Deep generative models can emulate the perceptual properties of complex image datasets, providing a latent representation of the data. However, manipulating such representation to perform meaningful and controllable transformations in the data space remains challenging without some form of supervision. While previous work has focused on exploiting statistical independence to \textit{disentangle} latent factors, we argue that such requirement can be advantageously relaxed and propose instead a non-statistical framework that relies on identifying a modular organization of the network, based on counterfactual manipulations. Our experiments support that modularity between groups of channels is achieved to a certain degree on a variety of generative models. This allowed the design of targeted interventions on complex image datasets, opening the way to applications such as computationally efficient style transfer and the automated assessment of robustness to contextual changes in pattern recognition systems. Deep generative models, by learning a non-linear function mapping a latent space to the space observations, have proven successful at designing realistic images in a variety of complex domains (objects, animals, human faces, interior scenes). In particular, two kinds of approaches emerged as state-of-the-art (SOTA): Generative Adversarial Networks (GAN) (Goodfellow et al., 2014) , and Variational Autoencoders (VAE) (Kingma & Welling, 2013; Rezende et al., 2014) . Efforts have been made to have such models produce disentangled latent representations that can control interpretable properties of images (Kulkarni et al., 2015; Higgins et al., 2017) . However, the resulting models are not necessarily mechanistic (or causal) in the sense that interpretable properties of an image cannot be ascribed to a particular part, a module, of the network architecture. Gaining access to a modular organization of generative models would benefit the interpretability and allow extrapolations, such as generating an object in a background that was not previously associated with this object, as illustrated in a preview of our experimental results in Fig. 1 . Such extrapolations are an integral part of human representational capabilities (consider common expressions such as "like an elephant in a china shop") and consistent with the modular organization of its visual system, comprising specialized regions encoding objects, faces and places (see e.g. GrillSpector & Malach (2004) ). Extrapolations moreover likely support adaptability to environmental changes and robust decision making (Dvornik et al., 2018) . How to leverage trained deep generative architectures to perform such extrapolations is an open problem, largely due to the non-linearities and high dimensionality that prevent interpretability of computations performed in successive layers. In this paper, we propose a causal framework to explore modularity, which relates to the causal principle of Independent Mechanisms, stating that the causal mechanisms contributing to the overall generating process do not influence nor inform each other (Peters et al., 2017) . 1 We study the effect of direct interventions in the network from the point of view that the mechanisms involved in generating data can be modified individually without affecting each other. This principle can be applied to generative models to assess how well they capture a causal mechanism (Besserve et al., 2018) . Causality allows to assay how an outcome would have changed, had some variables taken different values, referred to as a counterfactual (Pearl, 2009; Imbens & Rubin, 2015) . We use counterfactuals to assess the role of specific internal variables in the overall functioning of trained deep generative models, along with a rigorous definition of disentanglement in a causal framework. Then, we analyze this disentanglement in implemented models based on unsupervised counterfactual manipulations. We show empirically how VAEs and GANs trained on image databases exhibit modularity of their hidden units, encoding different features and allowing counterfactual editing of generated images. Related work. Our work relates to the interpretability of convolutional neural networks, which has been intensively investigated in discriminative architectures (Zeiler & Fergus, 2014; Dosovitskiy & Brox, 2016; Fong & Vedaldi, 2017; Zhang et al., 2017b; a) . Generative models require a different approach, as the downstream effect of changes in intermediate representations are high dimensional. InfoGANs. β-VAEs and other works (Chen et al., 2016; Mathieu et al., 2016; Kulkarni et al., 2015; Higgins et al., 2017) address supervised or unsupervised disentanglement of latent variables related to what we formalize as extrinsic disentanglement of transformations acting on data points. We introduce the novel concept of intrinsic disentanglement to uncover the internal organization of networks, arguing that many interesting transformations are statistically dependent and are thus unlikely to be disentangled in the latent space. This relates to Bau et al. (2018) who proposed a framework based on interventions on internal variables of a GAN which, in contrast to our fully unsupervised approach, requires semantic information. Higgins et al. (2018) suggest a definition of disentanglement based on group representation theory. Compared to this proposal, our approach (introduced independently in (Anonymous, 2018) ) is more flexible as it applies to arbitrary continuous transformations, free from the strong requirements of representation theory (see Appendix F). Finally, an interventional approach to disentanglement has also be taken by Suter et al. (2018) , who focuses on extrinsic disentanglement in a classical graphical model setting and develop measures of interventional robustness based on labeled data. We introduced a mathematical definition of disentanglement, related it to the causal notion of counterfactual and used it for the unsupervised characterization of the representation encoded by different groups of channels in deep generative architectures. We found evidence for interpretable modules of internal variables in four different generative models trained on two complex real world datasets. Our framework opens a way to a better understanding of complex generative architectures and applications such as the style transfer (Gatys et al., 2015) of controllable properties of generated images at low computational cost (no further optimization is required), and the automated assessment of robustness of object recognition systems to contextual changes. From a broader perspective, this research direction contributes to a better exploitation of deep neural networks obtained by costly and highly energy-consuming training procedures, by (1) enhancing their interpretability and (2) allowing them to be used for tasks their where not trained for. This offers a perspective on how more sustainable research in Artificial Intelligence could be fostered in the future.
We develop a framework to find modular internal representations in generative models and manipulate then to generate counterfactual examples.
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