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- ItemStability of deep neural networks via discrete rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Friz, Peter; Tapia, NikolasUsing rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.
- ItemUnified signature cumulants and generalized Magnus expansions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Friz, Peter; Hager, Paul; Tapia, NikolasThe signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative (``Hausdorff") variation of Riccati's equation. Many examples are given.
- ItemTransport and continuity equations with (very) rough noise(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bellinger, Carlo; Djurdjevac, Ana; Friz, Peter; Tapia, NikolasExistence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.