Path Development Network with Finite-dimensional Lie Group Representation
Hang Lou, Siran Li, Hao Ni
TL;DR
This paper tackles the dimensionality and data-adaptivity limitations of path-signature features by introducing a trainable path development layer that maps sequential data into a finite-dimensional matrix Lie group $G$ with Lie algebra $\mathfrak{g}$. By solving a Cartan development-like recurrence $z_{n+1}=z_n\exp(M_\theta(x_{n+1}-x_n))$, trained via manifold-aware backpropagation, the approach yields universal, characteristic features with enhanced stability for irregular time series. Empirical results across speech commands, character trajectories, sequential images, and non-Euclidean dynamics show that the development layer often surpasses signature baselines and significantly improves the performance and robustness of hybrid models such as LSTM+DEV, while maintaining compact parameter counts. The work represents a practical, scalable framework for time-series modelling on non-Euclidean spaces and suggests broad applicability to physics-inspired and geometry-aware learning tasks.
Abstract
Signature, lying at the heart of rough path theory, is a central tool for analysing controlled differential equations driven by irregular paths. Recently it has also found extensive applications in machine learning and data science as a mathematically principled, universal feature that boosts the performance of deep learning-based models in sequential data tasks. It, nevertheless, suffers from the curse of dimensionality when paths are high-dimensional. We propose a novel, trainable path development layer, which exploits representations of sequential data through finite-dimensional Lie groups, thus resulting in dimension reduction. Its backpropagation algorithm is designed via optimization on manifolds. Our proposed layer, analogous to recurrent neural networks (RNN), possesses an explicit, simple recurrent unit that alleviates the gradient issues. Our layer demonstrates its strength in irregular time series modelling. Empirical results on a range of datasets show that the development layer consistently and significantly outperforms signature features on accuracy and dimensionality. The compact hybrid model (stacking one-layer LSTM with the development layer) achieves state-of-the-art against various RNN and continuous time series models. Our layer also enhances the performance of modelling dynamics constrained to Lie groups. Code is available at https://github.com/PDevNet/DevNet.git.