树上去噪流的最小最大速率
The Minimax Rate of Denoising Flows on Trees
- Shenzhen MSU-BIT University（深圳北理莫斯科大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文完整解决了有根树上从高斯噪声恢复流的去噪问题，确定了最小最大风险为 $\min\{V^2H_T,\sigma^2k_0\}$，并提出了达到该速率的确定性指数加权聚合估计器，同时揭示了最小二乘法的次优性。
AI中文摘要：
等距回归和单纯形上的均值估计是最经典的几何约束去噪问题。Chatterjee 和 Lafferty 将两者推广到从高斯噪声中恢复有根树上的流，其中路径上的等距回归和星形上的单纯形是特例。他们的结果仅覆盖特殊树；在一般树上，最小最大风险在十年间一直未知。我们完整地解决了这个问题。在每棵具有 $n$ 个顶点的树上，对于每个预算 $V$ 和噪声水平 $\sigma$，最小最大风险的阶为 $\min\{V^2H_T,\\ \sigma^2k_0\}$，其中 $H_T$ 是直径，$k_0$ 是截断祖先剖面的交叉指数。一个确定性估计器，即对整数状态的指数加权聚合，能在 $O(n^2\log(n+1))$ 次算术运算内达到该速率。在已知预算下的最小二乘法，作为该模型的自然凸规划，在最坏情况下次优，其因子阶为 $(\log n)^{2/5}$。一个泛函，即截断祖先剖面，承载了速率、估计器和最小二乘比较。它充当覆盖半径、正锥中的填充熵以及精确计算背后的一维状态的范围。两个经典理论在树上合而为一，处处尖锐且可计算。
英文摘要：
Isotonic regression and mean estimation over a simplex are among the most classical denoising problems under geometric constraints. Chatterjee and Lafferty generalized both to the recovery of a flow on a rooted tree from Gaussian noise, with isotonic regression on a path and the simplex on a star. Their results reach only special trees; on a general tree the minimax risk remained unknown for a decade. We settle the question in full. On every tree with $n$ vertices, at every budget $V$ and noise level $σ$, the minimax risk is of order $\min\{V^2H_T,\ σ^2k_0\}$, with $H_T$ the diameter and $k_0$ the crossing index of a truncated ancestor profile. A deterministic estimator, an exponentially weighted aggregate over integer states, attains it in $O(n^2\log(n+1))$ arithmetic operations. Least squares at a known budget, the natural convex program for the model, is suboptimal in the worst case by a factor of order $(\log n)^{2/5}$. One functional, the truncated ancestor profile, carries the rate, the estimator, and the least squares comparison. It serves as covering radius, as packing entropy in the positive cone, and as the range of the one-dimensional state behind the exact computation. Two classical theories become one on trees, sharp and computable throughout.
