Performance of Hyperbolic Geometry Models on Top-N Recommendation Tasks

We introduce a simple autoencoder based on hyperbolic geometry for solving standard collaborative filtering problem. In contrast to many modern deep learning techniques, we build our solution using only a single hidden layer. Remarkably, even with such a minimalistic approach, we not only outperform the Euclidean counterpart but also achieve a competitive performance with respect to the current state-of-the-art. We additionally explore the effects of space curvature on the quality of hyperbolic models and propose an efficient data-driven method for estimating its optimal value.Comment: Accepted at ACM RecSys 2020; 7 page

Hyperbolic matrix factorization improves prediction of drug-target associations

Past research in computational systems biology has focused more on the development and applications of advanced statistical and numerical optimization techniques and much less on understanding the geometry of the biological space. By representing biological entities as points in a low dimensional Euclidean space, state-of-the-art methods for drug-target interaction (DTI) prediction implicitly assume the flat geometry of the biological space. In contrast, recent theoretical studies suggest that biological systems exhibit tree-like topology with a high degree of clustering. As a consequence, embedding a biological system in a flat space leads to distortion of distances between biological objects. Here, we present a novel matrix factorization methodology for drug-target interaction prediction that uses hyperbolic space as the latent biological space. When benchmarked against classical, Euclidean methods, hyperbolic matrix factorization exhibits superior accuracy while lowering embedding dimension by an order of magnitude. We see this as additional evidence that the hyperbolic geometry underpins large biological networks

Hyperbolic Deep Neural Networks: A Survey

Recently, there has been a rising surge of momentum for deep representation learning in hyperbolic spaces due to theirhigh capacity of modeling data like knowledge graphs or synonym hierarchies, possessing hierarchical structure. We refer to the model as hyperbolic deep neural network in this paper. Such a hyperbolic neural architecture potentially leads to drastically compact model withmuch more physical interpretability than its counterpart in Euclidean space. To stimulate future research, this paper presents acoherent and comprehensive review of the literature around the neural components in the construction of hyperbolic deep neuralnetworks, as well as the generalization of the leading deep approaches to the Hyperbolic space. It also presents current applicationsaround various machine learning tasks on several publicly available datasets, together with insightful observations and identifying openquestions and promising future directions

Structured Machine Learning for Robotics

Machine Learning has become the essential tool for automating tasks that consist in predicting the output associated to a certain input. However many modern algorithms are mainly developed for the simple cases of classification and regression. Structured prediction is the field concerned with predicting outputs consisting of complex objects such as graphs, orientations or sequences. While these objects are often of practical interest, they do not have many of the mathematical properties that allow to design principled and computationally feasible algorithms with traditional techniques. In this thesis we investigate and develop algorithms for learning manifold-valued functions in the context of structured prediction. Differentiable manifolds are a mathematical abstraction used in many domains to describe sets with continuous constraints and non-Euclidean geometric properties. By taking a structured prediction approach we show how to define statistically consistent estimators for predicting elements of a manifold, in constrast to traditional structured predition algorithms that are restricted to output sets with finite cardinality. We introduce a wide range of applications that leverage manifolds structures. Above all, we study the case of the hyperbolic manifold, a space suited for representing hierarchical data. By representing supervised datasets within hyperbolic space we show how it is possible to invent new concepts in a previously known hierarchy and show promising results in hierarchical classification. We also study how modern structured approaches can help with practical robotics tasks, either improving performances in behavioural pipelines or showing more robust predictions for constrained tasks. Specifically, we show how structured prediction can be used to tackle inverse kinematics problems of redundant robots, accounting for the constraints of the robotic joints. We also consider the task of biological motion detection and show that by leveraging the sequence structure of video streams we significantly reduce the latency of the application. Our studies are complemented by empirical evaluations on both synthetic and real data

Learning Neural Graph Representations in Non-Euclidean Geometries

The success of Deep Learning methods is heavily dependent on the choice of the data representation. For that reason, much of the actual effort goes into Representation Learning, which seeks to design preprocessing pipelines and data transformations that can support effective learning algorithms. The aim of Representation Learning is to facilitate the task of extracting useful information for classifiers and other predictor models. In this regard, graphs arise as a convenient data structure that serves as an intermediary representation in a wide range of problems. The predominant approach to work with graphs has been to embed them in an Euclidean space, due to the power and simplicity of this geometry. Nevertheless, data in many domains exhibit non-Euclidean features, making embeddings into Riemannian manifolds with a richer structure necessary. The choice of a metric space where to embed the data imposes a geometric inductive bias, with a direct impact on the performance of the models. This thesis is about learning neural graph representations in non-Euclidean geometries and showcasing their applicability in different downstream tasks. We introduce a toolkit formed by different graph metrics with the goal of characterizing the topology of the data. In that way, we can choose a suitable target embedding space aligned to the shape of the dataset. By virtue of the geometric inductive bias provided by the structure of the non-Euclidean manifolds, neural models can achieve higher performances with a reduced parameter footprint. As a first step, we study graphs with hierarchical structures. We develop different techniques to derive hierarchical graphs from large label inventories. Noticing the capacity of hyperbolic spaces to represent tree-like arrangements, we incorporate this information into an NLP model through hyperbolic graph embeddings and showcase the higher performance that they enable. Second, we tackle the question of how to learn hierarchical representations suited for different downstream tasks. We introduce a model that jointly learns task-specific graph embeddings from a label inventory and performs classification in hyperbolic space. The model achieves state-of-the-art results on very fine-grained labels, with a remarkable reduction of the parameter size. Next, we move to matrix manifolds to work on graphs with diverse structures and properties. We propose a general framework to implement the mathematical tools required to learn graph embeddings on symmetric spaces. These spaces are of particular interest given that they have a compound geometry that simultaneously contains Euclidean as well as hyperbolic subspaces, allowing them to automatically adapt to dissimilar features in the graph. We demonstrate a concrete implementation of the framework on Siegel spaces, showcasing their versatility on different tasks. Finally, we focus on multi-relational graphs. We devise the means to translate Euclidean and hyperbolic multi-relational graph embedding models into the space of symmetric positive definite (SPD) matrices. To do so we develop gyrocalculus in this geometry and integrate it with the aforementioned framework