Gary Bécigneul

Jovan Andonov, Octavian Ganea, Paulina Grnarova et al.

Language Modelling has been a central part of Natural Language Processing for a very long time and in the past few years LSTM-based language models have been the go-to method for commercial language modeling. Recently, it has been shown that when looking at language modelling from a matrix factorization point of view, the final Softmax layer limits the expressiveness of the model, by putting an upper bound on the rank of the resulting matrix. Additionally, a new family of neural networks based called NeuralODEs, has been introduced as a continuous alternative to Residual Networks. Moreover, it has been shown that there is a connection between these models and Normalizing Flows. In this work we propose a new family of language models based on NeuralODEs and the continuous analogue of Normalizing Flows and manage to improve on some of the baselines.

Benson Chen, Gary Bécigneul, Octavian-Eugen Ganea et al.

Current graph neural network (GNN) architectures naively average or sum node embeddings into an aggregated graph representation -- potentially losing structural or semantic information. We here introduce OT-GNN, a model that computes graph embeddings using parametric prototypes that highlight key facets of different graph aspects. Towards this goal, we successfully combine optimal transport (OT) with parametric graph models. Graph representations are obtained from Wasserstein distances between the set of GNN node embeddings and ``prototype'' point clouds as free parameters. We theoretically prove that, unlike traditional sum aggregation, our function class on point clouds satisfies a fundamental universal approximation theorem. Empirically, we address an inherent collapse optimization issue by proposing a noise contrastive regularizer to steer the model towards truly exploiting the OT geometry. Finally, we outperform popular methods on several molecular property prediction tasks, while exhibiting smoother graph representations.

Louis Abraham, Gary Bécigneul, Bernhard Schölkopf

We study the problem usually referred to as group testing in the context of COVID-19. Given $n$ samples taken from patients, how should we select mixtures of samples to be tested, so as to maximize information and minimize the number of tests? We consider both adaptive and non-adaptive strategies, and take a Bayesian approach with a prior both for infection of patients and test errors. We start by proposing a mathematically principled objective, grounded in information theory. We then optimize non-adaptive optimization strategies using genetic algorithms, and leverage the mathematical framework of adaptive sub-modularity to obtain theoretical guarantees for the greedy-adaptive method.

Calin Cruceru, Gary Bécigneul, Octavian-Eugen Ganea

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.

Ondrej Skopek, Octavian-Eugen Ganea, Gary Bécigneul

Euclidean geometry has historically been the typical "workhorse" for machine learning applications due to its power and simplicity. However, it has recently been shown that geometric spaces with constant non-zero curvature improve representations and performance on a variety of data types and downstream tasks. Consequently, generative models like Variational Autoencoders (VAEs) have been successfully generalized to elliptical and hyperbolic latent spaces. While these approaches work well on data with particular kinds of biases e.g. tree-like data for a hyperbolic VAE, there exists no generic approach unifying and leveraging all three models. We develop a Mixed-curvature Variational Autoencoder, an efficient way to train a VAE whose latent space is a product of constant curvature Riemannian manifolds, where the per-component curvature is fixed or learnable. This generalizes the Euclidean VAE to curved latent spaces and recovers it when curvatures of all latent space components go to 0.

Gregor Bachmann, Gary Bécigneul, Octavian-Eugen Ganea

Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical, that provide specific inductive biases useful for certain real-world data properties, e.g. scale-free, hierarchical or cyclical. However, the popular graph neural networks are currently limited in modeling data only via Euclidean geometry and associated vector space operations. Here, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) introducing a unified formalism that can interpolate smoothly between all geometries of constant curvature, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models smoothly recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, we outperform Euclidean GCNs in the tasks of node classification and distortion minimization for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature.

Octavian-Eugen Ganea, Yashas Annadani, Gary Bécigneul

We take steps towards understanding the "posterior collapse (PC)" difficulty in variational autoencoders (VAEs),~i.e. a degenerate optimum in which the latent codes become independent of their corresponding inputs. We rely on calculus of variations and theoretically explore a few popular VAE models, showing that PC always occurs for non-parametric encoders and decoders. Inspired by the popular noise contrastive estimation algorithm, we propose NC-VAE where the encoder discriminates between the latent codes of real data and of some artificially generated noise, in addition to encouraging good data reconstruction abilities. Theoretically, we prove that our model cannot reach PC and provide novel lower bounds. Our method is straightforward to implement and has the same run-time as vanilla VAE. Empirically, we showcase its benefits on popular image and text datasets.

Octavian-Eugen Ganea, Sylvain Gelly, Gary Bécigneul et al.

The Softmax function on top of a final linear layer is the de facto method to output probability distributions in neural networks. In many applications such as language models or text generation, this model has to produce distributions over large output vocabularies. Recently, this has been shown to have limited representational capacity due to its connection with the rank bottleneck in matrix factorization. However, little is known about the limitations of Linear-Softmax for quantities of practical interest such as cross entropy or mode estimation, a direction that we explore here. As an efficient and effective solution to alleviate this issue, we propose to learn parametric monotonic functions on top of the logits. We theoretically investigate the rank increasing capabilities of such monotonic functions. Empirically, our method improves in two different quality metrics over the traditional Linear-Softmax layer in synthetic and real language model experiments, adding little time or memory overhead, while being comparable to the more computationally expensive mixture of Softmaxes.

Alexandru Tifrea, Gary Bécigneul, Octavian-Eugen Ganea

Words are not created equal. In fact, they form an aristocratic graph with a latent hierarchical structure that the next generation of unsupervised learned word embeddings should reveal. In this paper, justified by the notion of delta-hyperbolicity or tree-likeliness of a space, we propose to embed words in a Cartesian product of hyperbolic spaces which we theoretically connect to the Gaussian word embeddings and their Fisher geometry. This connection allows us to introduce a novel principled hypernymy score for word embeddings. Moreover, we adapt the well-known Glove algorithm to learn unsupervised word embeddings in this type of Riemannian manifolds. We further explain how to solve the analogy task using the Riemannian parallel transport that generalizes vector arithmetics to this new type of geometry. Empirically, based on extensive experiments, we prove that our embeddings, trained unsupervised, are the first to simultaneously outperform strong and popular baselines on the tasks of similarity, analogy and hypernymy detection. In particular, for word hypernymy, we obtain new state-of-the-art on fully unsupervised WBLESS classification accuracy.

Gary Bécigneul, Octavian-Eugen Ganea

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

Octavian-Eugen Ganea, Gary Bécigneul, Thomas Hofmann

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks such as gated recurrent units. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks.

Octavian-Eugen Ganea, Gary Bécigneul, Thomas Hofmann

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry. Second, we view hierarchical relations as partial orders defined using a family of nested geodesically convex cones. We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces, and they canonically define the embedding learning process. Experiments show significant improvements of our method over strong recent baselines both in terms of representational capacity and generalization.

Gary Bécigneul

In machine learning and neuroscience, certain computational structures and algorithms are known to yield disentangled representations without us understanding why, the most striking examples being perhaps convolutional neural networks and the ventral stream of the visual cortex in humans and primates. As for the latter, it was conjectured that representations may be disentangled by being flattened progressively and at a local scale. An attempt at a formalization of the role of invariance in learning representations was made recently, being referred to as I-theory. In this framework and using the language of differential geometry, we show that pooling over a group of transformations of the input contracts the metric and reduces its curvature, and provide quantitative bounds, in the aim of moving towards a theoretical understanding on how to disentangle representations.