In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into
 NSFPAR ID:
 10359677
 Date Published:
 Journal Name:
 Information and Inference: A Journal of the IMA
 ISSN:
 20498772
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a prespecified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.$$L^2$$ ${L}^{2}$ 
Summary Sequential Monte Carlo algorithms are widely accepted as powerful computational tools for making inference with dynamical systems. A key step in sequential Monte Carlo is resampling, which plays the role of steering the algorithm towards the future dynamics. Several strategies have been used in practice, including multinomial resampling, residual resampling, optimal resampling, stratified resampling and optimal transport resampling. In onedimensional cases, we show that optimal transport resampling is equivalent to stratified resampling on the sorted particles, and both strategies minimize the resampling variance as well as the expected squared energy distance between the original and resampled empirical distributions. For general $d$dimensional cases, we show that if the particles are first sorted using the Hilbert curve, the variance of stratified resampling is $O(m^{(1+2/d)})$, an improvement over the best previously known rate of $O(m^{(1+1/d)})$, where $m$ is the number of resampled particles. We show that this improved rate is optimal for ordered stratified resampling schemes, as conjectured in Gerber et al. (2019). We also present an almostsure bound on the Wasserstein distance between the original and Hilbertcurveresampled empirical distributions. In light of these results, we show that for dimension $d>1$ the mean square error of sequential quasiMonte Carlo with $n$ particles can be $O(n^{14/\{d(d+4)\}})$ if Hilbert curve resampling is used and a specific lowdiscrepancy set is chosen. To our knowledge, this is the first known convergence rate lower than $o(n^{1})$.more » « less

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree biLipschitz embeds in some Euclidean space, with the ambient dimension and the biLipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244].more » « less

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