Optimization and Learning over Riemannian Manifolds

Learning over smooth nonlinear spaces has found wide applications. A principled approach for addressing such problems is to endow the search space with a Riemannian manifold geometry and numerical optimization can be performed intrinsically. Recent years have seen a surge of interest in leveraging Riemannian optimization for nonlinearly-constrained problems. This thesis investigates and improves on the existing algorithms for Riemannian optimization, with a focus on unified analysis frameworks and generic strategies. To this end, the first chapter systematically studies the choice of Riemannian geometries and their impacts on algorithmic convergence, on the manifold of positive definite matrices. The second chapter considers stochastic optimization on manifolds and proposes a unified framework for analyzing and improving the convergence of Riemannian variance reduction methods for nonconvex functions. The third chapter introduces a generic acceleration scheme based on the idea of extrapolation, which achieves optimal convergence rate asymptotically while being empirically efficient

From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond

Graph neural networks (GNNs) have demonstrated significant promise in modelling relational data and have been widely applied in various fields of interest. The key mechanism behind GNNs is the so-called message passing where information is being iteratively aggregated to central nodes from their neighbourhood. Such a scheme has been found to be intrinsically linked to a physical process known as heat diffusion, where the propagation of GNNs naturally corresponds to the evolution of heat density. Analogizing the process of message passing to the heat dynamics allows to fundamentally understand the power and pitfalls of GNNs and consequently informs better model design. Recently, there emerges a plethora of works that proposes GNNs inspired from the continuous dynamics formulation, in an attempt to mitigate the known limitations of GNNs, such as oversmoothing and oversquashing. In this survey, we provide the first systematic and comprehensive review of studies that leverage the continuous perspective of GNNs. To this end, we introduce foundational ingredients for adapting continuous dynamics to GNNs, along with a general framework for the design of graph neural dynamics. We then review and categorize existing works based on their driven mechanisms and underlying dynamics. We also summarize how the limitations of classic GNNs can be addressed under the continuous framework. We conclude by identifying multiple open research directions

Generalized Bures-Wasserstein Geometry for Positive Definite Matrices

This paper proposes a generalized Bures-Wasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three different ways: 1) by generalizing the Lyapunov operator in the metric, 2) by generalizing the orthogonal Procrustes distance, and 3) by generalizing the Wasserstein distance between the Gaussians. We show that they all lead to the same geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\mathbf{M}$ such that when $\mathbf{M} = \mathbf{I}$, we recover the BW geometry. We derive expressions for the distance, geodesic, exponential/logarithm maps, Levi-Civita connection, and sectional curvature under the generalized BW geometry. We also present applications and experiments that illustrate the efficacy of the proposed geometry

Rieoptax: Riemannian Optimization in JAX

We present Rieoptax, an open source Python library for Riemannian optimization in JAX. We show that many differential geometric primitives, such as Riemannian exponential and logarithm maps, are usually faster in Rieoptax than existing frameworks in Python, both on CPU and GPU. We support various range of basic and advanced stochastic optimization solvers like Riemannian stochastic gradient, stochastic variance reduction, and adaptive gradient methods. A distinguishing feature of the proposed toolbox is that we also support differentially private optimization on Riemannian manifolds

Generalized energy and gradient flow via graph framelets

In this work, we provide a theoretical understanding of the framelet-based graph neural networks through the perspective of energy gradient flow. By viewing the framelet-based models as discretized gradient flows of some energy, we show it can induce both low-frequency and high-frequency-dominated dynamics, via the separate weight matrices for different frequency components. This substantiates its good empirical performance on both homophilic and heterophilic graphs. We then propose a generalized energy via framelet decomposition and show its gradient flow leads to a novel graph neural network, which includes many existing models as special cases. We then explain how the proposed model generally leads to more flexible dynamics, thus potentially enhancing the representation power of graph neural networks

Generalized Laplacian Regularized Framelet GCNs

This paper introduces a novel Framelet Graph approach based on p-Laplacian GNN. The proposed two models, named p-Laplacian undecimated framelet graph convolution (pL-UFG) and generalized p-Laplacian undecimated framelet graph convolution (pL-fUFG) inherit the nature of p-Laplacian with the expressive power of multi-resolution decomposition of graph signals. The empirical study highlights the excellent performance of the pL-UFG and pL-fUFG in different graph learning tasks including node classification and signal denoising

Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challenges

Graph-based message-passing neural networks (MPNNs) have achieved remarkable success in both node and graph-level learning tasks. However, several identified problems, including over-smoothing (OSM), limited expressive power, and over-squashing (OSQ), still limit the performance of MPNNs. In particular, OSQ serves as the latest identified problem, where MPNNs gradually lose their learning accuracy when long-range dependencies between graph nodes are required. In this work, we provide an exposition on the OSQ problem by summarizing different formulations of OSQ from current literature, as well as the three different categories of approaches for addressing the OSQ problem. In addition, we also discuss the alignment between OSQ and expressive power and the trade-off between OSQ and OSM. Furthermore, we summarize the empirical methods leveraged from existing works to verify the efficiency of OSQ mitigation approaches, with illustrations of their computational complexities. Lastly, we list some open questions that are of interest for further exploration of the OSQ problem along with potential directions from the best of our knowledge