发表机构
Sorbonne Université; Institut universitaire de France（索邦大学; 法国高等学术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究布尔超立方体上傅里叶-索博列夫预算下的高维非参数回归，提出可解傅里叶度刻画极小极大风险，并扩展至单调回归及未知支撑情形。
AI 中文摘要
我们研究了在布尔超立方体\u000b{0,1}^d上，在傅里叶-索博列夫预算（即回归函数的总二次影响力上界B）下，基于样本量为n的高维非参数回归问题。核心对象是可解傅里叶度D*(d,n)，定义为最大的交互阶数D，使得阶数为D的傅里叶-沃尔什空间的维数不超过n。我们证明了在傅里叶-索博列夫椭球上的极小极大风险阶为B/D*(d,n)：估计可以做到统计上可见的交互阶数，而预算控制着由高阶交互携带的残差质量。然后，我们在相同预算下研究单调回归。对于支撑在已知的s个坐标上且s阶为log n的单调函数，我们得到了阶为B/s的匹配极小极大速率。最后，我们考虑一个未知支撑模型，其中函数是单调的且依赖于至多s个坐标；其极小极大风险由内在维度s以及选择活跃支撑的对数代价共同决定。
英文摘要
We study high-dimensional nonparametric regression on the Boolean hypercube \(\{0,1\}^d\), from a sample of size \(n\), under a Fourier--Sobolev budget, i.e., a bound \(B\) on the total quadratic influence of the regression function. The central object is the resolvable Fourier degree \(D^*(d,n)\), defined as the largest interaction order \(D\) such that the dimension of the degree-\(D\) Fourier--Walsh space does not exceed \(n\). We prove that the minimax risk over Fourier--Sobolev ellipsoids is of order \(B/D^*(d,n)\): estimation is possible up to the statistically visible interaction order, and the budget controls the residual mass carried by higher-order interactions. We then study monotone regression under the same budget. For monotone functions supported on a known set of \(s\) coordinates, with \(s\) of order \(\log n\), we obtain a matching minimax rate of order \(B/s\). Finally, we consider an unknown-support model in which the function is monotone and depends on at most \(s\) coordinates; its minimax risk is governed by the intrinsic dimension \(s\) together with the logarithmic cost of selecting the active support.
