Mapping between discrete and continuous distributions is a difficult task and many have had to resort to approximate or heuristical approaches. We propose a tessellation-based approach that directly learns quantization boundaries on a continuous space, complete with exact likelihood evaluations. This is done through constructing normalizing flows on convex polytopes parameterized through a differentiable Voronoi tessellation. Using a simple homeomorphism with an efficient log determinant Jacobian, we can then cheaply parameterize distributions on convex polytopes. We explore this approach in two application settings, mapping from discrete to continuous and vice versa. Firstly, a Voronoi dequantization allows automatically learning quantization boundaries in a multidimensional space. The location of boundaries and distances between regions can encode useful structural relations between the quantized discrete values. Secondly, a Voronoi mixture model has constant computation cost for likelihood evaluation regardless of the number of mixture components. Empirically, we show improvements over existing methods across a range of structured data modalities, and find that we can achieve a significant gain from just adding Voronoi mixtures to a baseline model.
There has been significant progress in creating machine learning models that identify objects in scenes along with their associated attributes and relationships; however, there is a large gap between the best models and human capabilities. One of the major reasons for this gap is the difficulty in collecting sufficient amounts of annotated relations and attributes for training these systems. While some attributes and relations are abundant, the distribution in the natural world and existing datasets is long tailed. In this paper, we address this problem by introducing a novel incremental active learning framework that asks for attributes and relations in visual scenes. While conventional active learning methods ask for labels of specific examples, we flip this framing to allow agents to ask for examples from specific categories. Using this framing, we introduce an active sampling method that asks for examples from the tail of the data distribution and show that it outperforms classical active learning methods on Visual Genome.
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.
We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, high-fidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of recurrent continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience.
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.
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.
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.
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.
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.
One of the hallmarks of human intelligence is the ability to compose learned knowledge into novel concepts which can be recognized without a single training example. In contrast, current state-of-the-art methods require hundreds of training examples for each possible category to build reliable and accurate classifiers. To alleviate this striking difference in efficiency, we propose a task-driven modular architecture for compositional reasoning and sample efficient learning. Our architecture consists of a set of neural network modules, which are small fully connected layers operating in semantic concept space. These modules are configured through a gating function conditioned on the task to produce features representing the compatibility between the input image and the concept under consideration. This enables us to express tasks as a combination of sub-tasks and to generalize to unseen categories by reweighting a set of small modules. Furthermore, the network can be trained efficiently as it is fully differentiable and its modules operate on small sub-spaces. We focus our study on the problem of compositional zero-shot classification of object-attribute categories. We show in our experiments that current evaluation metrics are flawed as they only consider unseen object-attribute pairs. When extending the evaluation to the generalized setting which accounts also for pairs seen during training, we discover that naive baseline methods perform similarly or better than current approaches. However, our modular network is able to outperform all existing approaches on two widely-used benchmark datasets.