[论文解读] Aggregated test of independence based on HSIC measures

Anouar Meynaoui, Béatrice Laurent+2 more|arXiv (Cornell University)|Feb 15, 2019

Random Matrices and Applications参考文献 36被引用 4

一句话总结

本文提出了一种基于HSIC的聚合检验方法，用于检验随机向量之间的独立性，该方法通过自适应选择多个高斯核的带宽，避免了手动选择核函数的需要。该方法在Sobolev球上实现了极小极大自适应最优性，通过理论边界推导出精确的非渐近分离率，并在数值研究中得到验证。

ABSTRACT

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level $\\alpha \\in (0,1)$. From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikol'skii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.

研究动机与目标

• 开发一种非渐近、自适应的独立性检验方法，避免在基于HSIC的方法中面临困难的带宽选择问题。
• 为聚合过程建立理论保证，包括Sobolev球上均匀分离率的精确上界。
• 推导出非渐近极小极大分离率的下界，证明该聚合检验在Sobolev正则性空间上具有极小极大自适应性。
• 通过数值研究评估聚合检验的实际性能，与现有的基于HSIC和非HSIC的独立性检验进行比较。

提出的方法

• 使用固定带宽的高斯核构建单个HSIC检验，每个检验在指定显著性水平α下控制第一类错误。
• 通过数据驱动的组合规则在多个带宽范围内聚合多个单个检验，以提高鲁棒性和自适应性。
• 采用置换方法近似聚合检验统计量的零分布，确保有效的大小控制。
• 推导出在光滑度为δ的Sobolev球上，聚合过程的非渐近均匀分离率的上界。
• 在相同的Sobolev类上建立非渐近极小极大分离率的下界，证明其在极小极大意义下的自适应性。
• 利用集中不等式和U-统计量的矩界，控制零假设下平方检验统计量的期望。

实验结果

研究问题

• RQ1 能否构建一种聚合HSIC检验，使其在未知依赖结构光滑度下自适应，而无需手动选择带宽？
• RQ2 在光滑度δ的Sobolev球上，聚合HSIC检验的非渐近均匀分离率是多少？
• RQ3 所提出的聚合检验在Sobolev球上是否具有非渐近意义下的极小极大最优性？
• RQ4 在有限样本下，该聚合检验与现有基于HSIC和非HSIC的独立性检验相比性能如何？
• RQ5 能否利用矩方法和U-统计量理论建立分离率的理论保证？

主要发现

• 聚合HSIC检验在Sobolev球上实现了与非渐近极小极大率相差一个常数因子的均匀分离率，证明了极小极大自适应性。
• 均匀分离率的上界为 n^{-(2δ+p+q)/(4δ+2p+2q)} 阶，其中δ为光滑度，(p,q)为维度，与已知的极小极大率一致。
• 推导出极小极大分离率的下界，确认该聚合过程在Sobolev球上具有极小极大意义下的最优性。
• 理论分析表明，零假设下平方检验统计量的期望被一致有界，确保了稳定性与大小控制。
• 数值研究表明，该聚合检验在各种场景下均优于使用固定带宽的标准HSIC检验及其他最先进的独立性检验。
• 该方法在不同光滑度水平和维度下均表现稳健，除带宽选择外无需额外调参，而带宽选择通过聚合实现自动化。