支持度恢复的改进最小二乘方法
Refined Least Squares for Support Recovery
摘要：
研究了基于噪声观测的精确支持度恢复问题，提出了改进的最小二乘（RLS）方法。给定一组噪声测量值$$ \myvec{y} = \myvec{X}\myvecθ^* + \myvecω,$$和$\myvec{X} \in \mathbb{R}^{N \times D}$，其中$\myvec{X} \in \mathbb{R}^{N \times D}$是（已知的）高斯矩阵，$\myvecω \in \mathbb{R}^N$是（未知的）高斯噪声向量，我们的目标是恢复（未知的）稀疏向量$\myvecθ^* \in \left\{-1,0,1\right\}^D$的支持。为了恢复对$\myvecθ^*$的支持，我们使用多个最小二乘解的平均值，每个最小二乘解都是基于整个方程组的子集计算的。通过识别平均最小二乘解的最显著系数来估计支持度。我们证明，在各种各样的设置，我们的方法优于国家的最先进的支持恢复算法。
英文摘要：
We study the problem of exact support recovery based on noisy observations and present Refined Least Squares (RLS). Given a set of noisy measurement $$ \myvec{y} = \myvec{X}\myvecθ^* + \myvecω,$$ and $\myvec{X} \in \mathbb{R}^{N \times D}$ which is a (known) Gaussian matrix and $\myvecω \in \mathbb{R}^N$ is an (unknown) Gaussian noise vector, our goal is to recover the support of the (unknown) sparse vector $\myvecθ^* \in \left\{-1,0,1\right\}^D$. To recover the support of the $\myvecθ^*$ we use an average of multiple least squares solutions, each computed based on a subset of the full set of equations. The support is estimated by identifying the most significant coefficients of the average least squares solution. We demonstrate that in a wide variety of settings our method outperforms state-of-the-art support recovery algorithms.
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