再论向量离群正则化的L21-norm鲁棒性
Revisiting L21-norm Robustness with Vector Outlier Regularization
摘要：
在许多实际应用中，数据通常包含离群值。一种流行的方法是使用L2，1范数函数作为鲁棒误差/损失函数。然而，L2，1范数函数的鲁棒性至今还没有得到很好的理解。本文提出了一种新的向量离群正则化（VOR）框架来理解和分析L2，1范数函数的鲁棒性。我们的VOR函数定义一个数据点为离群值，如果它在理论预测的阈值之外，并将其正则化-将其拉回到阈值线。然后，我们证明了L2，1函数是这种VOR的极限情况下，通常的最小二乘/L2误差函数的阈值缩小到零。VOR的一个有趣的特性是，离群值离其理论预测值有多远并不影响最终的正则化和分析结果。这个VOR性质揭示了L2，1范数函数最奇特的性质之一：离群值的影响似乎与它们的离群程度无关-如果离群值进一步远离内在流形/子空间，最终的分析结果不会改变。VOR为理解和分析L2，1范数函数的鲁棒性提供了一种新的途径。将VOR应用于矩阵分解，得到一种新的VORPCA模型。与基于迹范数的L21范数PCA进行了比较，证明了VORPCA的优越性。
英文摘要：
In many real-world applications, data usually contain outliers. One popular approach is to use L2,1 norm function as a robust error/loss function. However, the robustness of L2,1 norm function is not well understood so far. In this paper, we propose a new Vector Outlier Regularization (VOR) framework to understand and analyze the robustness of L2,1 norm function. Our VOR function defines a data point to be outlier if it is outside a threshold with respect to a theoretical prediction, and regularize it-pull it back to the threshold line. We then prove that L2,1 function is the limiting case of this VOR with the usual least square/L2 error function as the threshold shrinks to zero. One interesting property of VOR is that how far an outlier lies away from its theoretically predicted value does not affect the final regularization and analysis results. This VOR property unmasks one of the most peculiar property of L2,1 norm function: The effects of outliers seem to be independent of how outlying they are-if an outlier is moved further away from the intrinsic manifold/subspace, the final analysis results do not change. VOR provides a new way to understand and analyze the robustness of L2,1 norm function. Applying VOR to matrix factorization leads to a new VORPCA model. We give a comprehensive comparison with trace-norm based L21-norm PCA to demonstrate the advantages of VORPCA.
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