In many prediction and decision problems, the relevant inputs are path-valued covariates rather than static feature vectors. This paper studies asymptotic theory and empirical applications for path regression using signatures. We first establish \(L^2\) approximation rates for truncated signature representations. We prove a minimax-optimal approximation rate over a class of smooth coefficient functionals of observable It\^{o} diffusions. Building on this approximation theory, we then develop asymptotic results for three signature-based learning procedures: Signature-OLS, Signature-LASSO, and Signature-Logistic. These results establish asymptotic normality for least-squares path regression, sparse recovery for high-dimensional signature regression, and latent-score consistency for binary-response classification. Extensive empirical studies cover three real-data applications: foreign-exchange realized-volatility forecasting from intraday price paths, battery end-of-life prediction from early HPPC pulse paths, and epileptic seizure detection from short EEG windows. The empirical results show that signatures provide informative representations of path-valued covariates relative to handcrafted features.

Mr Oluwadamilola (Dami) Fasina will talk about; 'Learning with tensor paraproducts'

We discuss computational (Neural FIM) and analytical (tensor paraproducts) tools for learning structure of sets. In the first situation we focus on learning the metric amongst elements of a statistical manifold. To do so, we design a neural network which enables one to compute the Fisher information metric (FIM), so long the Jensen-Shannon divergences amongst probability distributions on the statistical manifold are preserved during training. In the second situation we focus on analyzing the structure of function compositions through separation of its low and high frequency components. This is accomplished by elaborating on J.M. Bony’s celebrated work on paraproducts by discretizing and allocating distinct scaling parameters along each dimension of the support of a function composition (with a prescribed regularity), permitting finer analytical control. A consequence of this extension is highlighted with a discussion of the regularity gains of kernels of integral operators.