Estimating properties of discrete distributions is a fundamental problem in statistical learning.

Let k be the collection of distributions over the alphabet[k]:= {1,2,...,k}, and [k] be the set of finite sequences over[k].

In previous work [12, 13], a method was introduced that leverages compressive sensing techniques tofind thefewest taxa thatfitsthefrequencyofshort sequences ofnucleotides (i.e., k-mers) in a given sample. Consider, for instance, an environment/sample made of s bacterial species but where two of them are almost identical: one would wish to say that the concentration vector is almost(s 1)-sparse rather thans-sparse!

Optimal Transport (OT, also known as the Wasserstein distance) is a popular metric for comparing probability distributions and has been successfully used in many machine-learning applications.In the semi-discrete $2$-Wasserstein problem, we wish to compute the cheapest way to transport all the mass from a continuous distribution $\mu$ to a discrete distribution $\nu$ in $\mathbb{R}^d$ for $d\ge 1$, where the cost of transporting unit mass between points $a$ and $b$ is $d(a,b)=||a-b||^2$. When both distributions are discrete, a simple combinatorial framework has been used to find the exact solution (see e.g.

Estimating properties of discrete distributions is a fundamental problem in statistical learning.

This field has experienced a surge in activity as researchers have produced numerous computational tools to analyze such data sets.

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