Related papers: Bi-invariant Two-Sample Tests in Lie Groups for Sh…

Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by…

Let $K$ denote a simply connected compact Lie group and let $G=K^{\mathbb C}$, the complexification. It is known that there exists an $LK$ bi-invariant probability measure on a natural hyperfunction completion of the complex loop group…

A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an…

Deep neural two-sample tests have recently shown strong power for detecting distributional differences between groups, yet their black-box nature limits interpretability and practical adoption in biomedical analysis. Moreover, most existing…

Hotelling's $T^2$-test for the mean of a multivariate normal distribution is one of the triumphs of classical multivariate analysis. It is uniformly most powerful among invariant tests, and admissible, proper Bayes, and locally and…

Hotelling's $T^2$ test is a classical approach for discriminating the means of two multivariate normal samples that share a population covariance matrix. Hotelling's test is not ideal for high-dimensional samples because the eigenvalues of…

We extend the work of Robinson and Turner to use hypothesis testing with persistence homology to test for measurable differences in shape between point clouds from three or more groups. Using samples of point clouds from three distinct…

We formulate and prove a twofold generalisation of Lie's second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with…

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…

Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups,…

Repeated observations have become increasingly common in biomedical research and longitudinal studies. For instance, wearable sensor devices are deployed to continuously track physiological and biological signals from each individual over…

We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of…

In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…

We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an…

A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the…

Symmetry principles underlie and guide scientific theory and research, from Curie's invariance formulation to modern applications across physics, chemistry, and mathematics. Building on a recent matrix Lie group measurement model, this…

Two-sample tests evaluate whether two samples are realizations of the same distribution (the null hypothesis) or two different distributions (the alternative hypothesis). We consider a new setting for this problem where sample features are…

Hotelling's T-squared test is a classical tool to test if the normal mean of a multivariate normal distribution is a specified one or the means of two multivariate normal means are equal. When the population dimension is higher than the…

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $…

In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional…