We study the completeness properties of the set of wavelets in L2(R). It is well-known that this set is not closed in the unit ball of L2(R). However, if one considers the metric inherited as a subspace (in the Fourier transform side) of L2(R, dξ) ∩ L(R∗, dξ |ξ| ), we do obtain a complete metric space.

Event-related potential (ERP) data in monolingual German speakers have shown that sentential metric expectancy violations elicit a biphasic ERP pattern consisting of an anterior negativity and a posterior positivity (P600). This pattern is comparable to that elicited by syntactic violations. However, proficient French late learners of German do not detect violations of metric expectancy in Germ...

This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L2-Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the L2-Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the L...

Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality w...

Several recent papers have treated the latent space of deep generative models, e.g., GANs or VAEs, as Riemannian manifolds. The argument is that operations such as interpolation are better done along geodesics that minimize path length not in the latent space but in the output space of the generator. However, this implicitly assumes that some simple metric such as L2 is meaningful in the output...

The solution to (3) through initial point φ0 is given by φτ = eφ0. Here e−iÂτ denotes the one-parameter group of unitary transformations generated by −iÂ, as described by Stone’s theorem. It follows that the integral curve through φ0 ∈ S2 will stay on the sphere. One concludes that, modulo the domain issues, the restriction of the vector field Aφ to the sphere S2 is a vector field on the sphere...

In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used for the nonlinear filtering problem, in relationship also to earlier project...