Seminaria

Lasso and ridge regression are well established and successful models for variance reduction and, for the lasso, variable selection. However, they come with a disadvantage of an increased bias in the estimator. In this seminar, I will talk about our general method that learns individual weights for each term in the regularisation penalty (e.g. lasso or ridge) with the goal to reduce the bias. To bound the amount of freedom for the model to choose the weights, a new regularisation term, that imposes a cost for choosing small weights, is introduced. If the form of this term is chosen wisely, the apparent doubling of the number of parameters vanishes, by means of solving for the weights in terms of the parameter estimates. We show that these estimators potentially keep the original estimators’ fundamental properties and experimentally verify that this can indeed reduce bias.

Quasiconvex subgroups are basic building blocks of hyperbolic groups, and relatively quasiconvex subgroups play a similar role in relatively hyperbolic groups. If $Q$ and $R$ are relatively quasiconvex subgroups of a relatively hyperbolic group $G$ then the intersection $Q \cap R$ will also be relatively quasiconvex, but the join $\langle Q,R \rangle$ may not be. I will discuss criteria for the existence of finite index subgroups $Q’ \leqslant_f Q$ and $R’ \leqslant_f R$ such that the ``virtual join’’ $\langle Q’, R’ \rangle$ is relatively quasiconvex. This is closely related to separability properties of $G$ and I will present applications to limit groups, Kleinian groups and fundamental groups of graphs of free groups with cyclic edge groups. The talk will be based on joint work with Lawk Mineh.

We review basic facts about the classical Laplace distribution and its asymmetric generalization. Both distributions naturally arise in connection with random summation and quantile regression settings, and offer an attractive and flexible alternative to the normal (Gaussian) distribution in a variety of setting, where the assumptions of symmetry and short tail are too restrictive. The growing popularity of the Laplace-based models in recent years is due to their fundamental properties, which include a sharp peak at the mode, heavier than Gaussian tails, existence of all moments, infinite divisibility, and, most importantly, random stability and approximation of geometric sums. Since the latter arise quite naturally, these distributions provide useful models in diverse areas, such as biology, economics, engineering, finance, geosciences, and physics. We review fundamental properties of these models, which give insight into their applicability in these areas, and discuss their various extensions, including Laplace-based time series and stochastic processes. This is a joint work with K. Podgorski.