用神经网络学习高维数据的几何结构，速度提升400倍且无需构建邻近图。
Riemannian Metric Matching for Scalable Geometric Modeling of Distributions
- 通过随机扰动构造条件期望，实现逐样本训练与恒定开销推理。
- 在图像数据上性能媲美甚至超越基于kNN的扩散几何方法，推理快400倍。
- 适合处理高维数据几何分析，尤其适用于邻居失效的场景。
高维数据集通常集中在低维结构附近，但传统几何估计依赖图和核函数，随数据规模和维度增长而变得低效。我们提出黎曼度量匹配：一种基于去噪概率框架的神经网络方法，用于学习数据的黎曼几何。具体地，我们学习了平方场算子（carré du champ），结合扩散几何理论，可访问完整的黎曼几何工具集以支持下游机器学习与统计任务。关键观察是，平方场算子可表示为数据随机扰动下的条件期望，从而实现逐样本训练与无需显式核函数构造的恒定成本、可摊销推理。实验表明，该方法在精度上媲美或优于基于kNN的扩散几何估计器，同时推理速度最高提升400倍，并可在高维图像上实现无图几何分析，克服了最近邻失效问题。
原文摘要 · Abstract (English)
High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.
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