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

arXiv:2508.12627 · stat.ML, cs.DS, cs.NA, math.NA, stat.CO, stat.ME · Submitted 2025-08-18 · Read on arXiv

stat.ML, cs.DS, cs.NA, math.NA, stat.CO, stat.ME

Submitted: 2025-08-18

Updated: 2026-08-26

Comments: Comments are welcome! 59 pages, 12 tables, 7 figures. An accompanying Python package is available at: https://libraries.io/pypi/u-stats or https://github.com/Amedar-Asterisk/U-Statistics-Python. Accepted by Statistics and Computing

Code: https://github.com/Amedar-Asterisk/U-Statistics-Python

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Causal Inference — 6× in this paper · explained in 4 episodes
• Core — 5× in this paper · explained in 2 episodes
• Degeneracy — 5× in this paper · explained in 3 episodes
• Benchmark — 4× in this paper · explained in 2 episodes
• Influence Functions — 4× in this paper · explained in 2 episodes
• Adjacency Matrix — 2× in this paper · explained in 1 episode
• Embeddings — 2× in this paper · explained in 2 episodes
• Sparsity — 2× in this paper · explained in 4 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Clustering — 1× in this paper · explained in 2 episodes
• Trace — 1× in this paper · explained in 2 episodes

Sources

• Fast convergence rates for dose-response estimation
• Doubly-robust inference and optimality in structure-agnostic models with smoothness
• Higher-Order Graphon Theory: Fluctuations, Degeneracies, and Inference
• Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference
• Counting Cycles with AI: Counting Cycles with AI: Computationally Efficient Equivalent Forms with Applications
• Perturbed Double Machine Learning: Nonstandard Inference Beyond the Parametric Length

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