A new computationally efficient dependence measure, and an adaptive
statistical test of independence, are proposed. The dependence measure is the
difference between analytic embeddings of the joint distribution and the
product of the marginals, evaluated at a finite set of locations (features).
These features are chosen so as to maximize a lower bound on the test power,
resulting in a test that is data-efficient, and that runs in linear time (with
respect to the sample size n). The optimized features can be interpreted as
evidence to reject the null hypothesis, indicating regions in the joint domain
where the joint distribution and the product of the marginals differ most.
Consistency of the independence test is established, for an appropriate choice
of features. In real-world benchmarks, independence tests using the optimized
features perform comparably to the state-of-the-art quadratic-time HSIC test,
and outperform competing O(n) and O(n log n) tests.

提出了一种新的计算有效的依赖度量和自适应统计独立性检验方法，其特点是数据高效运行时间为线性，并且在真实基准中表现良好。

具有分析核嵌入的独立性自适应测试

An Adaptive Test of Independence with Analytic Kernel Embeddings

Two semimetrics on probability distributions are proposed, given as the sum
of differences of expectations of analytic functions evaluated at spatial or
frequency locations (i.e, features). The features are chosen so as to maximize
the distinguishability of the distributions, by optimizing a lower bound on
test power for a statistical test using these features. The result is a
parsimonious and interpretable indication of how and where two distributions
differ locally. An empirical estimate of the test power criterion converges
with increasing sample size, ensuring the quality of the returned features. In
real-world benchmarks on high-dimensional text and image data, linear-time
tests using the proposed semimetrics achieve comparable performance to the
state-of-the-art quadratic-time maximum mean discrepancy test, while returning
human-interpretable features that explain the test results.

该研究提出了两个基于概率分布的半度量，通过优化测试功率的下界，使用空间或频率位置的解析函数的期望之差的总和作为距离度量来最大化概率分布的可分辨性，并返回可解释的特征，用于高维文本和图像数据上进行线性时间测试。

具有最大测试功效的可解释分布特征

Interpretable Distribution Features with Maximum Testing Power

A family of maximum mean discrepancy (MMD) kernel two-sample tests is
introduced. Members of the test family are called Block-tests or B-tests, since
the test statistic is an average over MMDs computed on subsets of the samples.
The choice of block size allows control over the tradeoff between test power
and computation time. In this respect, the $B$-test family combines favorable
properties of previously proposed MMD two-sample tests: B-tests are more
powerful than a linear time test where blocks are just pairs of samples, yet
they are more computationally efficient than a quadratic time test where a
single large block incorporating all the samples is used to compute a
U-statistic. A further important advantage of the B-tests is their
asymptotically Normal null distribution: this is by contrast with the
U-statistic, which is degenerate under the null hypothesis, and for which
estimates of the null distribution are computationally demanding. Recent
results on kernel selection for hypothesis testing transfer seamlessly to the
B-tests, yielding a means to optimize test power via kernel choice.

引入了一族最大平均差异 (MMD) 内核双样本检验，其中家族成员被称为 Block-tests 或 B-tests，检验统计量是样本子集上计算的 MMD 的平均，块大小的选择允许在测试功率和计算时间之间的权衡。B-test 家族结合了先前建议的 MMD 两个样本检验的有利特性：B-tests 比样本对成对大小的线性时间测试更强大，但它们比使用单个大块计算 U 统计量的二次时间测试更具计算效率。B-tests 的另一个重要优势是它们的渐近正常空分布，这与 U 统计量形成对比，在零假设下是退化的，计算零分布的估计是具有计算要求的。最近的关于假设检验的内核选择的结果可以无缝转移到 B-tests，通过内核选择优化测试功率的方法。