Aapo Hyvärinen

Aapo Hyvärinen, Ilyes Khemakhem, Ricardo Monti

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.

Aapo Hyvarinen, Ilyes Khemakhem, Hiroshi Morioka

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 due to 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.

Hiroshi Morioka, Aapo Hyvärinen

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.

Omar Chehab, Alexandre Gramfort, Aapo Hyvarinen

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.

Aapo Hyvärinen

This book uses the modern theory of artificial intelligence (AI) to understand human suffering or mental pain. Both humans and sophisticated AI agents process information about the world in order to achieve goals and obtain rewards, which is why AI can be used as a model of the human brain and mind. This book intends to make the theory accessible to a relatively general audience, requiring only some relevant scientific background. The book starts with the assumption that suffering is mainly caused by frustration. Frustration means the failure of an agent (whether AI or human) to achieve a goal or a reward it wanted or expected. Frustration is inevitable because of the overwhelming complexity of the world, limited computational resources, and scarcity of good data. In particular, such limitations imply that an agent acting in the real world must cope with uncontrollability, unpredictability, and uncertainty, which all lead to frustration. Fundamental in such modelling is the idea of learning, or adaptation to the environment. While AI uses machine learning, humans and animals adapt by a combination of evolutionary mechanisms and ordinary learning. Even frustration is fundamentally an error signal that the system uses for learning. This book explores various aspects and limitations of learning algorithms and their implications regarding suffering. At the end of the book, the computational theory is used to derive various interventions or training methods that will reduce suffering in humans. The amount of frustration is expressed by a simple equation which indicates how it can be reduced. The ensuing interventions are very similar to those proposed by Buddhist and Stoic philosophy, and include mindfulness meditation. Therefore, this book can be interpreted as an exposition of a computational theory justifying why such philosophies and meditation reduce human suffering.

Hermanni Hälvä, Jonathan So, Richard E. Turner et al.

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.

Omar Chehab, Alexandre Gramfort, Aapo Hyvarinen

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.

Omar Chehab, Aapo Hyvarinen, Andrej Risteski

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.

Hugo Richard, Luigi Gresele, Aapo Hyvärinen et al.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-session variability than other methods.On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset it reveals a clear sequence of evoked activity in sensor and source space. The code is freely available at https://github.com/hugorichard/multiviewica.

Hanlin Yu, Arto Klami, Aapo Hyvärinen et al.

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.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.

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.

Antti Hyttinen, Vitória Barin-Pacela, Aapo Hyvärinen

We consider independent component analysis of binary data. While fundamental in practice, this case has been much less developed than ICA for continuous data. We start by assuming a linear mixing model in a continuous-valued latent space, followed by a binary observation model. Importantly, we assume that the sources are non-stationary; this is necessary since any non-Gaussianity would essentially be destroyed by the binarization. Interestingly, the model allows for closed-form likelihood by employing the cumulative distribution function of the multivariate Gaussian distribution. In stark contrast to the continuous-valued case, we prove non-identifiability of the model with few observed variables; our empirical results imply identifiability when the number of observed variables is higher. We present a practical method for binary ICA that uses only pairwise marginals, which are faster to compute than the full multivariate likelihood. Experiments give insight into the requirements for the number of observed variables, segments, and latent sources that allow the model to be estimated.

Hugo Richard, Pierre Ablin, Bertrand Thirion et al.

We consider shared response modeling, a multi-view learning problem where one wants to identify common components from multiple datasets or views. We introduce Shared Independent Component Analysis (ShICA) that models each view as a linear transform of shared independent components contaminated by additive Gaussian noise. We show that this model is identifiable if the components are either non-Gaussian or have enough diversity in noise variances. We then show that in some cases multi-set canonical correlation analysis can recover the correct unmixing matrices, but that even a small amount of sampling noise makes Multiset CCA fail. To solve this problem, we propose to use joint diagonalization after Multiset CCA, leading to a new approach called ShICA-J. We show via simulations that ShICA-J leads to improved results while being very fast to fit. While ShICA-J is based on second-order statistics, we further propose to leverage non-Gaussianity of the components using a maximum-likelihood method, ShICA-ML, that is both more accurate and more costly. Further, ShICA comes with a principled method for shared components estimation. Finally, we provide empirical evidence on fMRI and MEG datasets that ShICA yields more accurate estimation of the components than alternatives.

Hermanni Hälvä, Sylvain Le Corff, Luc Lehéricy et al.

We introduce a new general identifiable framework for principled disentanglement referred to as Structured Nonlinear Independent Component Analysis (SNICA). Our contribution is to extend the identifiability theory of deep generative models for a very broad class of structured models. While previous works have shown identifiability for specific classes of time-series models, our theorems extend this to more general temporal structures as well as to models with more complex structures such as spatial dependencies. In particular, we establish the major result that identifiability for this framework holds even in the presence of noise of unknown distribution. Finally, as an example of our framework's flexibility, we introduce the first nonlinear ICA model for time-series that combines the following very useful properties: it accounts for both nonstationarity and autocorrelation in a fully unsupervised setting; performs dimensionality reduction; models hidden states; and enables principled estimation and inference by variational maximum-likelihood.

Hugo Richard, Pierre Ablin, Aapo Hyvärinen et al.

We consider a multi-view learning problem known as group independent component analysis (group ICA), where the goal is to recover shared independent sources from many views. The statistical modeling of this problem requires to take noise into account. When the model includes additive noise on the observations, the likelihood is intractable. By contrast, we propose Adaptive multiView ICA (AVICA), a noisy ICA model where each view is a linear mixture of shared independent sources with additive noise on the sources. In this setting, the likelihood has a tractable expression, which enables either direct optimization of the log-likelihood using a quasi-Newton method, or generalized EM. Importantly, we consider that the noise levels are also parameters that are learned from the data. This enables sources estimation with a closed-form Minimum Mean Squared Error (MMSE) estimator which weights each view according to its relative noise level. On synthetic data, AVICA yields better sources estimates than other group ICA methods thanks to its explicit MMSE estimator. On real magnetoencephalograpy (MEG) data, we provide evidence that the decomposition is less sensitive to sampling noise and that the noise variance estimates are biologically plausible. Lastly, on functional magnetic resonance imaging (fMRI) data, AVICA exhibits best performance in transferring information across views.

Ilyes Khemakhem, Ricardo Pio Monti, Robert Leech et al.

Two apparently unrelated fields -- normalizing flows and causality -- have recently received considerable attention in the machine learning community. In this work, we highlight an intrinsic correspondence between a simple family of autoregressive normalizing flows and identifiable causal models. We exploit the fact that autoregressive flow architectures define an ordering over variables, analogous to a causal ordering, to show that they are well-suited to performing a range of causal inference tasks, ranging from causal discovery to making interventional and counterfactual predictions. First, we show that causal models derived from both affine and additive autoregressive flows with fixed orderings over variables are identifiable, i.e. the true direction of causal influence can be recovered. This provides a generalization of the additive noise model well-known in causal discovery. Second, we derive a bivariate measure of causal direction based on likelihood ratios, leveraging the fact that flow models can estimate normalized log-densities of data. Third, we demonstrate that flows naturally allow for direct evaluation of both interventional and counterfactual queries, the latter case being possible due to the invertible nature of flows. Finally, throughout a series of experiments on synthetic and real data, the proposed method is shown to outperform current approaches for causal discovery as well as making accurate interventional and counterfactual predictions.

Hubert Banville, Omar Chehab, Aapo Hyvärinen et al.

Objective. Supervised learning paradigms are often limited by the amount of labeled data that is available. This phenomenon is particularly problematic in clinically-relevant data, such as electroencephalography (EEG), where labeling can be costly in terms of specialized expertise and human processing time. Consequently, deep learning architectures designed to learn on EEG data have yielded relatively shallow models and performances at best similar to those of traditional feature-based approaches. However, in most situations, unlabeled data is available in abundance. By extracting information from this unlabeled data, it might be possible to reach competitive performance with deep neural networks despite limited access to labels. Approach. We investigated self-supervised learning (SSL), a promising technique for discovering structure in unlabeled data, to learn representations of EEG signals. Specifically, we explored two tasks based on temporal context prediction as well as contrastive predictive coding on two clinically-relevant problems: EEG-based sleep staging and pathology detection. We conducted experiments on two large public datasets with thousands of recordings and performed baseline comparisons with purely supervised and hand-engineered approaches. Main results. Linear classifiers trained on SSL-learned features consistently outperformed purely supervised deep neural networks in low-labeled data regimes while reaching competitive performance when all labels were available. Additionally, the embeddings learned with each method revealed clear latent structures related to physiological and clinical phenomena, such as age effects. Significance. We demonstrate the benefit of self-supervised learning approaches on EEG data. Our results suggest that SSL may pave the way to a wider use of deep learning models on EEG data.

Ricardo Pio Monti, Ilyes Khemakhem, Aapo Hyvarinen

We posit that autoregressive flow models are well-suited to performing a range of causal inference tasks - ranging from causal discovery to making interventional and counterfactual predictions. In particular, we exploit the fact that autoregressive architectures define an ordering over variables, analogous to a causal ordering, in order to propose a single flow architecture to perform all three aforementioned tasks. We first leverage the fact that flow models estimate normalized log-densities of data to derive a bivariate measure of causal direction based on likelihood ratios. Whilst traditional measures of causal direction often require restrictive assumptions on the nature of causal relationships (e.g., linearity),the flexibility of flow models allows for arbitrary causal dependencies. Our approach compares favourably against alternative methods on synthetic data as well as on the Cause-Effect Pairs bench-mark dataset. Subsequently, we demonstrate that the invertible nature of flows naturally allows for direct evaluation of both interventional and counterfactual predictions, which require marginalization and conditioning over latent variables respectively. We present examples over synthetic data where autoregressive flows, when trained under the correct causal ordering, are able to make accurate interventional and counterfactual predictions

Luigi Gresele, Giancarlo Fissore, Adrián Javaloy et al.

Learning expressive probabilistic models correctly describing the data is a ubiquitous problem in machine learning. A popular approach for solving it is mapping the observations into a representation space with a simple joint distribution, which can typically be written as a product of its marginals -- thus drawing a connection with the field of nonlinear independent component analysis. Deep density models have been widely used for this task, but their maximum likelihood based training requires estimating the log-determinant of the Jacobian and is computationally expensive, thus imposing a trade-off between computation and expressive power. In this work, we propose a new approach for exact training of such neural networks. Based on relative gradients, we exploit the matrix structure of neural network parameters to compute updates efficiently even in high-dimensional spaces; the computational cost of the training is quadratic in the input size, in contrast with the cubic scaling of naive approaches. This allows fast training with objective functions involving the log-determinant of the Jacobian, without imposing constraints on its structure, in stark contrast to autoregressive normalizing flows.

Hermanni Hälvä, Aapo Hyvärinen

Recent advances in nonlinear Independent Component Analysis (ICA) provide a principled framework for unsupervised feature learning and disentanglement. The central idea in such works is that the latent components are assumed to be independent conditional on some observed auxiliary variables, such as the time-segment index. This requires manual segmentation of data into non-stationary segments which is computationally expensive, inaccurate and often impossible. These models are thus not fully unsupervised. We remedy these limitations by combining nonlinear ICA with a Hidden Markov Model, resulting in a model where a latent state acts in place of the observed segment-index. We prove identifiability of the proposed model for a general mixing nonlinearity, such as a neural network. We also show how maximum likelihood estimation of the model can be done using the expectation-maximization algorithm. Thus, we achieve a new nonlinear ICA framework which is unsupervised, more efficient, as well as able to model underlying temporal dynamics.

Hiroshi Morioka, Hermanni Hälvä, Aapo Hyvärinen

The nonlinear vector autoregressive (NVAR) model provides an appealing framework to analyze multivariate time series obtained from a nonlinear dynamical system. However, the innovation (or error), which plays a key role by driving the dynamics, is almost always assumed to be additive. Additivity greatly limits the generality of the model, hindering analysis of general NVAR processes which have nonlinear interactions between the innovations. Here, we propose a new general framework called independent innovation analysis (IIA), which estimates the innovations from completely general NVAR. We assume mutual independence of the innovations as well as their modulation by an auxiliary variable (which is often taken as the time index and simply interpreted as nonstationarity). We show that IIA guarantees the identifiability of the innovations with arbitrary nonlinearities, up to a permutation and component-wise invertible nonlinearities. We also propose three estimation frameworks depending on the type of the auxiliary variable. We thus provide the first rigorous identifiability result for general NVAR, as well as very general tools for learning such models.

Ilyes Khemakhem, Ricardo Pio Monti, Diederik P. Kingma et al.

We consider the identifiability theory of probabilistic models and establish sufficient conditions under which the representations learned by a very broad family of conditional energy-based models are unique in function space, up to a simple transformation. In our model family, the energy function is the dot-product between two feature extractors, one for the dependent variable, and one for the conditioning variable. We show that under mild conditions, the features are unique up to scaling and permutation. Our results extend recent developments in nonlinear ICA, and in fact, they lead to an important generalization of ICA models. In particular, we show that our model can be used for the estimation of the components in the framework of Independently Modulated Component Analysis (IMCA), a new generalization of nonlinear ICA that relaxes the independence assumption. A thorough empirical study shows that representations learned by our model from real-world image datasets are identifiable, and improve performance in transfer learning and semi-supervised learning tasks.

Hubert Banville, Isabela Albuquerque, Aapo Hyvärinen et al.

The supervised learning paradigm is limited by the cost - and sometimes the impracticality - of data collection and labeling in multiple domains. Self-supervised learning, a paradigm which exploits the structure of unlabeled data to create learning problems that can be solved with standard supervised approaches, has shown great promise as a pretraining or feature learning approach in fields like computer vision and time series processing. In this work, we present self-supervision strategies that can be used to learn informative representations from multivariate time series. One successful approach relies on predicting whether time windows are sampled from the same temporal context or not. As demonstrated on a clinically relevant task (sleep scoring) and with two electroencephalography datasets, our approach outperforms a purely supervised approach in low data regimes, while capturing important physiological information without any access to labels.

Hiroaki Sasaki, Takashi Takenouchi, Ricardo Monti et al.

Nonlinear independent component analysis (ICA) is a general framework for unsupervised representation learning, and aimed at recovering the latent variables in data. Recent practical methods perform nonlinear ICA by solving a series of classification problems based on logistic regression. However, it is well-known that logistic regression is vulnerable to outliers, and thus the performance can be strongly weakened by outliers. In this paper, we first theoretically analyze nonlinear ICA models in the presence of outliers. Our analysis implies that estimation in nonlinear ICA can be seriously hampered when outliers exist on the tails of the (noncontaminated) target density, which happens in a typical case of contamination by outliers. We develop two robust nonlinear ICA methods based on the γ-divergence, which is a robust alternative to the KL-divergence in logistic regression. The proposed methods are shown to have desired robustness properties in the context of nonlinear ICA. We also experimentally demonstrate that the proposed methods are very robust and outperform existing methods in the presence of outliers. Finally, the proposed method is applied to ICA-based causal discovery and shown to find a plausible causal relationship on fMRI data.

Wenkai Xu, Gang Niu, Aapo Hyvärinen et al.

Summarizing large-scaled directed graphs into small-scale representations is a useful but less studied problem setting. Conventional clustering approaches, which based on "Min-Cut"-style criteria, compress both the vertices and edges of the graph into the communities, that lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features which is useful for efficient interventions of population behavior. In this paper, we present a model, based on minimizing reconstruction error with non-negative constraints, which relates to a "Max-Cut" criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and on the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.

Ilyes Khemakhem, Diederik P. Kingma, Ricardo Pio Monti et al.

The framework of variational autoencoders allows us to efficiently learn deep latent-variable models, such that the model's marginal distribution over observed variables fits the data. Often, we're interested in going a step further, and want to approximate the true joint distribution over observed and latent variables, including the true prior and posterior distributions over latent variables. This is known to be generally impossible due to unidentifiability of the model. We address this issue by showing that for a broad family of deep latent-variable models, identification of the true joint distribution over observed and latent variables is actually possible up to very simple transformations, thus achieving a principled and powerful form of disentanglement. Our result requires a factorized prior distribution over the latent variables that is conditioned on an additionally observed variable, such as a class label or almost any other observation. We build on recent developments in nonlinear ICA, which we extend to the case with noisy, undercomplete or discrete observations, integrated in a maximum likelihood framework. The result also trivially contains identifiable flow-based generative models as a special case.

Takeru Matsuda, Masatoshi Uehara, Aapo Hyvarinen

Many statistical models are given in the form of non-normalized densities with an intractable normalization constant. Since maximum likelihood estimation is computationally intensive for these models, several estimation methods have been developed which do not require explicit computation of the normalization constant, such as noise contrastive estimation (NCE) and score matching. However, model selection methods for general non-normalized models have not been proposed so far. In this study, we develop information criteria for non-normalized models estimated by NCE or score matching. They are approximately unbiased estimators of discrepancy measures for non-normalized models. Simulation results and applications to real data demonstrate that the proposed criteria enable selection of the appropriate non-normalized model in a data-driven manner.

Ricardo Pio Monti, Kun Zhang, Aapo Hyvarinen

We consider the problem of inferring causal relationships between two or more passively observed variables. While the problem of such causal discovery has been extensively studied especially in the bivariate setting, the majority of current methods assume a linear causal relationship, and the few methods which consider non-linear dependencies usually make the assumption of additive noise. Here, we propose a framework through which we can perform causal discovery in the presence of general non-linear relationships. The proposed method is based on recent progress in non-linear independent component analysis and exploits the non-stationarity of observations in order to recover the underlying sources or latent disturbances. We show rigorously that in the case of bivariate causal discovery, such non-linear ICA can be used to infer the causal direction via a series of independence tests. We further propose an alternative measure of causal direction based on asymptotic approximations to the likelihood ratio, as well as an extension to multivariate causal discovery. We demonstrate the capabilities of the proposed method via a series of simulation studies and conclude with an application to neuroimaging data.

Saeed Saremi, Aapo Hyvarinen

We unify $\textit{kernel density estimation}$ and $\textit{empirical Bayes}$ and address a set of problems in unsupervised learning with a geometric interpretation of those methods, rooted in the $\textit{concentration of measure}$ phenomenon. Kernel density is viewed symbolically as $X\rightharpoonup Y$ where the random variable $X$ is smoothed to $Y= X+N(0,σ^2 I_d)$, and empirical Bayes is the machinery to denoise in a least-squares sense, which we express as $X \leftharpoondown Y$. A learning objective is derived by combining these two, symbolically captured by $X \rightleftharpoons Y$. Crucially, instead of using the original nonparametric estimators, we parametrize $\textit{the energy function}$ with a neural network denoted by $φ$; at optimality, $\nabla φ\approx -\nabla \log f$ where $f$ is the density of $Y$. The optimization problem is abstracted as interactions of high-dimensional spheres which emerge due to the concentration of isotropic gaussians. We introduce two algorithmic frameworks based on this machinery: (i) a "walk-jump" sampling scheme that combines Langevin MCMC (walks) and empirical Bayes (jumps), and (ii) a probabilistic framework for $\textit{associative memory}$, called NEBULA, defined à la Hopfield by the $\textit{gradient flow}$ of the learned energy to a set of attractors. We finish the paper by reporting the emergence of very rich "creative memories" as attractors of NEBULA for highly-overlapping spheres.

Hiroaki Sasaki, Aapo Hyvärinen

Conditional density estimation is a general framework for solving various problems in machine learning. Among existing methods, non-parametric and/or kernel-based methods are often difficult to use on large datasets, while methods based on neural networks usually make restrictive parametric assumptions on the probability densities. Here, we propose a novel method for estimating the conditional density based on score matching. In contrast to existing methods, we employ scalable neural networks, but do not make explicit parametric assumptions on densities. The key challenge in applying score matching to neural networks is computation of the first- and second-order derivatives of a model for the log-density. We tackle this challenge by developing a new neural-kernelized approach, which can be applied on large datasets with stochastic gradient descent, while the reproducing kernels allow for easy computation of the derivatives needed in score matching. We show that the neural-kernelized function approximator has universal approximation capability and that our method is consistent in conditional density estimation. We numerically demonstrate that our method is useful in high-dimensional conditional density estimation, and compares favourably with existing methods. Finally, we prove that the proposed method has interesting connections to two probabilistically principled frameworks of representation learning: Nonlinear sufficient dimension reduction and nonlinear independent component analysis.

Ricardo Pio Monti, Aapo Hyvärinen

Connectivity estimation is challenging in the context of high-dimensional data. A useful preprocessing step is to group variables into clusters, however, it is not always clear how to do so from the perspective of connectivity estimation. Another practical challenge is that we may have data from multiple related classes (e.g., multiple subjects or conditions) and wish to incorporate constraints on the similarities across classes. We propose a probabilistic model which simultaneously performs both a grouping of variables (i.e., detecting community structure) and estimation of connectivities between the groups which correspond to latent variables. The model is essentially a factor analysis model where the factors are allowed to have arbitrary correlations, while the factor loading matrix is constrained to express a community structure. The model can be applied on multiple classes so that the connectivities can be different between the classes, while the community structure is the same for all classes. We propose an efficient estimation algorithm based on score matching, and prove the identifiability of the model. Finally, we present an extension to directed (causal) connectivities over latent variables. Simulations and experiments on fMRI data validate the practical utility of the method.

Aapo Hyvarinen, Hiroaki Sasaki, Richard E. Turner

Nonlinear ICA is a fundamental problem for unsupervised representation learning, emphasizing the capacity to recover the underlying latent variables generating the data (i.e., identifiability). Recently, the very first identifiability proofs for nonlinear ICA have been proposed, leveraging the temporal structure of the independent components. Here, we propose a general framework for nonlinear ICA, which, as a special case, can make use of temporal structure. It is based on augmenting the data by an auxiliary variable, such as the time index, the history of the time series, or any other available information. We propose to learn nonlinear ICA by discriminating between true augmented data, or data in which the auxiliary variable has been randomized. This enables the framework to be implemented algorithmically through logistic regression, possibly in a neural network. We provide a comprehensive proof of the identifiability of the model as well as the consistency of our estimation method. The approach not only provides a general theoretical framework combining and generalizing previously proposed nonlinear ICA models and algorithms, but also brings practical advantages.

Saeed Saremi, Arash Mehrjou, Bernhard Schölkopf et al.

Density estimation is a fundamental problem in statistical learning. This problem is especially challenging for complex high-dimensional data due to the curse of dimensionality. A promising solution to this problem is given here in an inference-free hierarchical framework that is built on score matching. We revisit the Bayesian interpretation of the score function and the Parzen score matching, and construct a multilayer perceptron with a scalable objective for learning the energy (i.e. the unnormalized log-density), which is then optimized with stochastic gradient descent. In addition, the resulting deep energy estimator network (DEEN) is designed as products of experts. We present the utility of DEEN in learning the energy, the score function, and in single-step denoising experiments for synthetic and high-dimensional data. We also diagnose stability problems in the direct estimation of the score function that had been observed for denoising autoencoders.

Takeru Matsuda, Aapo Hyvarinen

We develop a general method for estimating a finite mixture of non-normalized models. Here, a non-normalized model is defined to be a parametric distribution with an intractable normalization constant. Existing methods for estimating non-normalized models without computing the normalization constant are not applicable to mixture models because they contain more than one intractable normalization constant. The proposed method is derived by extending noise contrastive estimation (NCE), which estimates non-normalized models by discriminating between the observed data and some artificially generated noise. We also propose an extension of NCE with multiple noise distributions. Then, based on the observation that conventional classification learning with neural networks is implicitly assuming an exponential family as a generative model, we introduce a method for clustering unlabeled data by estimating a finite mixture of distributions in an exponential family. Estimation of this mixture model is attained by the proposed extensions of NCE where the training data of neural networks are used as noise. Thus, the proposed method provides a probabilistically principled clustering method that is able to utilize a deep representation. Application to image clustering using a deep neural network gives promising results.

Hiroaki Sasaki, Takafumi Kanamori, Aapo Hyvärinen et al.

Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical challenge both in mode-seeking clustering and density ridge estimation is accurate estimation of the ratios of the first- and second-order density derivatives to the density. A naive approach takes a three-step approach of first estimating the data density, then computing its derivatives, and finally taking their ratios. However, this three-step approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator, and division by the estimated density could significantly magnify the estimation error. To cope with these problems, we propose a novel estimator for the \emph{density-derivative-ratios}. The proposed estimator does not involve density estimation, but rather \emph{directly} approximates the ratios of density derivatives of any order. Moreover, we establish a convergence rate of the proposed estimator. Based on the proposed estimator, novel methods both for mode-seeking clustering and density ridge estimation are developed, and the respective convergence rates to the mode and ridge of the underlying density are also established. Finally, we experimentally demonstrate that the developed methods significantly outperform existing methods, particularly for relatively high-dimensional data.

Aapo Hyvarinen, Hiroshi Morioka

Nonlinear independent component analysis (ICA) provides an appealing framework for unsupervised feature learning, but the models proposed so far are not identifiable. Here, we first propose a new intuitive principle of unsupervised deep learning from time series which uses the nonstationary structure of the data. Our learning principle, time-contrastive learning (TCL), finds a representation which allows optimal discrimination of time segments (windows). Surprisingly, we show how TCL can be related to a nonlinear ICA model, when ICA is redefined to include temporal nonstationarities. In particular, we show that TCL combined with linear ICA estimates the nonlinear ICA model up to point-wise transformations of the sources, and this solution is unique --- thus providing the first identifiability result for nonlinear ICA which is rigorous, constructive, as well as very general.

Hiroaki Sasaki, Michael U. Gutmann, Hayaru Shouno et al.

The statistical dependencies which independent component analysis (ICA) cannot remove often provide rich information beyond the linear independent components. It would thus be very useful to estimate the dependency structure from data. While such models have been proposed, they usually concentrated on higher-order correlations such as energy (square) correlations. Yet, linear correlations are a most fundamental and informative form of dependency in many real data sets. Linear correlations are usually completely removed by ICA and related methods, so they can only be analyzed by developing new methods which explicitly allow for linearly correlated components. In this paper, we propose a probabilistic model of linear non-Gaussian components which are allowed to have both linear and energy correlations. The precision matrix of the linear components is assumed to be randomly generated by a higher-order process and explicitly parametrized by a parameter matrix. The estimation of the parameter matrix is shown to be particularly simple because using score matching, the objective function is a quadratic form. Using simulations with artificial data, we demonstrate that the proposed method improves identifiability of non-Gaussian components by simultaneously learning their correlation structure. Applications on simulated complex cells with natural image input, as well as spectrograms of natural audio data show that the method finds new kinds of dependencies between the components.

Shohei Shimizu, Aapo Hyvarinen, Yoshinobu Kawahara

Structural equation models and Bayesian networks have been widely used to analyze causal relations between continuous variables. In such frameworks, linear acyclic models are typically used to model the datagenerating process of variables. Recently, it was shown that use of non-Gaussianity identifies a causal ordering of variables in a linear acyclic model without using any prior knowledge on the network structure, which is not the case with conventional methods. However, existing estimation methods are based on iterative search algorithms and may not converge to a correct solution in a finite number of steps. In this paper, we propose a new direct method to estimate a causal ordering based on non-Gaussianity. In contrast to the previous methods, our algorithm requires no algorithmic parameters and is guaranteed to converge to the right solution within a small fixed number of steps if the data strictly follows the model.

Hiroaki Sasaki, Aapo Hyvärinen, Masashi Sugiyama

Mean shift clustering finds the modes of the data probability density by identifying the zero points of the density gradient. Since it does not require to fix the number of clusters in advance, the mean shift has been a popular clustering algorithm in various application fields. A typical implementation of the mean shift is to first estimate the density by kernel density estimation and then compute its gradient. However, since good density estimation does not necessarily imply accurate estimation of the density gradient, such an indirect two-step approach is not reliable. In this paper, we propose a method to directly estimate the gradient of the log-density without going through density estimation. The proposed method gives the global solution analytically and thus is computationally efficient. We then develop a mean-shift-like fixed-point algorithm to find the modes of the density for clustering. As in the mean shift, one does not need to set the number of clusters in advance. We empirically show that the proposed clustering method works much better than the mean shift especially for high-dimensional data. Experimental results further indicate that the proposed method outperforms existing clustering methods.

Bharath Sriperumbudur, Kenji Fukumizu, Arthur Gretton et al.

In this paper, we consider an infinite dimensional exponential family, $\mathcal{P}$ of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space, $H$ and show it to be quite rich in the sense that a broad class of densities on $\mathbb{R}^d$ can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in $\mathcal{P}$. The main goal of the paper is to estimate an unknown density, $p_0$ through an element in $\mathcal{P}$. Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between $p_0$ and $\mathcal{P}$, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. Instead, we propose an estimator, $\hat{p}_n$ based on minimizing the \emph{Fisher divergence}, $J(p_0\Vert p)$ between $p_0$ and $p\in \mathcal{P}$, which involves solving a simple finite-dimensional linear system. When $p_0\in\mathcal{P}$, we show that the proposed estimator is consistent, and provide a convergence rate of $n^{-\min\left\{\frac{2}{3},\frac{2β+1}{2β+2}\right\}}$ in Fisher divergence under the smoothness assumption that $\log p_0\in\mathcal{R}(C^β)$ for some $β\ge 0$, where $C$ is a certain Hilbert-Schmidt operator on $H$ and $\mathcal{R}(C^β)$ denotes the image of $C^β$. We also investigate the misspecified case of $p_0\notin\mathcal{P}$ and show that $J(p_0\Vert\hat{p}_n)\rightarrow \inf_{p\in\mathcal{P}}J(p_0\Vert p)$ as $n\rightarrow\infty$, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non-parametric kernel density estimator, and that the advantage with the proposed estimator grows as $d$ increases.

Kun Zhang, Heng Peng, Laiwan Chan et al.

Model selection based on classical information criteria, such as BIC, is generally computationally demanding, but its properties are well studied. On the other hand, model selection based on parameter shrinkage by $\ell_1$-type penalties is computationally efficient. In this paper we make an attempt to combine their strengths, and propose a simple approach that penalizes the likelihood with data-dependent $\ell_1$ penalties as in adaptive Lasso and exploits a fixed penalization parameter. Even for finite samples, its model selection results approximately coincide with those based on information criteria; in particular, we show that in some special cases, this approach and the corresponding information criterion produce exactly the same model. One can also consider this approach as a way to directly determine the penalization parameter in adaptive Lasso to achieve information criteria-like model selection. As extensions, we apply this idea to complex models including Gaussian mixture model and mixture of factor analyzers, whose model selection is traditionally difficult to do; by adopting suitable penalties, we provide continuous approximators to the corresponding information criteria, which are easy to optimize and enable efficient model selection.

Tatsuya Tashiro, Shohei Shimizu, Aapo Hyvarinen et al.

We consider learning a causal ordering of variables in a linear non-Gaussian acyclic model called LiNGAM. Several existing methods have been shown to consistently estimate a causal ordering assuming that all the model assumptions are correct. But, the estimation results could be distorted if some assumptions actually are violated. In this paper, we propose a new algorithm for learning causal orders that is robust against one typical violation of the model assumptions: latent confounders. The key idea is to detect latent confounders by testing independence between estimated external influences and find subsets (parcels) that include variables that are not affected by latent confounders. We demonstrate the effectiveness of our method using artificial data and simulated brain imaging data.

Shohei Shimizu, Aapo Hyvarinen, Yutaka Kano et al.

In recent years, several methods have been proposed for the discovery of causal structure from non-experimental data (Spirtes et al. 2000; Pearl 2000). Such methods make various assumptions on the data generating process to facilitate its identification from purely observational data. Continuing this line of research, we show how to discover the complete causal structure of continuous-valued data, under the assumptions that (a) the data generating process is linear, (b) there are no unobserved confounders, and (c) disturbance variables have non-gaussian distributions of non-zero variances. The solution relies on the use of the statistical method known as independent component analysis (ICA), and does not require any pre-specified time-ordering of the variables. We provide a complete Matlab package for performing this LiNGAM analysis (short for Linear Non-Gaussian Acyclic Model), and demonstrate the effectiveness of the method using artificially generated data.

Patrik O. Hoyer, Aapo Hyvarinen, Richard Scheines et al.

An important task in data analysis is the discovery of causal relationships between observed variables. For continuous-valued data, linear acyclic causal models are commonly used to model the data-generating process, and the inference of such models is a well-studied problem. However, existing methods have significant limitations. Methods based on conditional independencies (Spirtes et al. 1993; Pearl 2000) cannot distinguish between independence-equivalent models, whereas approaches purely based on Independent Component Analysis (Shimizu et al. 2006) are inapplicable to data which is partially Gaussian. In this paper, we generalize and combine the two approaches, to yield a method able to learn the model structure in many cases for which the previous methods provide answers that are either incorrect or are not as informative as possible. We give exact graphical conditions for when two distinct models represent the same family of distributions, and empirically demonstrate the power of our method through thorough simulations.

Kun Zhang, Aapo Hyvarinen

By taking into account the nonlinear effect of the cause, the inner noise effect, and the measurement distortion effect in the observed variables, the post-nonlinear (PNL) causal model has demonstrated its excellent performance in distinguishing the cause from effect. However, its identifiability has not been properly addressed, and how to apply it in the case of more than two variables is also a problem. In this paper, we conduct a systematic investigation on its identifiability in the two-variable case. We show that this model is identifiable in most cases; by enumerating all possible situations in which the model is not identifiable, we provide sufficient conditions for its identifiability. Simulations are given to support the theoretical results. Moreover, in the case of more than two variables, we show that the whole causal structure can be found by applying the PNL causal model to each structure in the Markov equivalent class and testing if the disturbance is independent of the direct causes for each variable. In this way the exhaustive search over all possible causal structures is avoided.

Tatsuya Tashiro, Shohei Shimizu, Aapo Hyvarinen et al.

We consider to learn a causal ordering of variables in a linear non-Gaussian acyclic model called LiNGAM. Several existing methods have been shown to consistently estimate a causal ordering assuming that all the model assumptions are correct. But, the estimation results could be distorted if some assumptions actually are violated. In this paper, we propose a new algorithm for learning causal orders that is robust against one typical violation of the model assumptions: latent confounders. We demonstrate the effectiveness of our method using artificial data.

Miika Pihlaja, Michael Gutmann, Aapo Hyvarinen

We introduce a new family of estimators for unnormalized statistical models. Our family of estimators is parameterized by two nonlinear functions and uses a single sample from an auxiliary distribution, generalizing Maximum Likelihood Monte Carlo estimation of Geyer and Thompson (1992). The family is such that we can estimate the partition function like any other parameter in the model. The estimation is done by optimizing an algebraically simple, well defined objective function, which allows for the use of dedicated optimization methods. We establish consistency of the estimator family and give an expression for the asymptotic covariance matrix, which enables us to further analyze the influence of the nonlinearities and the auxiliary density on estimation performance. Some estimators in our family are particularly stable for a wide range of auxiliary densities. Interestingly, a specific choice of the nonlinearity establishes a connection between density estimation and classification by nonlinear logistic regression. Finally, the optimal amount of auxiliary samples relative to the given amount of the data is considered from the perspective of computational efficiency.

Kun Zhang, Aapo Hyvarinen

Separation of the sources and analysis of their connectivity have been an important topic in EEG/MEG analysis. To solve this problem in an automatic manner, we propose a two-layer model, in which the sources are conditionally uncorrelated from each other, but not independent; the dependence is caused by the causality in their time-varying variances (envelopes). The model is identified in two steps. We first propose a new source separation technique which takes into account the autocorrelations (which may be time-varying) and time-varying variances of the sources. The causality in the envelopes is then discovered by exploiting a special kind of multivariate GARCH (generalized autoregressive conditional heteroscedasticity) model. The resulting causal diagram gives the effective connectivity between the separated sources; in our experimental results on MEG data, sources with similar functions are grouped together, with negative influences between groups, and the groups are connected via some interesting sources.