Deep Learning (DL) revolutionized the field of AI, transformed many industries and opened new markets. Early DL successes focused on discriminative models for classification and segmentation. Recently, interest has surged in generative DL that directly models data distributions in an unsupervised fashion. This has enabled numerous applications such as chatbots, image synthesis drug discovery, weather forecasting, antigen-specific antibody design, and protein function & structure prediction. Still, most effective DL architectures rely on Euclidean assumptions, limiting their applicability to grid-structured data only, such as images, audio, or data with a well-established vector embedding, e.g. language. Increasingly, real-world data resides either on non-Euclidean spaces -- e.g., graphs such as social networks, molecular structures, and manifolds such as 3D meshes -- or mixed-geometry (i.e., data with both Euclidean and non-Euclidean components) -- e.g, EEG data which can be best represented as spatio-temporal graphs (non-Euclidean) of electric signals (Euclidean), biological data such as proteins which are described by the language of amino acids (Euclidean) and a physical atomic structure (non-Euclidean). While DL has made progress on discriminative models on graphs, generative modelling on non-Euclidean or mixed-geometry data remains a major open challenge. Developing generative DL techniques for data of arbitrary underlying geometry offers immense opportunities across science, engineering, and beyond.
MAGAL aims to bridge this critical gap by developing a mathematically grounded framework for building generative models for data with arbitrary geometries (i.e., Euclidean, non-Euclidean and mixed). By pioneering principled techniques to handle data of mixed geometry, MAGAL could transform and streamline the design and application of generative modelling across disciplines.
I aspire to establish a common mathematical framework to phrase neural-block conception, analysis, stability, acceleration, interpretability, and verification. This will be the outcome of interpreting the operations of network blocks such as self-attention, squeeze-and excitation, non-local layers, etc. through the lenses of tensor calculus. This provides us with the rich analytical tools of tensor algebra, allowing us, for example, to see verification through the prism of analytic functions. Most importantly, we will pursue applications beyond the established domains of vision, language, and speech, exploring applications of Deep Neural Networks (DNNs) to domains of non-Euclidean data. In those cases, the brute data-driven approaches fail, and the inductive bias and control offered by tensor decompositions will allow practitioners of those domains to learn highly accurate, interpretable deep learning models.
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