# Learning with Expected Signatures: Theory and Applications

arXiv: https://arxiv.org/abs/2505.20465 · HTML: https://mlsanity.com/paper/2505.20465/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2025.

## Built on by (most cited first, 3 of 3)

- [PaperFit: Vision-in-the-Loop Typesetting Optimization for Scientific Documents](https://mlsanity.com/paper/2605.10341/) (2026)
- [Anticipatory Reinforcement Learning: From Generative Path-Laws to Distributional Value Functions](https://mlsanity.com/paper/2604.04662/) (2026)
- [Generative Path-Law Jump-Diffusion: Sequential MMD-Gradient Flows and Generalisation Bounds in Marcus-Signature RKHS](https://mlsanity.com/paper/2604.05008/) (2026)

## Abstract (arXiv)

The expected signature maps a collection of data streams to a lower dimensional representation, with a remarkable property: the resulting feature tensor can fully characterize the data generating distribution. This "model-free" embedding has been successfully leveraged to build multiple domain-agnostic machine learning (ML) algorithms for time series and sequential data. The convergence results proved in this paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value, allowing for a more complete probabilistic interpretation of expected signature-based ML methods. Moreover, when the data generating process is a martingale, we suggest a simple modification of the expected signature estimator with significantly lower mean squared error and empirically demonstrate how it can be effectively applied to improve predictive performance.