%DIFDELCMD < {\em %%% Johnson-Lindenstrauss (JL) Transform is one of the most popular methods for dimension reduction. In this paper, we study the effectiveness of JL transform for solving the high dimensional optimization problems with  outliers. We focus on two fundamental optimization problems: {\em k-center clustering with outliers} and {\em SVM with outliers}. In general, the time complexity for dealing with outliers in high dimensional space could be very large.{\em   Based on some novel insights in geometry, we prove that the complexities of these two problems can be significantly reduced through JL transform . In the experiments, we compare JL transform with several other well known dimension reduction methods, and study their performances on synthetic and real datasets.
%DIFDELCMD < {\em %%% In this paper, we consider robust optimization problems in high dimensions. Because a real-world dataset may contain significant noise or even specially crafted samples from some attacker, we are particularly interested in the optimization problems with arbitrary outliers. We focus on two fundamental optimization problems: {\em SVM with outliers} and {\em k-center clustering with outliers}. They are in fact extremely challenging combinatorial optimization problems, since we cannot impose any restriction on the outliers. Therefore, their computational complexities are quite high especially when we consider the instances in high dimensional spaces. The{\em Johnson-Lindenstrauss (JL) Transform is one of the most popular methods for dimension reduction. Though the JL transform has been widely studied in the past decades, its effectiveness for dealing with outliers has never been investigated before (to the best of our knowledge). Based on some novel insights from the geometry, we prove that the complexities of these two problems can be significantly reduced through the JL transform. Moreover, we prove that the solution in the dimensionality-reduced space can be efficiently recovered in the original \mathbb{R . In the experiments, we compare JL transform with several other well known dimension reduction methods, and study their performances on synthetic and real datasets.
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Johnson-Lindenstrauss (JL) Transform
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study the effectiveness of JL transform for solving the high dimensional
consider robust optimization problems in high dimensions. Because a real-world dataset may contain significant noise or even specially crafted samples from some attacker, we are particularly interested in the
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In general, the time complexity for dealing with outliers
They are in fact extremely challenging combinatorial optimization problems, since we cannot impose any restriction on the outliers. Therefore, their computational complexities are quite high especially when we consider the instances
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space could be very large.
spaces. The
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Johnson-Lindenstrauss (JL) Transform
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is one of the most popular methods for dimension reduction. Though the JL transform has been widely studied in the past decades, its effectiveness for dealing with outliers has never been investigated before (to the best of our knowledge).
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from the
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JL transform
the JL transform. Moreover, we prove that the solution in the dimensionality-reduced space can be efficiently recovered in the original \mathbb{R