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首页> 外文会议>16th Annual Conference on Learning Theory and 7th Kernel Workshop, COLT/Kernel 2003 Aug 24-27, 2003 Washington, DC, USA >Using a Linear Fit to Determine Monotonicity Directions
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Using a Linear Fit to Determine Monotonicity Directions

机译：使用线性拟合确定单调性方向

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摘要

Let f be a function on R~d that is monotonic in every variable. There are 2~d possible assignments to the directions of monotonicity (two per variable). We provide sufficient conditions under which the optimal linear model obtained from a least squares regression on f will identify the monotonicity directions correctly. We show that when the input dimensions are independent, the linear fit correctly identifies the monotonicity directions. We provide an example to illustrate that in the general case, when the input dimensions are not independent, the linear fit may not identify the directions correctly. However, when the inputs are jointly Gaussian, as is often assumed in practice, the linear fit will correctly identify the monotonicity directions, even if the input dimensions are dependent. Gaussian densities are a special case of a more general class of densities (Mahalanobis densities) for which the result holds. Our results hold when f is a classification or regression function. If a finite data set is sampled from the function, we show that if the exact linear regression would have yielded the correct monotonicity directions, then the sample regression will also do so asymptotically (in a probabilistic sense). This result holds even if the data are noisy.
机译：设f是R〜d上的一个函数，在每个变量中都是单调的。单调性方向有2〜d个可能的分配（每个变量两个）。我们提供了充分的条件，在这些条件下，从f的最小二乘回归获得的最佳线性模型将正确识别单调性方向。我们表明，当输入维是独立的时，线性拟合可正确识别单调性方向。我们提供一个示例来说明在一般情况下，当输入尺寸不独立时，线性拟合可能无法正确识别方向。但是，当输入为联合高斯时（如通常在实践中假设的那样），即使输入尺寸是相关的，线性拟合也将正确地识别单调性方向。高斯密度是结果适用的更一般的一类密度（马哈拉诺比斯密度）的特例。当f是分类或回归函数时，我们的结果成立。如果从函数中采样了一个有限的数据集，我们将表明，如果精确的线性回归将产生正确的单调性方向，那么样本回归也将渐近地（从概率意义上）进行。即使数据嘈杂，该结果仍然成立。

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