用量子力学方法定义新概率度量，提升高维分布比较精度
Quantum-inspired probability metrics define a complete, universal space for statistical learning
- 将概率分布嵌入量子态空间，构造新型概率度量
- 在生成模型任务中替代MMD，显著提升性能表现
- 适合需要精确比较高维分布的研究者使用
比较概率分布是自然科学、社会科学和计算科学中的核心挑战。现有方法如最大均值差异（MMD）在高维及非紧致域中表现不佳。本文提出量子概率度量（QPM），通过将概率测度嵌入希尔伯特空间上的正定迹一算子空间构建。该方法扩展了基于核的方法，并克服了MMD在非紧致空间中的不完备性。作为积分概率度量（IPM），QPM能统一逼近ℝⁿ上所有有界一致连续函数，对高维中细微分布差异具有更强敏感性。对于经验分布，可通过特征值方法快速计算，且具备解析梯度，适用于学习与优化。尽管大样本下计算复杂度为O(n³)（相比MMD的O(n²)更高），但在经典生成建模任务中作为MMD的即插即用替代方案，仍可显著提升性能。该方法融合量子力学与经典概率论，为分析与操作概率测度提供了强大工具。
原文摘要 · Abstract (English)
Comparing probability distributions is a core challenge across the natural, social, and computational sciences. Existing methods, such as Maximum Mean Discrepancy (MMD), struggle in high-dimensional and non-compact domains. Here we introduce quantum probability metrics (QPMs), derived by embedding probability measures in the space of quantum states: positive, unit-trace operators on a Hilbert space. This construction extends kernel-based methods and overcomes the incompleteness of MMD on non-compact spaces. Viewed as an integral probability metric (IPM), QPMs have dual functions that uniformly approximate all bounded, uniformly continuous functions on $\mathbb{R}^n$, offering enhanced sensitivity to subtle distributional differences in high dimensions. For empirical distributions, QPMs are readily calculated using eigenvalue methods, with analytic gradients suited for learning and optimization. Although computationally more intensive for large sample sizes ($O(n^3)$ vs. $O(n^2)$), QPMs can significantly improve performance as a drop-in replacement for MMD, as demonstrated in a classic generative modeling task. By combining the rich mathematical framework of quantum mechanics with classical probability theory, this approach lays the foundation for powerful tools to analyze and manipulate probability measures.
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