Noisy recovery from random linear observations: Sharp minimax rates under elliptical constraints
研究了在椭圆约束下从随机线性观测中恢复参数的极小极大最优速率，推导出刻画速率的泛函，适用于参数和非参数回归问题。
Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the random operator, and elliptical norms that define the error metric and the parameter space. This nonasymptotic result is sharp up to an explicit universal constant, and it becomes asymptotically exact as the radius of the parameter space is allowed to grow. We demonstrate the generality of the result by applying it to both parametric and nonparametric regression problems.