A Unified Geometric Deep Learning Framework for Financial Time Series: Integrating Temporal, Graph, and Signature-Based Representations

Financial time series exhibit complex statistical properties, in- cluding nonlinear dependencies, non-stationarity, and evolv- ing relationships among assets. Traditional econometric mod- els and purely temporal machine learning approaches often struggle to capture these characteristics as they focus mainly on sequential dynamics while overlooking the underlying ge- ometric structure of financial data. This thesis investigates the role of geometric representations in financial time series analysis and proposes a unified framework that integrates temporal, geometric, and algebraic perspectives for model- ing, forecasting, and generating financial data. The central idea is that financial time series contain latent structural in- formation that can be captured through graph-based repre- sentations and signature-based features. The thesis introduces a signature-based similarity measure as an alternative to correlation matrices, improving community detection and portfolio construction. It then proposes a novel Time–Geometric forecasting model that combines temporal deep learning architectures with Graph Neural Networks to capture both temporal and geometric patterns within the time series data. Finally, the framework is extended to generative modeling and higher-order dependency analysis through graph- based generative adversarial networks and hypergraph rep- resentations derived directly from multivariate time series. Overall, this work shows that incorporating geometric struc- ture into financial time series modeling provides a power- ful framework for understanding, forecasting, and generat- ing complex financial data, highlighting the potential of geo- metric deep learning in quantitative finance.