Sorting an array is a fundamental routine in machine learning, one that is
used to compute rank-based statistics, cumulative distribution functions
(CDFs), quantiles, or to select closest neighbors and labels. The sorting
function is however piece-wise constant (the sorting permutation of a vector
does not change if the entries of that vector are infinitesimally perturbed)
and therefore has no gradient information to back-propagate. We propose a
framework to sort elements that is algorithmically differentiable. We leverage
the fact that sorting can be seen as a particular instance of the optimal
transport (OT) problem on $\mathbb{R}$, from input values to a predefined array
of sorted values (e.g. $1,2,\dots,n$ if the input array has $n$ elements).
Building upon this link , we propose generalized CDFs and quantile operators by
varying the size and weights of the target presorted array. Because this
amounts to using the so-called Kantorovich formulation of OT, we call these
quantities K-sorts, K-CDFs and K-quantiles. We recover differentiable
algorithms by adding to the OT problem an entropic regularization, and
approximate it using a few Sinkhorn iterations. We call these operators
S-sorts, S-CDFs and S-quantiles, and use them in various learning settings: we
benchmark them against the recently proposed neuralsort [Grover et al. 2019],
propose applications to quantile regression and introduce differentiable
formulations of the top-k accuracy that deliver state-of-the art performance.

提出了一种基于最优传输问题的算法可微分排序框架，包括基于 Kantorovich 公式的算子对于 CDFs 和 quantiles 的推广，以及基于这一框架在量子回归和神经网络中的应用

可微排名与排序的最优输运算法

Differentiable Ranks and Sorting using Optimal Transport

In this work in progress, we study the asymptotic behaviour of the
$p$-quantile of the Beta distribution, i.e. the quantity $q$ defined implicitly
by $\int_0^q t^{a - 1} (1 - t)^{b - 1} \text{d} t = p B (a, b)$, as a function
of the first parameter $a$. In particular, we derive asymptotic expansions of
and $q$ and its logarithm at $0$ and $\infty$. Moreover, we provide some
relations between Bell and N{\o}rlund Polynomials, a generalisation of
Bernoulli numbers. Finally, we provide Maple and Sage algorithms for computing
the terms of the asymptotic expansions.

本文探讨 Beta 分布的 p - 分位数的渐近行为及与 Nørlund 多项式的关系，提供了算法来计算其渐近展开式。

Beta 分布的倒数的渐近展开

Asymptotic expansions of the inverse of the Beta distribution

Expected Shortfall (ES) in several variants has been proposed as remedy for
the defi-ciencies of Value-at-Risk (VaR) which in general is not a coherent
risk measure. In fact, most definitions of ES lead to the same results when
applied to continuous loss distributions. Differences may appear when the
underlying loss distributions have discontinuities. In this case even the
coherence property of ES can get lost unless one took care of the details in
its definition. We compare some of the definitions of Expected Shortfall,
pointing out that there is one which is robust in the sense of yielding a
coherent risk measure regardless of the underlying distributions. Moreover,
this Expected Shortfall can be estimated effectively even in cases where the
usual estimators for VaR fail.
Key words: Expected Shortfall; Risk measure; worst conditional expectation;
tail con-ditional expectation; value-at-risk (VaR); conditional value-at-risk
(CVaR); tail mean; co-herence; quantile; sub-additivity.

比较预期损失（ES）的定义，指出有一种预期损失是健壮的，并且即使在具有不连续性的基础损失分布中也能保持一致的风险度量，而且即使在 Value-at-Risk (VaR) 估计方法失败的情况下，该损失也可以有效地估计。