arXiv 1503.083932015-03math.STcs.ITmath.ITstat.TH

SLOPE对未知稀疏性的自适应性和渐近极大极小性

SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

Weijie Su, Emmanuel Candes

备注：: To appear in the Annals of Statistics

摘要：

我们考虑高维稀疏回归问题，我们观察到$y = X β+ z$，其中$X$是一个$n \times p$设计矩阵和$z$是一个$n$维向量的独立高斯误差，每个方差$σ^2$。我们的重点是最近引入的SLOPE估计（（Bogdan等人，2014）），它使用秩相关惩罚$\sum_{1 \le i \le p} λ_i |\hat β|_{(i)}$来正则化最小二乘估计值，其中$|\hat β|_{(i)}$是拟合系数的第$i$个最大幅度。在高斯设计下，其中$X$的项是i.i.d.~ $\mathcal{N}(0, 1/n)$，我们表明，SLOPE，与权重$λ_i$刚刚约等于$σ\cdot Φ^{-1}(1-iq/(2p))$（$Φ^{-1}(α)$是$α$的第四分位数的标准正常和$q$是一个固定的数字在$(0,1)$）实现了平方误差的估计服从\[ \sup_{\| β\|_0 \le k} \,\, \mathbb{P} \left(\| \hatβ_{\text{SLOPE}} - β\|^2 > (1+ε) \, 2σ^2 k \log(p/k) \right) \longrightarrow 0 \]作为维度$p$增加到$\infty$，其中$ε> 0$是一个任意小常数。这在$\ell_0$ -稀疏水平上的弱假设下成立，即$k/p \rightarrow 0$和$(k\log p)/n \rightarrow 0$，并且在这是任何估计器可以实现的最佳可能误差的意义上是尖锐的。一个显着的特点是，SLOPE不需要任何知识的程度的稀疏，但自动适应，以产生最佳的总平方误差在广泛的$\ell_0$ -稀疏类。我们不知道任何其他估计与此属性。

英文摘要：

We consider high-dimensional sparse regression problems in which we observe $y = X β+ z$, where $X$ is an $n \times p$ design matrix and $z$ is an $n$-dimensional vector of independent Gaussian errors, each with variance $σ^2$. Our focus is on the recently introduced SLOPE estimator ((Bogdan et al., 2014)), which regularizes the least-squares estimates with the rank-dependent penalty $\sum_{1 \le i \le p} λ_i |\hat β|_{(i)}$, where $|\hat β|_{(i)}$ is the $i$th largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of $X$ are i.i.d.~$\mathcal{N}(0, 1/n)$, we show that SLOPE, with weights $λ_i$ just about equal to $σ\cdot Φ^{-1}(1-iq/(2p))$ ($Φ^{-1}(α)$ is the $α$th quantile of a standard normal and $q$ is a fixed number in $(0,1)$) achieves a squared error of estimation obeying \[ \sup_{\| β\|_0 \le k} \,\, \mathbb{P} \left(\| \hatβ_{\text{SLOPE}} - β\|^2 > (1+ε) \, 2σ^2 k \log(p/k) \right) \longrightarrow 0 \] as the dimension $p$ increases to $\infty$, and where $ε> 0$ is an arbitrary small constant. This holds under a weak assumption on the $\ell_0$-sparsity level, namely, $k/p \rightarrow 0$ and $(k\log p)/n \rightarrow 0$, and is sharp in the sense that this is the best possible error any estimator can achieve. A remarkable feature is that SLOPE does not require any knowledge of the degree of sparsity, and yet automatically adapts to yield optimal total squared errors over a wide range of $\ell_0$-sparsity classes. We are not aware of any other estimator with this property.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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