  The path signature, a mathematically principled and universal feature of
sequential data, leads to a performance boost of deep learning-based models in
various sequential data tasks as a complimentary feature. However, it suffers
from the curse of dimensionality when the path dimension is high. To tackle
this problem, we propose a novel, trainable path development layer, which
exploits representations of sequential data with the help of finite-dimensional
matrix Lie groups. We also design the backpropagation algorithm of the
development layer via an optimisation method on manifolds known as
trivialisation. Numerical experiments demonstrate that the path development
consistently and significantly outperforms, in terms of accuracy and
dimensionality, signature features on several empirical datasets. Moreover,
stacking the LSTM with the development layer with a suitable matrix Lie group
is empirically proven to alleviate the gradient issues of LSTMs and the
resulting hybrid model achieves the state-of-the-art performance.
