Conference Agenda

Geometric representations for multivariate extremes, derived from the shapes of scaled sample clouds and their so-called limit sets, are becoming an increasingly popular modelling tool. Recent work has shown that limit sets connect several existing extremal dependence concepts and offer a high degree of practical utility for inference of multivariate extremes. However, existing geometric approaches are limited to low-dimensional settings, and some of these techniques make strong assumptions about the form of the limit set. In this talk, we introduce DeepGauge - the first deep learning approach for limit set estimation. By leveraging the predictive power and computational scalability of neural networks, we construct asymptotically-justified yet highly flexible semi-parametric models for extremal dependence. Unlike existing techniques, DeepGauge can be applied in high-dimensional settings and does not impose any assumptions on the resulting limit set estimates. Moreover, we also introduce a range of novel theoretical results pertaining to the geometric framework and our limit set estimator. We showcase the efficacy of our deep approach by modelling the complex extremal dependence between metocean variables sampled from the North Sea.

The notion of group invariance helps neural networks recognize patterns and features under geometric transformations. Indeed, it has been shown that group invariance can largely improve deep learning performances in practice, where such transformations are very common. This research studies affine invariance on continuous-domain convolutional neural networks. While existing research only considers isometric invariance or similarity invariance so far, we focus on the full structure of affine transforms generated by the generalized linear group $\mathrm{GL}_2(\mathbb{R})$. We introduce a criterion to assess the similarity of two input signals under affine transformations. Then, we investigate the convolution of lifted signals and compute the corresponding integration over $G_2$ (the affine Lie group $\mathbb{R}^2 \ltimes \mathrm{GL}_2(\mathbb{R})$). Our research could eventually extend the scope of geometrical transformations that practical deep-learning pipelines can handle.

Raster data cubes collect measurements of a spatio-temporal random field at regularly spaced points and equidistant times. We design an ensemble forecasting methodology for cubes generated by an influenced mixed moving average field with finite second-order moments. The latter does not have, in general, a known predictive distribution. We then use the setting and the causal embedding discussed in [1] and employ a deep (stochastic) neural network to determine ensemble forecasts. The parameter's distribution of the network is assumed to be Gaussian and determined by minimizing the PAC-Bayesian bound for $\theta$-lex weakly dependent data proven in [1].

[1] I.V. Curato, O. Furat, L. Proietti and B. Ströh, (2024), Mixed moving average field guided learning for spatio-temporal data, arXiv:2301.00736.