Thesis title: Spectral and Geometric Approaches in Deep Learning: From Shapes to Graphs and Biological Structures
Our environment is fundamentally structured and geometric, a quality reflected in the data we collect. By exploring the symmetries and geometric properties inherent in complex systems, we can significantly improve the performance of deep learning models, even when data availability is limited. This dissertation investigates the integration of geometric principles into learning tasks, with a particular emphasis on real-world applications. Based on the evolving field of Geometric Deep Learning, this work extends traditional Euclidean models, exploring advanced geometries that better capture intrinsic data properties and foster more efficient learning processes.
We begin with the extension of classical tools from differential geometry and probability theory to compute quantiles on manifolds. These methods are applied to diverse real-world datasets, including geophysical phenomena and dihedral angles in proteins, demonstrating their practical utility and versatility.
From a geometry processing perspective, spectral representation emerges as a cornerstone of this exploration. Its utility is investigated in both shape modelling and graph isomorphism. In shape modelling, we develop novel techniques that integrate eigenvalues across multiple levels, enabling the extraction of rich multiscale features from 3D shapes. In graph isomorphism, spectral representation is employed to identify correspondences between graphs and their sub-isomorphic components, leading to significant advancements in learning tasks. Notably, these approaches find valuable applications in neural network representational spaces, where spectral maps provide meaningful and computationally efficient advantages.
The culmination of this research lies in applying geometric deep learning to biochemistry, with a focus on the intricate structures of small molecules and proteins. A standout example is the application of geometric deep learning to antibody-antigen interactions, where intrinsic graph structures and surface characteristics are leveraged to achieve superior performance. By embedding geometric insights into the learning framework, we achieve significant improvements in the decoding of biological complexity and predictive accuracy.
In conclusion, this dissertation highlights the transformative potential of integrating geometric principles into machine learning frameworks, with a particular focus on analyzing biological data. Through innovative methodologies and empirical validation, it makes significant contributions to theoretical advances and practical applications in geometry processing, representation learning, and computational biology.