对称性能发挥多大作用？稀疏函数数据分析中的相变与对称性选择
How far can symmetry help? Phase transitions and symmetry selection in sparse functional data analysis
AI总结：
本文研究稀疏函数数据分析中对称性对协方差曲面相变的影响，推导了阈值的位移规律，通过数值验证了相关结论，为该领域的对称性选择提供了理论依据。
AI中文摘要：
在稀疏函数数据分析中，n条曲线每条在m个随机点上被观测，协方差曲面会发生尖锐的相变：若协方差具有光滑性β，当m超过阈值m*_n ≍ n^{1/(2β)}时，风险会从二维非参数率降至参数率n^{-1}。本文确定了定义域的对称性对该相变的影响：保持过程与设计的q阶循环群会将阈值移至n^{1/(2β)}q^{-1/2}；该指数为平方根，因对称性作用于可用配对数量（其对方差呈二次影响），而参数下限不受群平均影响。该位移会饱和：当轨道比带宽更细时，根据泊松求和及核自相关的正定性，缩减因子为min(q,c_K/h)，超过该点后率会降至一维非参数率。因此，即使是完整圆群这类旋转对称性，也无法将阈值降至n^{1/(4β)}以下；该下限可由本文估计量从上方向逼近，而本文下界则可在多项式因子内从下方向逼近。故在对数尺度上，对称性可将阈值从经典阈值向常数采样推进一半。统一下界基于平稳子类的保正填充，缩小间隙的问题留待后续研究。若对称性仅为近似，风险会产生近似项，且设计平面会分裂为三个区域，其中一个区域无法通过额外采样触及；控制该区域的对称谱在傅里叶坐标中明确，且留出法可选择对称水平，其领先常数在饱和前为1，饱和后在绝对常数范围内。上述所有规律均通过数值验证。相比之下，定义域的重新参数化不会改变阈值。
英文摘要：
In sparse functional data analysis, where $n$ curves are each observed at $m$ random points, the covariance surface undergoes a sharp phase transition: if the covariance has smoothness $β$, the risk drops from the two-dimensional nonparametric rate to the parametric rate $n^{-1}$ once $m$ exceeds $m^*_n \asymp n^{1/(2β)}$. We determine what a symmetry of the domain does to that transition. A cyclic group of order $q$ preserving process and design displaces the threshold to $n^{1/(2β)}q^{-1/2}$; the exponent is a square root because symmetry acts on the number of usable pairs, which enters the variance quadratically, while the parametric floor is untouched by group averaging. The displacement saturates: once the orbit is finer than the bandwidth the reduction factor is $\min(q,c_K/h)$, by Poisson summation and positive definiteness of the kernel autocorrelation, and beyond that point the rate collapses to the one-dimensional nonparametric rate. Hence no rotation symmetry, even the full circle group, lowers the threshold below $n^{1/(4β)}$; this floor is attained from above by our estimator and from below, up to a polynomial factor, by our lower bounds. Symmetry thus takes one halfway on a logarithmic scale from the classical threshold to constant sampling. The uniform lower bound rests on a positivity-preserving packing of the stationary sub-class; closing the gap is left open. If the symmetry is only approximate, the risk acquires an approximation term and the design plane splits into three regimes, one unreachable by additional sampling; the symmetry spectrum governing it is explicit in Fourier coordinates, and a hold-out procedure selects the symmetry level with leading constant one below saturation and within an absolute constant beyond. All laws above are confirmed numerically. Reparametrisation of the domain, by contrast, leaves the threshold unchanged.
