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We compute the Wiener chaos decomposition of the signature for a class of Gaussian processes, which contains fractional Brownian motion (fBm) with Hurst parameter H ∈ (1 / 4 , 1) . At level 0, our result yields an expression for the expected signature of such processes, which determines their law (Chevyrev and Lyons in Ann Probab 44(6):4049–4082, 2016). In particular, this formula simultaneously extends both the one for 1 / 2 < H -fBm (Baudoin and Coutin in Stochast Process Appl 117(5):550–574, 2007) and the one for Brownian motion ( H = 1 / 2 ) (Fawcett 2003), to the general case H > 1 / 4 , thereby resolving an established open problem. Other processes studied include continuous and centred Gaussian semimartingales. [ABSTRACT FROM AUTHOR]

Based on the recent development of the framework of Volterra rough paths (Harang and Tindel in Stoch Process Appl 142:34–78, 2021), we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with H > 1 2 and for the standard Brownian motion. The Volterra kernel k(t, s) is allowed to be singular, and behaving similar to | t - s | - γ for some γ ≥ 0 . The construction is done in both the Stratonovich and Itô senses. It is based on a modified Garsia–Rodemich–Romsey lemma which is of interest in its own right, as well as tools from Malliavin calculus. A discussion of challenges and potential extensions is provided. [ABSTRACT FROM AUTHOR]

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver's moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in R d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in R d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations. [ABSTRACT FROM AUTHOR]

We investigate the functioning of a classifying biological neural network from the perspective of statistical learning theory, modelled, in a simplified setting, as a continuous-time stochastic recurrent neural network (RNN) with the identity activation function. In the purely stochastic (robust) regime, we give a generalisation error bound that holds with high probability, thus showing that the empirical risk minimiser is the best-in-class hypothesis. We show that RNNs retain a partial signature of the paths they are fed as the unique information exploited for training and classification tasks. We argue that these RNNs are easy to train and robust and support these observations with numerical experiments on both synthetic and real data. We also show a trade-off phenomenon between accuracy and robustness. [ABSTRACT FROM AUTHOR]