Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separable. This is true in arbitrary dimension, and for families of distributions generated through perturbations of shifts and scalings of a fixed distribution. We also prove conditions under which the $L^2$ distance of the LOT embedding between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only compute $N$ optimal transport maps to define the $N^2$ pairwise distances between $N$ distributions. We demonstrate the benefits of LOT on a number of distribution classification problems.

We propose two adaptations of linear Hermite subdivisions schemes to operate on manifold-valued data based on a Log-exp approach and on projection, respectively. Furthermore, we introduce a new proximity condition, which bounds the difference between a linear Hermite subdivision scheme and its manifold-valued analogue. Verification of this condition gives the main result: The manifold-valued Hermite subdivision scheme constructed from a C 1-convergent linear scheme is also C 1 , if certain technical conditions are met.

We study many properties of level-dependent Hermite subdivision, focusing on schemes preserving polynomial and exponential data. We specifically consider interpolatory schemes, which give rise to level-dependent multiresolution analyses through a prediction-correction approach. A result on the decay of the associated multiwavelet coefficients, corresponding to an uniformly continuous and differentiable function, is derived. It makes use of the approximation of any such function with a generalized Taylor formula expressed in terms of polynomials and exponentials.

In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order d allows for d factorizations of the subdivision operator with respect to the Gregory operators: A new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the d-th factorization provides a "convergence from contractivity" method for showing C d -convergence of the associated Hermite subdivision scheme. The power of our factorization framework lies in the reduction of computational effort for large d: In order to prove C d -convergence, up to now, d factorization steps were needed, while our method requires only one step, independently of d. Furthermore, in this paper, we show by an example that the spectral condition is not equivalent to the reproduction of polynomials.

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