论文标题
关于一般顺序样条回归的相变
On a phase transition in general order spline regression
论文作者
论文摘要
在高斯序列模型中，$ y =θ_0 + \ varepsilon $ in $ \ mathbb {r}^n $中，我们研究了通过免费开口的（通用的）键（d，d_0，k）$θ（d，d_0，k）近似信号$θ_0$的基本限制。这里$ d $是样条的程度，$ d_0 $是每个内结的可分配性顺序，而$ k $是最大数量的零件。我们表明，给定任何整数$ d \ geq 0 $和$ d_0 \ in \ { - 1,0，\ ldots，d-1 \} $，$θ（d，d_0，k）$的最小值估计率以下相位转移： \ inf _ {\widetildeθ} \ sup_ {θ\inθ（d，d_0，k）} \ Mathbb {e}_θ\ | \wideTildeθ -tideTildeθ -θ\ |^2 \ |^2 \ |^2 \ asymp_d asymp_d asemp_d begin k \ log（en/k），＆k \ geq k_0+1。 \ end {cases} \ end {aligned} \ end {equation*}过渡边界$ k_0 $，它采用表格$ \ lfloor {（d + 1）/（d-d_0）/（d-d_0）\ rfloor} + 1 $，证明了正常性参数$ d_0 $ $ d_0 $的关键作用$ d_0 $ d_0 $。 $ \ log（en）$费率。我们进一步表明，一旦鼓励了额外的“ $ d $单调”形状约束（包括$ d = 0 $的单调性和$ d = 1 $的凸度），可以消除上述相变，并且可以实现所有$ k $的$ k \ log \ log \ log \ log \ log \ log \ log（16n/k）$。这些结果为开发$ \ ell_0 $ - 二元化（形状约束）样条回归过程提供了理论支持，作为$ \ ell_1 $ - 和$ \ ell_2 $ penalizatizatization的有用替代方案。
In the Gaussian sequence model $Y= θ_0 + \varepsilon$ in $\mathbb{R}^n$, we study the fundamental limit of approximating the signal $θ_0$ by a class $Θ(d,d_0,k)$ of (generalized) splines with free knots. Here $d$ is the degree of the spline, $d_0$ is the order of differentiability at each inner knot, and $k$ is the maximal number of pieces. We show that, given any integer $d\geq 0$ and $d_0\in\{-1,0,\ldots,d-1\}$, the minimax rate of estimation over $Θ(d,d_0,k)$ exhibits the following phase transition: \begin{equation*} \begin{aligned} \inf_{\widetildeθ}\sup_{θ\inΘ(d,d_0, k)}\mathbb{E}_θ\|\widetildeθ - θ\|^2 \asymp_d \begin{cases} k\log\log(16n/k), & 2\leq k\leq k_0,\\ k\log(en/k), & k \geq k_0+1. \end{cases} \end{aligned} \end{equation*} The transition boundary $k_0$, which takes the form $\lfloor{(d+1)/(d-d_0)\rfloor} + 1$, demonstrates the critical role of the regularity parameter $d_0$ in the separation between a faster $\log \log(16n)$ and a slower $\log(en)$ rate. We further show that, once encouraging an additional '$d$-monotonicity' shape constraint (including monotonicity for $d = 0$ and convexity for $d=1$), the above phase transition is eliminated and the faster $k\log\log(16n/k)$ rate can be achieved for all $k$. These results provide theoretical support for developing $\ell_0$-penalized (shape-constrained) spline regression procedures as useful alternatives to $\ell_1$- and $\ell_2$-penalized ones.
