揭示回归与分类的等价性，可用来评估数据可回归难度并设计新训练方法。
On the Equivalence of Regression and Classification
- 将回归样本映射为分类问题，实现一对一等价转换。
- 基于等价性提出新回归公式，提升模型泛化能力。
- 可用于无须建模直接评估数据可回归性，适合机器学习初学者。
回归与分类之间的正式联系一直不明确。尽管支持向量回归中使用了 $ orm{w}$ 的边界最大化项，但其作用仅被视作正则化。本文证明：当 $M$ 个样本位于超平面上时，回归问题与包含 $2M$ 个样本的线性可分分类任务存在一一对应关系。在等价分类任务上进行边界最大化，导出了不同于传统方法的回归公式。利用该等价性，我们提出了一个“可回归性”度量，可在不先训练模型的情况下估计数据集的回归难度。此外，我们还基于等价性训练神经网络，学习一个线性化映射，将输入变量变换到线性回归有效的空间。
原文摘要 · Abstract (English)
A formal link between regression and classification has been tenuous. Even though the margin maximization term $\|w\|$ is used in support vector regression, it has at best been justified as a regularizer. We show that a regression problem with $M$ samples lying on a hyperplane has a one-to-one equivalence with a linearly separable classification task with $2M$ samples. We show that margin maximization on the equivalent classification task leads to a different regression formulation than traditionally used. Using the equivalence, we demonstrate a ``regressability'' measure, that can be used to estimate the difficulty of regressing a dataset, without needing to first learn a model for it. We use the equivalence to train neural networks to learn a linearizing map, that transforms input variables into a space where a linear regressor is adequate.
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