用切片矩特征高效表示神经网络分布，比传统池化更精准
EMPEROR: Efficient Moment-Preserving Representation of Distributions
- 通过多方向投影并拟合轻量高斯混合模型，编码特征分布
- 理论保证唯一性，有限样本误差随切片数和样本数最优下降
- 适合需要分布感知的图像、文本等多模态任务
我们提出EMPEROR（高效矩保持分布表示），一种数学严谨且计算高效的框架，用于表示神经网络中出现的高维概率测度。与启发式全局池化不同，EMPEROR通过统计矩编码特征分布。该方法基于切片矩理论：将特征投影到多个方向，对每个投影拟合轻量一维高斯混合模型（GMM），并将切片参数聚合为紧凑描述符。我们通过Carleman条件和Cramér-Wold定理建立确定性保证，确保GMM由其切片矩唯一确定，并推导出随切片数和样本数最优缩放的有限样本误差界。实验表明，EMPEROR在多种数据模态下捕捉的分布信息比常见池化方案更丰富，同时保持计算效率与广泛适用性。
原文摘要 · Abstract (English)
We introduce EMPEROR (Efficient Moment-Preserving Representation of Distributions), a mathematically rigorous and computationally efficient framework for representing high-dimensional probability measures arising in neural network representations. Unlike heuristic global pooling operations, EMPEROR encodes a feature distribution through its statistical moments. Our approach leverages the theory of sliced moments: features are projected onto multiple directions, lightweight univariate Gaussian mixture models (GMMs) are fit to each projection, and the resulting slice parameters are aggregated into a compact descriptor. We establish determinacy guarantees via Carleman's condition and the Cramér-Wold theorem, ensuring that the GMM is uniquely determined by its sliced moments, and we derive finite-sample error bounds that scale optimally with the number of slices and samples. Empirically, EMPEROR captures richer distributional information than common pooling schemes across various data modalities, while remaining computationally efficient and broadly applicable.
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