On computing and the complexity of computing higher-order $U$-statistics, exactly

Higher-order $U$-statistics abound in fields such as statistics, machine learning, and computer science, but are known to be highly time-consuming to compute in practice. Despite their widespread appearance, a comprehensive study of their computational complexity is surprisingly lacking. This paper aims to fill this gap by presenting several results related to the computational aspect of $U$-statistics. First, we derive a useful decomposition from a $m$-th order $U$-statistic to a linear combination of $V$-statistics with orders not exceeding $m$, which are generally more feasible to compute. Second, we explore the connection between exactly computing $V$-statistics and Einstein summation, a tool often used in computational mathematics and quantum computing to accelerate tensor computations. Third, we provide an optimistic estimate of the time complexity for exactly computing $U$-statistics, based on the treewidth of a particular graph associated with the $U$-statistic kernel. The above ingredients lead to (1) a new, much more runtime-efficient algorithm to exactly compute general higher-order $U$-statistics, and (2) a more streamlined characterization of runtime complexity of computing $U$-statistics. We develop an accompanying open-source package called \texttt{u-stats} in both Python (https://github.com/zrq1706/U-Statistics-Python) and R (https://github.com/cxy0714/U-Statistics-R). We demonstrate through three examples in statistics that \texttt{u-stats} achieves impressive runtime performance compared to existing benchmarks. This paper also aspires to achieve two goals: (1) to capture the interest of researchers in both statistics and other related areas to further advance the algorithmic development of $U$-statistics and (2) to lift the burden of implementing higher-order $U$-statistics from practitioners.