B. Drabant | Searcharxiv

arXiv2022

Regression based thresholds in principal loading analysis

Principal loading analysis is a dimension reduction method that discards variables which have only a small distorting effect on the covariance matrix. As a special case, principal loading analysis discards variables that are not correlated with the remaining ones. In multivariate linear regression on the other hand, predictors that are neither correlated with both the remaining predictors nor with the dependent variables have a regression coefficients equal to zero. Hence, if the goal is to select a number of predictors, variables that do not correlate are discarded as it is also done in principal loading analysis. That both methods select the same variables occurs not only for the special case of zero correlation however. We contribute conditions under which both methods share the same variable selection. Further, we extend those conditions to provide a choice for the threshold in principal loading analysis which only follows recommendations based on simulation results so far.

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arXiv1998

Actions of Multiplier Hopf Algebras

For an action $α$ of a group $G$ on an algebra $R$ (over $\Bbb C$), the crossed product $R\times_αG$ is the vector space of $R$-valued functions with finite support in $G$, together with the twisted convolution product given by $$(ξη)(p) = \sum_{q \in G} ξ(q) α_q (η(q^{-1}p))$$ where $p\in G$. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra $A$ on an algebra $R$, it is possible to make the tensor product $R\ot A$ into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by $R# A$. In the group case, the action $α$ of $G$ on $R$ yields an action of the group algebra $\Bbb C G$ as a Hopf algebra on $R$ and the crossed $R\times_αG$ coincides with the smash product $R# \Bbb C G$. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf algebras to formulate this duality theorem. We prove a result in the case of an algebraic quantum group and its dual. The more general case is only stated and will be proven in a separate paper on coactions. These duality theorems for actions are substantial generalizations of the corresponding theorem for Hopf algebras. Also the techniques that are used here to prove this result are slightly different and simpler.

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