用共享支撑的离散化方法加速概率测度学习，显著降低计算成本。
Scalable Learning from Probability Measures with Mean Measure Quantization
- 所有输入测度统一用共享支撑的K点离散化近似
- 理论证明量化后测度具有一致性，下游任务收敛性有保障
- 实测性能接近独立量化，但运行时间大幅减少
我们研究数据为概率测度时的统计学习问题。最优传输（OT）常用于比较和操作此类对象，但当测度支撑较大时计算成本过高。本文提出一种基于量化的方法：将所有输入测度近似为共享相同支撑的K点离散测度。理论上建立了量化测度的一致性，并推导了基于量化测度的多个OT下游任务的收敛性保证。在合成与真实数据集上的数值实验表明，该方法性能可媲美独立量化，同时显著降低运行时间。
原文摘要 · Abstract (English)
We consider statistical learning problems in which data are observed as a set of probability measures. Optimal transport (OT) is a popular tool to compare and manipulate such objects, but its computational cost becomes prohibitive when the measures have large support. We study a quantization-based approach in which all input measures are approximated by $K$-point discrete measures sharing a common support. We establish consistency of the resulting quantized measures. We further derive convergence guarantees for several OT-based downstream tasks computed from the quantized measures. Numerical experiments on synthetic and real datasets demonstrate that the proposed approach achieves performance comparable to individual quantization while substantially reducing runtime.
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