发表机构
The Ohio State University（俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在MTW正曲率流形上平滑经验分布时，内蕴、环境及潜在空间平滑的Wasserstein误差权衡，证明小尺度内蕴平滑更优，并给出带宽选择规则与潜在平滑有利条件。
AI 中文摘要
对经验分布进行平滑可以填补观测之间的空隙，但周围空间中添加的噪声也会使概率质量远离支撑数据的流形。我们通过内蕴热平滑、环境高斯平滑以及编码器-解码器潜在空间中的平滑的Wasserstein误差界来研究这种权衡。在具有正交叉曲率的Ma-Trudinger-Wang条件和所述传输正则性假设下，我们证明了足够小的内蕴平滑优于未平滑的经验测度。我们的环境界量化了周围维数增加如何限制该界所建议的平滑尺度。对于潜在平滑，分析表明其收益取决于重建精度以及编码器-解码器对如何沿流形保持运动并响应其他方向上的噪声。该界的二次近似产生了带宽选择规则以及在此比较中有利于潜在平滑的条件。受控的合成实验检验了预测的带宽和几何趋势。一项探索性的MNIST研究说明了潜在平滑相对于像素空间扰动的效果。
英文摘要
Smoothing empirical data can fill gaps between observations, while noise in the surrounding space can move mass away from the manifold supporting the data. We derive Wasserstein error bounds for ambient Gaussian smoothing and for smoothing through an encoder--decoder representation. For a compact connected smooth manifold without boundary and a smooth positive target density, established heat-flow and entropy estimates imply that sufficiently small intrinsic heat smoothing improves the squared intrinsic Wasserstein error. Coupling this estimate with ambient noise yields a bound whose quadratic term includes ambient dimension and a curvature correction. Our main latent bound tracks reconstruction error, tangential distortion, and the decoder's response to orthogonal latent noise. Its quadratic surrogate gives bandwidth rules and explicit conditions under which the latent upper-bound surrogate is smaller than the ambient one. These surrogate comparisons do not assert an ordering of the exact Wasserstein errors. Synthetic experiments examine the predicted scaling and latent geometry on spheres and a product of spheres.
Comments43 pages
