Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

Cetacean Translation Initiative: a roadmap to deciphering the communication of sperm whales

The past decade has witnessed a groundbreaking rise of machine learning for human language analysis, with current methods capable of automatically accurately recovering various aspects of syntax and semantics - including sentence structure and grounded word meaning - from large data collections. Recent research showed the promise of such tools for analyzing acoustic communication in nonhuman species. We posit that machine learning will be the cornerstone of future collection, processing, and analysis of multimodal streams of data in animal communication studies, including bioacoustic, behavioral, biological, and environmental data. Cetaceans are unique non-human model species as they possess sophisticated acoustic communications, but utilize a very different encoding system that evolved in an aquatic rather than terrestrial medium. Sperm whales, in particular, with their highly-developed neuroanatomical features, cognitive abilities, social structures, and discrete click-based encoding make for an excellent starting point for advanced machine learning tools that can be applied to other animals in the future. This paper details a roadmap toward this goal based on currently existing technology and multidisciplinary scientific community effort. We outline the key elements required for the collection and processing of massive bioacoustic data of sperm whales, detecting their basic communication units and language-like higher-level structures, and validating these models through interactive playback experiments. The technological capabilities developed by such an undertaking are likely to yield cross-applications and advancements in broader communities investigating non-human communication and animal behavioral research.

Shape My Face: Registering 3D Face Scans by Surface-to-Surface Translation

Existing surface registration methods focus on fitting in-sample data with little to no generalization ability and require both heavy pre-processing and careful hand-tuning. In this paper, we cast the registration task as a surface-to-surface translation problem, and design a model to reliably capture the latent geometric information directly from raw 3D face scans. We introduce Shape-My-Face (SMF), a powerful encoder-decoder architecture based on an improved point cloud encoder, a novel visual attention mechanism, graph convolutional decoders with skip connections, and a specialized mouth model that we smoothly integrate with the mesh convolutions. Compared to the previous state-of-the-art learning algorithms for non-rigid registration of face scans, SMF only requires the raw data to be rigidly aligned (with scaling) with a pre-defined face template. Additionally, our model provides topologically-sound meshes with minimal supervision, offers faster training time, has orders of magnitude fewer trainable parameters, is more robust to noise, and can generalize to previously unseen datasets. We extensively evaluate the quality of our registrations on diverse data. We demonstrate the robustness and generalizability of our model with in-the-wild face scans across different modalities, sensor types, and resolutions. Finally, we show that, by learning to register scans, SMF produces a hybrid linear and non-linear morphable model that can be used for generation, shape morphing, and expression transfer through manipulation of the latent space, including in-the-wild. We train SMF on a dataset of human faces comprising 9 large-scale databases on commodity hardware.

Utilising Graph Machine Learning within Drug Discovery and Development

Graph Machine Learning (GML) is receiving growing interest within the pharmaceutical and biotechnology industries for its ability to model biomolecular structures, the functional relationships between them, and integrate multi-omic datasets - amongst other data types. Herein, we present a multidisciplinary academic-industrial review of the topic within the context of drug discovery and development. After introducing key terms and modelling approaches, we move chronologically through the drug development pipeline to identify and summarise work incorporating: target identification, design of small molecules and biologics, and drug repurposing. Whilst the field is still emerging, key milestones including repurposed drugs entering in vivo studies, suggest graph machine learning will become a modelling framework of choice within biomedical machine learning.

Tuning Word2vec for Large Scale Recommendation Systems

Word2vec is a powerful machine learning tool that emerged from Natural Lan-guage Processing (NLP) and is now applied in multiple domains, including recom-mender systems, forecasting, and network analysis. As Word2vec is often used offthe shelf, we address the question of whether the default hyperparameters are suit-able for recommender systems. The answer is emphatically no. In this paper, wefirst elucidate the importance of hyperparameter optimization and show that un-constrained optimization yields an average 221% improvement in hit rate over thedefault parameters. However, unconstrained optimization leads to hyperparametersettings that are very expensive and not feasible for large scale recommendationtasks. To this end, we demonstrate 138% average improvement in hit rate with aruntime budget-constrained hyperparameter optimization. Furthermore, to makehyperparameter optimization applicable for large scale recommendation problemswhere the target dataset is too large to search over, we investigate generalizinghyperparameters settings from samples. We show that applying constrained hy-perparameter optimization using only a 10% sample of the data still yields a 91%average improvement in hit rate over the default parameters when applied to thefull datasets. Finally, we apply hyperparameters learned using our method of con-strained optimization on a sample to the Who To Follow recommendation serviceat Twitter and are able to increase follow rates by 15%.

Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting

While Graph Neural Networks (GNNs) have achieved remarkable results in a variety of applications, recent studies exposed important shortcomings in their ability to capture the structure of the underlying graph. It has been shown that the expressive power of standard GNNs is bounded by the Weisfeiler-Lehman (WL) graph isomorphism test, from which they inherit proven limitations such as the inability to detect and count graph substructures. On the other hand, there is significant empirical evidence, e.g. in network science and bioinformatics, that substructures are often informative for downstream tasks, suggesting that it is desirable to design GNNs capable of leveraging this important source of information. To this end, we propose a novel topologically-aware message passing scheme based on subgraph isomorphism counting. We show that our architecture allows incorporating domain-specific inductive biases and that it is strictly more expressive than the WL test. Importantly, in contrast to recent works on the expressivity of GNNs, we do not attempt to adhere to the WL hierarchy; this allows us to retain multiple attractive properties of standard GNNs such as locality and linear complexity, while being able to disambiguate even hard instances of graph isomorphism. We extensively evaluate our method on graph classification and regression tasks and show state-of-the-art results on multiple datasets including molecular graphs and social networks.

Geometrically Principled Connections in Graph Neural Networks

Graph convolution operators bring the advantages of deep learning to a variety of graph and mesh processing tasks previously deemed out of reach. With their continued success comes the desire to design more powerful architectures, often by adapting existing deep learning techniques to non-Euclidean data. In this paper, we argue geometry should remain the primary driving force behind innovation in the emerging field of geometric deep learning. We relate graph neural networks to widely successful computer graphics and data approximation models: radial basis functions (RBFs). We conjecture that, like RBFs, graph convolution layers would benefit from the addition of simple functions to the powerful convolution kernels. We introduce affine skip connections, a novel building block formed by combining a fully connected layer with any graph convolution operator. We experimentally demonstrate the effectiveness of our technique and show the improved performance is the consequence of more than the increased number of parameters. Operators equipped with the affine skip connection markedly outperform their base performance on every task we evaluated, i.e., shape reconstruction, dense shape correspondence, and graph classification. We hope our simple and effective approach will serve as a solid baseline and help ease future research in graph neural networks.

Learning Interpretable Disease Self-Representations for Drug Repositioning

Drug repositioning is an attractive cost-efficient strategy for the development of treatments for human diseases. Here, we propose an interpretable model that learns disease self-representations for drug repositioning. Our self-representation model represents each disease as a linear combination of a few other diseases. We enforce proximity in the learnt representations in a way to preserve the geometric structure of the human phenome network - a domain-specific knowledge that naturally adds relational inductive bias to the disease self-representations. We prove that our method is globally optimal and show results outperforming state-of-the-art drug repositioning approaches. We further show that the disease self-representations are biologically interpretable.

Learning interpretable disease self-representations for drug repositioning

Drug repositioning is an attractive cost-efficient strategy for the development of treatments for human diseases. Here, we propose an interpretable model that learns disease self-representations for drug repositioning. Our self-representation model represents each disease as a linear combination of a few other diseases. We enforce the proximity between diseases to preserve the geometric structure of the human phenome network - a domain-specific knowledge that naturally adds relational inductive bias to the disease self-representations. We prove that our method is globally optimal and show results outperforming state-of-the-art drug repositioning approaches. We further show that the disease self-representations are biologically interpretable.

Transferability of Spectral Graph Convolutional Neural Networks

This paper focuses on spectral graph convolutional neural networks (ConvNets), where filters are defined as elementwise multiplication in the frequency domain of a graph. In machine learning settings where the dataset consists of signals defined on many different graphs, the trained ConvNet should generalize to signal on graphs unseen in the training set. It is thus important to transfer filters from one graph to the other. Transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs describe the same phenomenon, then a single filter/ConvNet should have similar repercussions on both graphs. This paper aims at debunking the common misconception that spectral filters are not transferable. We show that if two graphs discretize the same continuous metric space, then a spectral filter/ConvNet has approximately the same repercussion on both graphs. Our analysis is more permissive than the standard analysis. Transferability is typically described as the robustness of the filter to small graph perturbations and re-indexing of the vertices. Our analysis accounts also for large graph perturbations. We prove transferability between graphs that can have completely different dimensions and topologies, only requiring that both graphs discretize the same underlying continuous space.