基于Cauchy随机投影的$l_1$降维的非线性估计和尾界
Nonlinear Estimators and Tail Bounds for Dimension Reduction in $l_1$ Using Cauchy Random Projections
摘要：
对于$l_1$中的降维，{ \em柯西随机投影}的方法将原始数据矩阵$\mathbf{A} \in\mathbb{R}^{n\times D}$与随机矩阵$\mathbf{R} \in \mathbb{R}^{D\times k}$（$k\ll\min(n,D)$）相乘，该随机矩阵的条目是i.i.d.标准柯西C（0，1）样本。由于不可能的结果，人们不能希望从$\mathbf{B} = \mathbf{AR} \in \mathbb{R}^{n\times k}$中恢复$\mathbf{A}$中的成对$l_1$距离，使用线性估计而不产生大的误差。然而，非线性估计器在数据流计算、信息检索、学习和数据挖掘中的某些应用仍然是有用的。我们提出了三种类型的非线性估计：偏差校正的样本中位数估计，偏差校正的几何平均估计，偏差校正的最大似然估计。样本中位数估计量和几何均值估计量是渐近等价的（如$k\to \infty$），但后者在小$k$时更准确。我们得到了几何均值估计的显式尾界，并建立了类似于$l_1$中降维的Johnson-Lindenstrauss（JL）引理，该引理比$l_2$中降维的经典JL引理弱.渐近地，无论是样本中位数估计和几何平均估计是约80%的效率相比，最大似然估计（MLE）。我们分析的MLE的时刻，并提出近似的分布的MLE的逆高斯。
英文摘要：
For dimension reduction in $l_1$, the method of {\em Cauchy random projections} multiplies the original data matrix $\mathbf{A} \in\mathbb{R}^{n\times D}$ with a random matrix $\mathbf{R} \in \mathbb{R}^{D\times k}$ ($k\ll\min(n,D)$) whose entries are i.i.d. samples of the standard Cauchy C(0,1). Because of the impossibility results, one can not hope to recover the pairwise $l_1$ distances in $\mathbf{A}$ from $\mathbf{B} = \mathbf{AR} \in \mathbb{R}^{n\times k}$, using linear estimators without incurring large errors. However, nonlinear estimators are still useful for certain applications in data stream computation, information retrieval, learning, and data mining. We propose three types of nonlinear estimators: the bias-corrected sample median estimator, the bias-corrected geometric mean estimator, and the bias-corrected maximum likelihood estimator. The sample median estimator and the geometric mean estimator are asymptotically (as $k\to \infty$) equivalent but the latter is more accurate at small $k$. We derive explicit tail bounds for the geometric mean estimator and establish an analog of the Johnson-Lindenstrauss (JL) lemma for dimension reduction in $l_1$, which is weaker than the classical JL lemma for dimension reduction in $l_2$. Asymptotically, both the sample median estimator and the geometric mean estimators are about 80% efficient compared to the maximum likelihood estimator (MLE). We analyze the moments of the MLE and propose approximating the distribution of the MLE by an inverse Gaussian.
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