Scalable Machine Learning Algorithms using Path Signatures (2506.17634v2)

Abstract: The interface between stochastic analysis and machine learning is a rapidly evolving field, with path signatures - iterated integrals that provide faithful, hierarchical representations of paths - offering a principled and universal feature map for sequential and structured data. Rooted in rough path theory, path signatures are invariant to reparameterization and well-suited for modelling evolving dynamics, long-range dependencies, and irregular sampling - common challenges in real-world time series and graph data. This thesis investigates how to harness the expressive power of path signatures within scalable machine learning pipelines. It introduces a suite of models that combine theoretical robustness with computational efficiency, bridging rough path theory with probabilistic modelling, deep learning, and kernel methods. Key contributions include: Gaussian processes with signature kernel-based covariance functions for uncertainty-aware time series modelling; the Seq2Tens framework, which employs low-rank tensor structure in the weight space for scalable deep modelling of long-range dependencies; and graph-based models where expected signatures over graphs induce hypo-elliptic diffusion processes, offering expressive yet tractable alternatives to standard graph neural networks. Further developments include Random Fourier Signature Features, a scalable kernel approximation with theoretical guarantees, and Recurrent Sparse Spectrum Signature Gaussian Processes, which combine Gaussian processes, signature kernels, and random features with a principled forgetting mechanism for multi-horizon time series forecasting with adaptive context length. We hope this thesis serves as both a methodological toolkit and a conceptual bridge, and provides a useful reference for the current state of the art in scalable, signature-based learning for sequential and structured data.

• The paper integrates path signatures, a powerful feature representation from rough path theory, into scalable machine learning algorithms to improve modeling of sequential and structured data.
• It proposes methods like integrating signature kernels into scalable Gaussian Processes via sparse variational inference and introducing Random Fourier Signature Features to handle computational challenges.
• Numerical results across various domains demonstrate the efficacy and scalability of these methods, suggesting potential for widespread adoption in fields like finance, healthcare, and time series analysis.

Overview of Scalable Machine Learning Algorithms Using Path Signatures

The paper "Scalable Machine Learning Algorithms Using Path Signatures" focuses on integrating path signatures into scalable machine learning algorithms to enhance sequential and structured data modeling. Path signatures, derived from rough path theory, offer a sophisticated feature representation for sequences, capturing complex dynamics and ensuring invariance under reparameterization.

Key Concepts and Contributions

Path Signatures: Introduced as hierarchical features, path signatures essentially convert a path into a series of tensors. Each tensor corresponds to an iterated integral of the path's coordinates, providing a structured encoding of the path's geometry. This feature representation is particularly advantageous for sequential data, offering robustness to changes in time sampling and enabling tree-like equivalence. The paper addresses computational barriers related to the dimensionality growth of tensors with innovative algorithms, ensuring feasibility in practical applications.

Gaussian Processes: The work combines path signature features with Gaussian processes to enhance probabilistic modeling of sequential data. By embedding signature kernels within Gaussian process models, the paper enables expressive modeling while offering computational scalability via sparse variational inference techniques. This approach is shown to improve performance in probabilistic time series classification tasks.

Seq2Tens Framework: This framework merges signature features with deep learning by stacking low-rank linear layers. This mitigates computational costs while maintaining expressiveness, facilitating applications in time series classification, healthcare mortality prediction, and generative modeling.

Graph Representation: Extends path signatures to graph data using hypo-elliptic diffusions, enabling scalable architectures to capture both global and local graph structures. This extension outperforms conventional graph neural networks in tasks demanding long-range reasoning.

Random Fourier Signature Features: This section explores scalable random feature-based approximations for signature kernels. These approximations address computational limitations for large datasets while retaining competitive performance. The introduction of Random Fourier Signature Features reduces the quadratic complexity traditionally associated with signature kernels.

Recurrent Sparse Spectrum Signature Gaussian Processes: This combines Random Fourier Signature Features with Gaussian Processes, incorporating a forgetting mechanism for adaptable time series forecasting.

Numerical Results and Implications

Throughout the paper, strong numerical results are demonstrated across various domains, showcasing the efficacy and scalability of the proposed methodologies. These contributions offer significant practical implications, from improved handling of large-scale data to enhanced modeling capabilities in complex sequential settings.

Theoretical and Practical Implications

Theoretically, the unifying theme of the paper bridges the gap between the mathematical elegance of path signatures and practical deployment in machine learning models. The results obtained provide robust evidence of the viability and advantages of integrating path signatures in various sequence modeling contexts, offering universal approximation properties.

Practically, the scalability of these methods implies potential widespread adoption in fields requiring efficient sequential data processing. This includes time series forecasting in finance, healthcare analytics, and more. The paper hints at future directions in AI, particularly how path signatures might inspire further innovations in representing complex data patterns across applications.

Future Directions

While the paper showcases substantial improvements, future work could explore deeper integration of path signatures across varying state spaces and the optimization of kernel hyperparameters for bespoke applications. The research also opens avenues for leveraging path signatures in AI models that handle non-standard data structures, potentially revolutionizing approaches to sequence-related AI tasks.

In summary, the paper presents a comprehensive framework for harnessing path signatures in scalable ML models, melding theoretical insights with practical application methodologies and setting the stage for future advances in sequence modeling.