通过自动发现变量组合，显著提升符号回归的准确性与效率。
Dimension Reduction for Symbolic Regression
- 搜索小规模变量组合替换，用函数依赖性验证有效性。
- 迭代降维后，多种先进符号回归算法性能显著提升。
- 适合处理复杂公式、变量冗余的科学建模任务。
符号回归的解由输入变量和有限函数符号集合中的运算符构成。评估符号回归算法的关键指标是：从有限样本中恢复公式（至符号等价）的能力。公式越复杂（变量和运算符越多），恢复难度越高。自然界的符号公式中，变量常以固定组合出现，可将组合替换为新变量，从而降低维度。但有效替换的发现极具挑战。本文提出在小规模替换表达式空间中搜索，并通过函数依赖性测试验证其有效性。该迭代降维方法可与任意符号回归方法结合使用。实验表明，该方法能可靠识别有效替换，显著提升多种前沿符号回归算法的性能。
原文摘要 · Abstract (English)
Solutions of symbolic regression problems are expressions that are composed of input variables and operators from a finite set of function symbols. One measure for evaluating symbolic regression algorithms is their ability to recover formulae, up to symbolic equivalence, from finite samples. Not unexpectedly, the recovery problem becomes harder when the formula gets more complex, that is, when the number of variables and operators gets larger. Variables in naturally occurring symbolic formulas often appear only in fixed combinations. This can be exploited in symbolic regression by substituting one new variable for the combination, effectively reducing the number of variables. However, finding valid substitutions is challenging. Here, we address this challenge by searching over the expression space of small substitutions and testing for validity. The validity test is reduced to a test of functional dependence. The resulting iterative dimension reduction procedure can be used with any symbolic regression approach. We show that it reliably identifies valid substitutions and significantly boosts the performance of different types of state-of-the-art symbolic regression algorithms.
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