基于L1回归的对抗性稀疏噪声压缩感知
Compressed Sensing with Adversarial Sparse Noise via L1 Regression
摘要：
我们提出了一个简单而有效的算法解决问题的\emph{sparse robust linear regression}。在这个问题中，人们希望从被稀疏噪声破坏的线性测量来估计稀疏向量$w^* \in \mathbb{R}^n$，该稀疏噪声可以任意地改变测量响应$y$的逆向选择的$η$分数，以及向响应引入有界范数噪声。对于高斯测量，我们表明，一个简单的算法基于L1回归可以成功地估计$w^*$的任何$η< η_0 \approx 0.239$，这个阈值是紧的算法。该算法所需的测量数量是$O(k \log \frac{n}{k})$，用于$k$ -稀疏估计，其在没有任何稀疏噪声的情况下所需数量的常数因子内。在这三个属性中，我们展示了-估计稀疏和密集$w^*$的能力;对离群值的大常数分数的容忍度;以及对抗性而不是分布性的容忍度（例如，高斯）密集噪声-据我们所知，以前没有一个结果达到两个以上。
英文摘要：
We present a simple and effective algorithm for the problem of \emph{sparse robust linear regression}. In this problem, one would like to estimate a sparse vector $w^* \in \mathbb{R}^n$ from linear measurements corrupted by sparse noise that can arbitrarily change an adversarially chosen $η$ fraction of measured responses $y$, as well as introduce bounded norm noise to the responses. For Gaussian measurements, we show that a simple algorithm based on L1 regression can successfully estimate $w^*$ for any $η< η_0 \approx 0.239$, and that this threshold is tight for the algorithm. The number of measurements required by the algorithm is $O(k \log \frac{n}{k})$ for $k$-sparse estimation, which is within constant factors of the number needed without any sparse noise. Of the three properties we show---the ability to estimate sparse, as well as dense, $w^*$; the tolerance of a large constant fraction of outliers; and tolerance of adversarial rather than distributional (e.g., Gaussian) dense noise---to the best of our knowledge, no previous result achieved more than two.
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