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Collaborating Authors

Nickel, Maximilian


Moser Flow: Divergence-based Generative Modeling on Manifolds

arXiv.org Artificial Intelligence

We are interested in learning generative models for complex geometries described via manifolds, such as spheres, tori, and other implicit surfaces. Current extensions of existing (Euclidean) generative models are restricted to specific geometries and typically suffer from high computational costs. We introduce Moser Flow (MF), a new class of generative models within the family of continuous normalizing flows (CNF). MF also produces a CNF via a solution to the change-of-variable formula, however differently from other CNF methods, its model (learned) density is parameterized as the source (prior) density minus the divergence of a neural network (NN). The divergence is a local, linear differential operator, easy to approximate and calculate on manifolds. Therefore, unlike other CNFs, MF does not require invoking or backpropagating through an ODE solver during training. Furthermore, representing the model density explicitly as the divergence of a NN rather than as a solution of an ODE facilitates learning high fidelity densities. Theoretically, we prove that MF constitutes a universal density approximator under suitable assumptions. Empirically, we demonstrate for the first time the use of flow models for sampling from general curved surfaces and achieve significant improvements in density estimation, sample quality, and training complexity over existing CNFs on challenging synthetic geometries and real-world benchmarks from the earth and climate sciences.


Learning Neural Event Functions for Ordinary Differential Equations

arXiv.org Machine Learning

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and differentiated through. Neural Event ODEs are capable of modeling discrete (instantaneous) changes in a continuous-time system, without prior knowledge of when these changes should occur or how many such changes should exist. We test our approach in modeling hybrid discrete- and continuous- systems such as switching dynamical systems and collision in multi-body systems, and we propose simulation-based training of point processes with applications in discrete control.


CURI: A Benchmark for Productive Concept Learning Under Uncertainty

arXiv.org Artificial Intelligence

Humans can learn and reason under substantial uncertainty in a space of infinitely many concepts, including structured relational concepts ("a scene with objects that have the same color") and ad-hoc categories defined through goals ("objects that could fall on one's head"). In contrast, standard classification benchmarks: 1) consider only a fixed set of category labels, 2) do not evaluate compositional concept learning and 3) do not explicitly capture a notion of reasoning under uncertainty. We introduce a new few-shot, meta-learning benchmark, Compositional Reasoning Under Uncertainty (CURI) to bridge this gap. CURI evaluates different aspects of productive and systematic generalization, including abstract understandings of disentangling, productive generalization, learning boolean operations, variable binding, etc. Importantly, it also defines a model-independent "compositionality gap" to evaluate the difficulty of generalizing out-of-distribution along each of these axes. Extensive evaluations across a range of modeling choices spanning different modalities (image, schemas, and sounds), splits, privileged auxiliary concept information, and choices of negatives reveal substantial scope for modeling advances on the proposed task. All code and datasets will be available online.


Riemannian Continuous Normalizing Flows

arXiv.org Machine Learning

Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic spaces, most normalizing flows implicitly assume a flat geometry, making them either misspecified or ill-suited in these situations. To overcome this problem, we introduce Riemannian continuous normalizing flows, a model which admits the parametrization of flexible probability measures on smooth manifolds by defining flows as the solution to ordinary differential equations. We show that this approach can lead to substantial improvements on both synthetic and real-world data when compared to standard flows or previously introduced projected flows.


Hyperbolic Graph Neural Networks

Neural Information Processing Systems

Learning from graph-structured data is an important task in machine learning and artificial intelligence, for which Graph Neural Networks (GNNs) have shown great promise. Motivated by recent advances in geometric representation learning, we propose a novel GNN architecture for learning representations on Riemannian manifolds with differentiable exponential and logarithmic maps. We develop a scalable algorithm for modeling the structural properties of graphs, comparing Euclidean and hyperbolic geometry. In our experiments, we show that hyperbolic GNNs can lead to substantial improvements on various benchmark datasets. Papers published at the Neural Information Processing Systems Conference.


Learning Multivariate Hawkes Processes at Scale

arXiv.org Machine Learning

Multivariate Hawkes Processes (MHPs) are an important class of temporal point processes that have enabled key advances in understanding and predicting social information systems. However, due to their complex modeling of temporal dependencies, MHPs have proven to be notoriously difficult to scale, what has limited their applications to relatively small domains. In this work, we propose a novel model and computational approach to overcome this important limitation. By exploiting a characteristic sparsity pattern in real-world diffusion processes, we show that our approach allows to compute the exact likelihood and gradients of an MHP -- independently of the ambient dimensions of the underlying network. We show on synthetic and real-world datasets that our model does not only achieve state-of-the-art predictive results, but also improves runtime performance by multiple orders of magnitude compared to standard methods on sparse event sequences. In combination with easily interpretable latent variables and influence structures, this allows us to analyze diffusion processes at previously unattainable scale.


Reducing the Rank in Relational Factorization Models by Including Observable Patterns

Neural Information Processing Systems

Tensor factorizations have become popular methods for learning from multi-relational data. In this context, the rank of a factorization is an important parameter that determines runtime as well as generalization ability. To determine conditions under which factorization is an efficient approach for learning from relational data, we derive upper and lower bounds on the rank required to recover adjacency tensors. Based on our findings, we propose a novel additive tensor factorization model for learning from latent and observable patterns in multi-relational data and present a scalable algorithm for computing the factorization. Experimentally, we show that the proposed approach does not only improve the predictive performance over pure latent variable methods but that it also reduces the required rank --- and therefore runtime and memory complexity --- significantly.


Hyperbolic Graph Neural Networks

arXiv.org Machine Learning

Learning from graph-structured data is an important task in machine learning and artificial intelligence, for which Graph Neural Networks (GNNs) have shown great promise. Motivated by recent advances in geometric representation learning, we propose a novel GNN architecture for learning representations on Riemannian manifolds with differentiable exponential and logarithmic maps. We develop a scalable algorithm for modeling the structural properties of graphs, comparing Euclidean and hyperbolic geometry. In our experiments, we show that hyperbolic GNNs can lead to substantial improvements on various benchmark datasets.


Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry

arXiv.org Artificial Intelligence

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincar\'e-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincar\'e embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic space can reveal important aspects of a company's organizational structure as well as reveal historical relationships between language families.


Fast Linear Model for Knowledge Graph Embeddings

arXiv.org Machine Learning

This paper shows that a simple baseline based on a Bag-of-Words (BoW) representation learns surprisingly good knowledge graph embeddings. By casting knowledge base completion and question answering as supervised classification problems, we observe that modeling co-occurences of entities and relations leads to state-of-the-art performance with a training time of a few minutes using the open sourced library fastText.