Final Report Summary - ESIG (Creating rigorous mathematical and computational tools that can summarise high dimensional data streams in terms of their effects.)
Newton introduced calculus to model the interactions of systems, and introduced simple examples of controlled differential systems. However, as the number of interacting elements increases the trajectories become “rough” and too complex to model on normal scales using differential calculus. In 1942 Itô introduced stochastic calculus and accommodated Brownian forcing. Rough path theory generalized classical deterministic calculus to provide a systematic technology for modelling the interactions between complex evolving systems rich enough to capture Itô’s approach and more. Key to the development of Rough Path theory was the clean minimal identification of the “missing terms” in the description of a complex stream x needed to describe the stream well enough to predict its effects. This mathematically precise description or transform of the stream, the signature, was the focus of this ERC advanced grant project.
The project had two clear overall objectives. The first was to better understand the signature as a mathematical object; the second was to develop its use as a practical tool for applications. There has been strategic progress (completely exceeding the expectations of the PI) in both directions. Fundamental results explain how the signature is a faithful description of a data stream(Boedihardjo, H., Geng, X., Lyons, T. and Yang, D., 2016. The signature of a rough path: uniqueness. Advances in Mathematics, 293). Over the period of the project, the signature has become an effective tool in engineering data science for the analysis of complex multimodal data streams in in the recognition of Chinese handwriting on mobile devices (it has recognised billions of gestures and the app has been downloaded over a million times). The PI collaborated with vision experts, to use this approach with deep learning to classify actions in video material and obtain the state of the art (Yang, W., Lyons, T., Ni, H., Schmid, C., Jin, L. and Chang, J., 2017. Leveraging the Path Signature for Skeleton-based Human Action Recognition. arXiv preprint arXiv:1707.03993.).
The project spans from the purest maths to innovative approaches to describing data streams. It has been very exciting for the PI. Without the support of the ERC, the brilliant ECR colleagues it provided the PI, and the freedom it gave to the PI to be independent, the fundamental progress in understanding the signature as a transform and the momentum into applications simply would not have happened.
The project had two clear overall objectives. The first was to better understand the signature as a mathematical object; the second was to develop its use as a practical tool for applications. There has been strategic progress (completely exceeding the expectations of the PI) in both directions. Fundamental results explain how the signature is a faithful description of a data stream(Boedihardjo, H., Geng, X., Lyons, T. and Yang, D., 2016. The signature of a rough path: uniqueness. Advances in Mathematics, 293). Over the period of the project, the signature has become an effective tool in engineering data science for the analysis of complex multimodal data streams in in the recognition of Chinese handwriting on mobile devices (it has recognised billions of gestures and the app has been downloaded over a million times). The PI collaborated with vision experts, to use this approach with deep learning to classify actions in video material and obtain the state of the art (Yang, W., Lyons, T., Ni, H., Schmid, C., Jin, L. and Chang, J., 2017. Leveraging the Path Signature for Skeleton-based Human Action Recognition. arXiv preprint arXiv:1707.03993.).
The project spans from the purest maths to innovative approaches to describing data streams. It has been very exciting for the PI. Without the support of the ERC, the brilliant ECR colleagues it provided the PI, and the freedom it gave to the PI to be independent, the fundamental progress in understanding the signature as a transform and the momentum into applications simply would not have happened.