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

Towards music perception by redundancy reduction and unsupervised learning in probabilistic models

PhDThe study of music perception lies at the intersection of several disciplines: perceptual psychology and cognitive science, musicology, psychoacoustics, and acoustical signal processing amongst others. Developments in perceptual theory over the last fifty years have emphasised an approach based on Shannon’s information theory and its basis in probabilistic systems, and in particular, the idea that perceptual systems in animals develop through a process of unsupervised learning in response to natural sensory stimulation, whereby the emerging computational structures are well adapted to the statistical structure of natural scenes. In turn, these ideas are being applied to problems in music perception. This thesis is an investigation of the principle of redundancy reduction through unsupervised learning, as applied to representations of sound and music. In the first part, previous work is reviewed, drawing on literature from some of the fields mentioned above, and an argument presented in support of the idea that perception in general and music perception in particular can indeed be accommodated within a framework of unsupervised learning in probabilistic models. In the second part, two related methods are applied to two different low-level representations. Firstly, linear redundancy reduction (Independent Component Analysis) is applied to acoustic waveforms of speech and music. Secondly, the related method of sparse coding is applied to a spectral representation of polyphonic music, which proves to be enough both to recognise that the individual notes are the important structural elements, and to recover a rough transcription of the music. Finally, the concepts of distance and similarity are considered, drawing in ideas about noise, phase invariance, and topological maps. Some ecologically and information theoretically motivated distance measures are suggested, and put in to practice in a novel method, using multidimensional scaling (MDS), for visualising geometrically the dependency structure in a distributed representation.Engineering and Physical Science Research Counci

Non-perturbative and finite N corrections on matrix holography

In this thesis, we study one-dimensional, matrix-valued quantum mechanical models. These models, exhibit numerous characteristics and can be used to study various problems. One among these problems is their gravitational nature, giving some hints about a quantum theory of gravity. The latter emanates from string theory since string theory already contains the matrix models. Therefore, by studying these models, we can understand aspects of a quantum gravitational theory, unraveling this way, some mechanisms present only in quantum gravity. To attack this problem and due to the difficulty of getting analytic solutions because of the matrix character of the model, we deploy some numerical approaches based on Monte Carlo simulation. In this way, we can understand gravitational features non-perturbatively, equipping us with a powerful tool to explore quantum gravity in regimes not studied in the past

Behavior quantification as the missing link between fields: Tools for digital psychiatry and their role in the future of neurobiology

The great behavioral heterogeneity observed between individuals with the same psychiatric disorder and even within one individual over time complicates both clinical practice and biomedical research. However, modern technologies are an exciting opportunity to improve behavioral characterization. Existing psychiatry methods that are qualitative or unscalable, such as patient surveys or clinical interviews, can now be collected at a greater capacity and analyzed to produce new quantitative measures. Furthermore, recent capabilities for continuous collection of passive sensor streams, such as phone GPS or smartwatch accelerometer, open avenues of novel questioning that were previously entirely unrealistic. Their temporally dense nature enables a cohesive study of real-time neural and behavioral signals. To develop comprehensive neurobiological models of psychiatric disease, it will be critical to first develop strong methods for behavioral quantification. There is huge potential in what can theoretically be captured by current technologies, but this in itself presents a large computational challenge -- one that will necessitate new data processing tools, new machine learning techniques, and ultimately a shift in how interdisciplinary work is conducted. In my thesis, I detail research projects that take different perspectives on digital psychiatry, subsequently tying ideas together with a concluding discussion on the future of the field. I also provide software infrastructure where relevant, with extensive documentation. Major contributions include scientific arguments and proof of concept results for daily free-form audio journals as an underappreciated psychiatry research datatype, as well as novel stability theorems and pilot empirical success for a proposed multi-area recurrent neural network architecture.Comment: PhD thesis cop

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity