**Project Heads**

*Christian Bayer, Peter Friz*

**Project Members**

Nikolas Tapia (TU / WIAS)

**Project Duration**

01.01.2019 – 31.12.2020

**Located at**

TU Berlin / WIAS

**Project Webpages**

**Selected Publications
**

**C. Bayer**,**P. K. Friz**,**N. Tapia**.*Robustness of Residual Networks via Rough Path techniques*. (2020) WIAS preprint 2732.- C. Bellingeri, A. Djurdjevac,
**P. K. Friz**,**N. Tapia**.*Transport and continuity equations with (very) rough noise*. (2020). arXiv 2002.10432 [math.AP]. - E. Celledoni, P. E. Lystad,
**N. Tapia**.*Signatures in Shape Analysis: an Efficient Approach to Motion Identification*. (2019) In*Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science vol. 11712*. F. Nielsen, F. Barbaresco eds. doi: 10.1007/978-3-030-26980-7_3. - J. Diehl, K. Ebrahimi-Fard,
**N. Tapia**.*Time-warping invariants of multidimensional time series*. Acta Appl. Math 171(1):265–290 (2020). doi: 10.1007/s10440-020-00333-x. - J. Diehl, K. Ebrahimi-Fard,
**N. Tapia**.*Iterated-sums signature, quasisymmetric functions and time series analysis*. Sém. Lothar. Combin. 84B (2020), Article 84B.86, 12 pp. - J. Diehl, K. Ebrahimi-Fard,
**N. Tapia**.*Generalized iterated-sums signatures*. (2020) arXiv:2012.04597 [math.RA]. - J. Diehl, R. Preiß, M. Ruddy,
**N. Tapia**.*The moving frame method for iterated-integrals: orthogonal invariants*. (2020) arXiv:2012.05880 [math.DG] **P. K. Friz**, M. Hairer,*A Course on Rough Paths*. 2nd ed. Universitext (2020). Springer.- P. Hager,
**P. K. Friz**,**N. Tapia**.*Unified signature cumulants and generalized Magnus expansions*. (2021) arXiv:2102.03345 [math.PR]. **N. Tapia**, L. Zambotti.*The geometry of the space of branched Rough Paths*. Proc. London Math. Soc. 121(2):220–251 (2020). doi: 10.1112/plms.12311.

**Selected Pictures
**

The weights are taken by an actual trained network from He et al.

One can observe that since the weights turn out to have high 1-variation, our analysis yield sharper control for higher values of p.

The picture shows that even in this case choosing p>1 can improve a priori bounds.

One observes that due to the high variability of the trained weights, a priori knowledge of the deviation is better if we are allowed to choose p>1.

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