Uncoupled isotonic regression via minimum Wasserstein deconvolution
Uncoupled isotonic regression via minimum Wasserstein deconvolution
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DOI：
10.1093/imaiai/iaz006
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
P. Rigollet;J. Weed
中科院分区：
文献类型：
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作者：
P. Rigollet;J. Weed
保序回归是形状约束估计中的一个标准问题，其目标是从独立对$(xi，yi)$中估计一个未知的非递减回归函数$f$，其中${mathbb{E}}[yi]=f(Xi)，i=1，ldots n$。虽然这个问题在统计和计算上都很好地理解了，但对它的非耦合对应问题却知之甚少，其中只给出了无序集合${x_1，\ldots，x_n$和${y_1，\ldots，y_n$。在这项工作中，我们利用最优传输理论的工具，在弱矩条件下推导出$y_i上的极小极大速率，并给出了一个实现最优速率的有效算法。上界和下界都使用矩匹配变元，这些变元也与学习分布和反卷积的混合有关。
Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown non-decreasing regression function $f$ from independent pairs $(x_i, y_i)$ where ${\mathbb{E}}[y_i]=f(x_i), i=1, \ldots n$. While this problem is well understood both statistically and computationally, much less is known about its uncoupled counterpart, where one is given only the unordered sets $\{x_1, \ldots , x_n\}$ and $\{y_1, \ldots , y_n\}$. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on $y_i$ and to give an efficient algorithm achieving optimal rates. Both upper and lower bounds employ moment-matching arguments that are also pertinent to learning mixtures of distributions and deconvolution.