论文标题
在$ \ mathbb {r}^d $中排列和未链接单调回归：一种基于混合建模和最佳传输的方法
Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport
论文作者
论文摘要
假设我们在$ \ mathbb {r}^d $中有一个回归问题y和$ \ mathbb {r}^d $ in $ \ mathbb {r}^d $，对于$ d \ geq 1 $。在置换或未链接回归中，我们可以访问X和Y上的单独的无序数据，而不是在通常的回归中（x，y） - pairs上的数据。到目前为止，在文献中，$ d = 1 $引起了人们的关注，例如，参见Rigollet和Weed的最新论文[信息与推理，8，619--717]和Balabdaoui等。 [J。马赫。学习。 Res。，22（172），1--60]。在本文中，我们考虑使用$ d \ geq 1 $的一般多元设置。我们表明，回归函数周期性单调性的概念足以在排列/未链接回归模型中识别和估计。我们研究了置换回归设置中的排列恢复，并根据基于Kiefer-Wolfowitz [Ann。数学。 Statist。，27，887--906]非参数最大似然估计器和最佳运输理论的技术。我们在相关的均方根上为高斯噪声提供明确的上限。与以前在情况下的工作一样，$ d = 1 $，置换/未链接设置涉及在基础反卷积问题中加生根的缓慢（对数）收敛速率。数值研究证实了我们的理论分析，并表明所提出的方法至少与上述先前工作中的方法相提并论$ d = 1 $，同时在计算复杂性方面实现了大量降低。
Suppose that we have a regression problem with response variable Y in $\mathbb{R}^d$ and predictor X in $\mathbb{R}^d$, for $d \geq 1$. In permuted or unlinked regression we have access to separate unordered data on X and Y, as opposed to data on (X,Y)-pairs in usual regression. So far in the literature the case $d=1$ has received attention, see e.g., the recent papers by Rigollet and Weed [Information & Inference, 8, 619--717] and Balabdaoui et al. [J. Mach. Learn. Res., 22(172), 1--60]. In this paper, we consider the general multivariate setting with $d \geq 1$. We show that the notion of cyclical monotonicity of the regression function is sufficient for identification and estimation in the permuted/unlinked regression model. We study permutation recovery in the permuted regression setting and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz [Ann. Math. Statist., 27, 887--906] nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error for Gaussian noise. As in previous work on the case $d = 1$, the permuted/unlinked setting involves slow (logarithmic) rates of convergence rooting in the underlying deconvolution problem. Numerical studies corroborate our theoretical analysis and show that the proposed approach performs at least on par with the methods in the aforementioned prior work in the case $d = 1$ while achieving substantial reductions in terms of computational complexity.
