低次多项式阈值函数的正则性引理和低权逼近器
A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions
摘要：
给出了布尔立方体{-1，1}^n上d次多项式阈值函数的一个正则性引理.这一结果表明，每个degree-d PTF可以分解成一个常数的子函数，使得几乎所有的子函数是接近正则PTF。这里，“正则PTF”是PTF符号（p（x）），其中每个变量对多项式p（x）的影响是p的总影响的一小部分。作为这个正则性引理的应用，我们证明了对于任何常数d \geq 1，\eps \geq 0，n个变量上的每一个degree-d PTF都可以通过具有总幅度为O（n^d）的整数权重的常度PTF来近似于精度eps。这个重量界被证明是最佳的常数因子。
英文摘要：
We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d \geq 1, \eps \geq 0, every degree-d PTF over n variables has can be approximated to accuracy eps by a constant-degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.
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