固定基数子集上的非参数回归：极小极大风险与随机设计效应
Nonparametric Regression on Fixed-Cardinality Subsets:Minimax Risk and Random-Design Effects
- Sorbonne Université（索邦大学）
- Institut universitaire de France（法国高等研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究固定基数子集作为协变量的非参数回归，利用Johnson图调和分解定义Johnson-Sobolev尺度，推导调和投影风险界并建立极小极大最优性，同时分析随机设计下的秩亏效应，提出中心化补全估计器以实现常数最优。
AI中文摘要：
我们研究以子集为协变量的回归问题。响应变量是输入子集的未知函数，观测值由均匀采样的子集（每个子集恰好包含来自大小为d的基集合中的k个项）处的带噪声评估组成。该问题出现在组合筛选、捆绑偏好建模以及其他结果依赖于规定大小集合的场景中。固定基数约束耦合了成员坐标，因此标准乘积域上的交互和平滑性概念不能直接沿用。我们利用Johnson图的调和分解来定义内在的Johnson-Sobolev尺度，并确定低阶交互空间的精确维度。我们推导了调和投影的有限样本风险界，并在Johnson-Sobolev能量预算相对于噪声方差受控时建立了非渐近极小极大特征。我们还证明，当经验Gram矩阵条件数足够好时，稳定的最小二乘法消除了经验投影中依赖于信号的波动。对于任意信噪比，秩亏分析量化了随机设计未能识别的分量。综合来看，秩亏分析和中心化补全估计器在min(k,d-k)和平滑阶数固定时产生了常数意义上匹配的极小极大界。补充材料中的数值实验展示了不同交互轮廓下的估计，并区分了观测噪声、随机设计波动和不完全覆盖的影响。
英文摘要：
We study regression with subsets as covariates. The response is an unknown function of the input subset, and observations consist of noisy evaluations at uniformly sampled subsets, each containing exactly \(k\) items from a ground set of size \(d\). This problem arises in combination screening, bundle preference modeling, and other settings in which outcomes depend on collections of prescribed size. The fixed-cardinality constraint couples the membership coordinates, so standard product-domain notions of interaction and smoothness cannot be imported unchanged. We use the harmonic decomposition of the Johnson graph to define an intrinsic Johnson--Sobolev scale and determine the exact dimensions of low-order interaction spaces. We derive finite-sample risk bounds for harmonic projection and establish a nonasymptotic minimax characterization when the Johnson--Sobolev energy budget is controlled relative to the noise variance. We also show that stable least squares removes the signal-dependent fluctuation of empirical projection when the empirical Gram matrix is sufficiently well conditioned. For arbitrary signal-to-noise ratios, a rank-deficiency analysis quantifies the components left unidentified by the random design. Together, the rank-deficiency analysis and a centered completion estimator yield minimax bounds that match up to constants whenever \(\min(k,d-k)\) and the smoothness order are fixed. Numerical experiments reported in the Supplementary Material illustrate estimation under different interaction profiles and distinguish the effects of observation noise, random-design fluctuation, and incomplete coverage.
