R ^n上线性函数的无分布检验
Distribution-Free Testing of Linear Functions on R^n
摘要：
我们研究了测试函数f：R ^n-> R是否是线性的问题（即，加性和齐次），其中函数之间的距离是相对于R上的未知概率分布来测量的。我们证明了，给定查询访问f，采样访问未知分布以及标准高斯分布，并且eps> 0，我们可以区分加法函数和eps-远离加法函数的函数，其查询时间为O（（1/eps）log（1/eps）），与n无关。此外，在假设f是连续函数的情况下，可以使用相同数量的查询将加和性测试器扩展为用于线性的无分布测试器。另一方面，我们表明，如果我们只允许获得采样点上的f值，那么任何无分布测试程序都需要Omega（n）样本，即使底层分布是标准高斯分布。
英文摘要：
We study the problem of testing whether a function f:R^n->R is linear (i.e., both additive and homogeneous) in the distribution-free property testing model, where the distance between functions is measured with respect to an unknown probability distribution over R. We show that, given query access to f, sampling access to the unknown distribution as well as the standard Gaussian, and eps>0, we can distinguish additive functions from functions that are eps-far from additive functions with O((1/eps)log(1/eps)) queries, independent of n. Furthermore, under the assumption that f is a continuous function, the additivity tester can be extended to a distribution-free tester for linearity using the same number of queries. On the other hand, we show that if we are only allowed to get values of f on sampled points, then any distribution-free tester requires Omega(n) samples, even if the underlying distribution is the standard Gaussian.
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