Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop
improved algorithms for matrix problems that arise in scientific computing,
data science, machine learning, etc. Determinantal Point Processes (DPPs), a
seemingly unrelated topic in pure and applied mathematics, is a class of
stochastic point processes with probability distribution characterized by
sub-determinants of a kernel matrix. Recent work has uncovered deep and
fruitful connections between DPPs and RandNLA which lead to new guarantees and
improved algorithms that are of interest to both areas. We provide an overview
of this exciting new line of research, including brief introductions to RandNLA
and DPPs, as well as applications of DPPs to classical linear algebra tasks
such as least squares regression, low-rank approximation and the Nystr\"om
method. For example, random sampling with a DPP leads to new kinds of unbiased
estimators for least squares, enabling more refined statistical and inferential
understanding of these algorithms; a DPP is, in some sense, an optimal
randomized algorithm for the Nystr\"om method; and a RandNLA technique called
leverage score sampling can be derived as the marginal distribution of a DPP.
We also discuss recent algorithmic developments, illustrating that, while not
quite as efficient as standard RandNLA techniques, DPP-based algorithms are
only moderately more expensive.

本研究概述了随机数值线性代数 (Randomized Numerical Linear Algebra) 和行列式点过程 (Determinantal Point Processes) 两个看似不相关的数学领域之间新的深入而有益的联系，以及这些联系带来的新保证和改进的算法，并且介绍了 DPP 对于经典线性代数任务，如最小二乘回归、低秩逼近和 Nystrom 方法的应用。

随机数值线性代数中的行列式点过程

Determinantal Point Processes in Randomized Numerical Linear Algebra

Linguistic typology studies the range of structures present in human
language. The main goal of the field is to discover which sets of possible
phenomena are universal, and which are merely frequent. For example, all
languages have vowels, while most---but not all---languages have an /u/ sound.
In this paper we present the first probabilistic treatment of a basic question
in phonological typology: What makes a natural vowel inventory? We introduce a
series of deep stochastic point processes, and contrast them with previous
computational, simulation-based approaches. We provide a comprehensive suite of
experiments on over 200 distinct languages.

本文首次利用基于概率的深度随机点过程解决音韵类型学中的基础问题：什么构成了自然元音库，作者通过超过 200 种不同语言的广泛实验以及与之前的计算机模拟方法的对比，提出了全面的理论。