of representation learning and causal discovery. Yet, finding practical identifiability conditions that guarantee a unique solution is crucial for its practical applicability. Most approaches so far have been based on assumptions on the latent causal mechanisms, such as temporal causality, or existence of supervision or interventions; these can be too restrictive in actual applications. Here, we show identifiability based on novel, weak constraints, which requires no temporal structure, intervention, nor weak supervision. The approach is based on assuming the observational mixing exhibits a suitable grouping of the observational variables. We also propose a novel self-supervised estimation framework consistent with the model, prove its statistical consistency, and experimentally show its superior CRL performances compared to the state-of-the-art baselines. We further demonstrate its robustness against latent confounders and causal cycles.

Abstract Introduction There has been a growing interest in studying brain activity under naturalistic conditions. However, the relationship between individual differences in ongoing brain activity and psychological characteristics is not well understood. We investigated this connection, focusing on the association between oscillatory activity in the brain and individually characteristic dispositional traits. Given the variability of unconstrained resting states among individuals, we devised a paradigm that could harmonize the state of mind across all participants. Methods We constructed task contrasts that included focused attention (FA), self‐centered future planning, and rumination on anxious thoughts triggered by visual imagery. Magnetoencephalography was recorded from 28 participants under these 3 conditions for a duration of 16 min. The oscillatory power in the alpha and beta bands was converted into spatial contrast maps, representing the difference in brain oscillation power between the two conditions. We performed permutation cluster tests on these spatial contrast maps. Additionally, we applied penalized canonical correlation analysis (CCA) to study the relationship between brain oscillation patterns and behavioral traits. Results The data revealed that the FA condition, as compared to the other conditions, was associated with higher alpha and beta power in the temporal areas of the left hemisphere and lower alpha and beta power in the parietal areas of the right hemisphere. Interestingly, the penalized CCA indicated that behavioral inhibition was positively correlated, whereas anxiety was negatively correlated, with a pattern of high oscillatory power in the bilateral precuneus and low power in the bilateral temporal regions. This unique association was found in the anxious‐thoughts condition when contrasted with the focused‐attention condition. Conclusion Our findings suggest individual temperament traits significantly affect brain engagement in naturalistic conditions. This research underscores the importance of considering individual traits in neuroscience and offers an effective method for analyzing brain activity and psychological differences.

Magnetoencephalography (MEG) is a functional brain imaging modality, which measures the weak magnetic field arising from neuronal activity. The source amplitudes and locations are estimated from the sensor data by solving an ill-posed inverse problem. Commonly used solutions for these problems operate on data from individual subjects. Combining the measurements of multiple subjects has been suggested to increase the spatial resolution of MEG by leveraging the intersubject differences for increased information. In this article, we compare 3 multisubject analysis methods on a retinotopic mapping dataset recorded from 20 subjects. The compared methods are eLORETA with source-space averaging, minimum Wasserstein estimates (MWE), and MWE with source-space averaging. The results were quantified by the geodesic distances between early (60-100 ms) MEG peak activations and fMRI-based retinotopic target points in the primary visual cortex (V1). By increasing the subject count from 1 to 10, the median distances decreased by 6.6-9.4 mm (33-46%) compared with the single-subject median distances of around 20 mm. The observed peak activation locations with multisubject analysis also comply better with the established retinotopic maps of the primary visual cortex. Our results suggest that higher spatial accuracy can be achieved by pooling data from multiple subjects. The strength of MWE lies in individualized and sparse source estimates, but in our data, averaging eLORETA estimates across individuals in source space outperformed MWE in spatial accuracy.

There has been a growing interest in studying brain activity under naturalistic conditions. However, the relationship between individual differences in ongoing brain activity and psychological characteristics is not well understood. We investigated this connection, focusing on the association between oscillatory activity in the brain and individually characteristic dispositional traits. Given the variability of unconstrained resting states among individuals, we devised a paradigm that could harmonize the state of mind across all participants. We constructed task contrasts that included focused attention (FA), self-centered future planning, and rumination on anxious thoughts triggered by visual imagery. Magnetoencephalography was recorded from 28 participants under these 3 conditions for a duration of 16 min. The oscillatory power in the alpha and beta bands was converted into spatial contrast maps, representing the difference in brain oscillation power between the two conditions. We performed permutation cluster tests on these spatial contrast maps. Additionally, we applied penalized canonical correlation analysis (CCA) to study the relationship between brain oscillation patterns and behavioral traits. The data revealed that the FA condition, as compared to the other conditions, was associated with higher alpha and beta power in the temporal areas of the left hemisphere and lower alpha and beta power in the parietal areas of the right hemisphere. Interestingly, the penalized CCA indicated that behavioral inhibition was positively correlated, whereas anxiety was negatively correlated, with a pattern of high oscillatory power in the bilateral precuneus and low power in the bilateral temporal regions. This unique association was found in the anxious-thoughts condition when contrasted with the focused-attention condition. Our findings suggest individual temperament traits significantly affect brain engagement in naturalistic conditions. This research underscores the importance of considering individual traits in neuroscience and offers an effective method for analyzing brain activity and psychological differences.

A central problem in unsupervised deep learning is how to find useful representations of high-dimensional data, sometimes called "disentanglement." Most approaches are heuristic and lack a proper theoretical foundation. In linear representation learning, independent component analysis (ICA) has been successful in many applications areas, and it is principled, i.e., based on a well-defined probabilistic model. However, extension of ICA to the nonlinear case has been problematic because of the lack of identifiability, i.e., uniqueness of the representation. Recently, nonlinear extensions that utilize temporal structure or some auxiliary information have been proposed. Such models are in fact identifiable, and consequently, an increasing number of algorithms have been developed. In particular, some self-supervised algorithms can be shown to estimate nonlinear ICA, even though they have initially been proposed from heuristic perspectives. This paper reviews the state of the art of nonlinear ICA theory and algorithms.

Resting-state magnetoencephalography (MEG) data show complex but structured spatiotemporal patterns. How-ever, the neurophysiological basis of these signal patterns is not fully known and the underlying signal sources are mixed in MEG measurements. Here, we developed a method based on the nonlinear independent component analysis (ICA), a generative model trainable with unsupervised learning, to learn representations from resting -state MEG data. After being trained with a large dataset from the Cam-CAN repository, the model has learned to represent and generate patterns of spontaneous cortical activity using latent nonlinear components, which reflects principal cortical patterns with specific spectral modes. When applied to the downstream classification task of audio-visual MEG, the nonlinear ICA model achieves competitive performance with deep neural networks despite limited access to labels. We further validate the generalizability of the model across different datasets by applying it to an independent neurofeedback dataset for decoding the subject's attentional states, providing a real-time feature extraction and decoding mindfulness and thought-inducing tasks with an accuracy of around 70% at the individual level, which is much higher than obtained by linear ICA or other baseline methods. Our results demonstrate that nonlinear ICA is a valuable addition to existing tools, particularly suited for unsupervised representation learning of spontaneous MEG activity which can then be applied to specific goals or tasks when labelled data are scarce.

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

Causal discovery methods typically extract causal relations between multiple nodes (variables) based on univariate observations of each node. However, one frequently encounters situations where each node is multivariate, i.e. has multiple observational modalities. Furthermore, the observed modalities may be generated through an unknown mixing process, so that some original latent variables are entangled inside the nodes. In such a multimodal case, the existing frameworks cannot be applied. To analyze such data, we propose a new causal representation learning framework called connectivity-contrastive learning (CCL). CCL disentangles the observational mixing and extracts a set of mutually independent latent components, each having a separate causal structure between the nodes. The actual learning proceeds by a novel self-supervised learning method in which the pretext task is to predict the label of a pair of nodes from the observations of the node pairs. We present theorems which show that CCL can indeed identify both the latent components and the multimodal causal structure under weak technical assumptions, up to some indeterminacy. Finally, we experimentally show its superior causal discovery performance compared to state-of-the-art baselines, in particular demonstrating robustness against latent confounders.

An old problem in multivariate statistics is that linear Gaussian models are often unidentifiable, i.e. some parameters cannot be uniquely estimated. In factor (component) analysis, an orthogonal rotation of the factors is unidentifiable, while in linear regression, the direction of effect cannot be identified. For such linear models, non-Gaussianity of the (latent) variables has been shown to provide identifiability. In the case of factor analysis, this leads to independent component analysis, while in the case of the direction of effect, non-Gaussian versions of structural equation modelling solve the problem. More recently, we have shown how even general nonparametric nonlinear versions of such models can be estimated. Non-Gaussianity is not enough in this case, but assuming we have time series, or that the distributions are suitably modulated by some observed auxiliary variables, the models are identifiable. This paper reviews the identifiability theory for the linear and nonlinear cases, considering both factor analytic models and structural equation models.

Self-supervised learning is an increasingly popular approach to unsupervised learning, achieving state-of-the-art results. A prevalent approach consists in contrasting data points and noise points within a classification task: this requires a good noise distribution which is notoriously hard to specify. While a comprehensive theory is missing, it is widely assumed that the optimal noise distribution should in practice be made equal to the data distribution, as in Generative Adversarial Networks (GANs). We here empirically and theoretically challenge this assumption. We turn to Noise-Contrastive Estimation (NCE) which grounds this self-supervised task as an estimation problem of an energy-based model of the data. This ties the optimality of the noise distribution to the sample efficiency of the estimator, which is rigorously defined as its asymptotic variance, or mean-squared error. In the special case where the normalization constant only is unknown, we show that NCE recovers a family of Importance Sampling estimators for which the optimal noise is indeed equal to the data distribution. However, in the general case where the energy is also unknown, we prove that the optimal noise density is the data density multiplied by a correction term based on the Fisher score. In particular, the optimal noise distribution is different from the data distribution, and is even from a different family. Nevertheless, we soberly conclude that the optimal noise may be hard to sample from, and the gain in efficiency can be modest compared to choosing the noise distribution equal to the data's.

Learning a parametric model of a data distribution is a well-known statistical problem that has seen renewed interest as it is brought to scale in deep learning. Framing the problem as a self-supervised task, where data samples are discriminated from noise samples, is at the core of state-of-the-art methods, beginning with Noise-Contrastive Estimation (NCE). Yet, such contrastive learning requires a good noise distribution, which is hard to specify; domain-specific heuristics are therefore widely used. While a comprehensive theory is missing, it is widely assumed that the optimal noise should in practice be made equal to the data, both in distribution and proportion. This setting underlies Generative Adversarial Networks (GANs) in particular. Here, we empirically and theoretically challenge this assumption on the optimal noise. We show that deviating from this assumption can actually lead to better statistical estimators, in terms of asymptotic variance. In particular, the optimal noise distribution is different from the data’s and even from a different family.