轮廓算子：一维投影下低秩测度的可辨识性
The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections
浏览论文内容
中文总结 AI 辅助
本文提出轮廓算子框架，证明2k个一维投影边际可最优识别紧支撑秩≤k符号测度，并引入轮廓混合估计（SME）实现高效非参数密度估计，性能优于多种基线。
中文摘要 AI 辅助
结构化恢复现象，如压缩感知中的受限等距性质，已表明高维对象通常可以从极低维的线性测量中重建。本文为$\mathbb{R}^2$上的低秩符号测度开发了一个类似的恢复框架，这里的低秩符号测度定义为可以表示为具有一维因子的乘积测度的有限和的测度。该框架基于线性算子，称为“轮廓算子”，它将一个测度映射到固定的有限集合的一维线性前推。主要结果表明，适当选择的$2k$个投影边际足以识别每个紧支撑的秩$\le k$符号测度，这个数量是最优的，并且投影方向不能任意选择。该框架还扩展到更高维的乘积测度之和，通过建立充分条件，使得成对边际的集合能够识别完整模型。在此框架基础上，引入了一种计算高效的估计器，称为“轮廓混合估计”（SME），用于通过将数据的一维投影边际与对应的经验边际在Wasserstein距离上匹配，从数据构建低秩经验测度。当与一维密度估计器结合时，SME产生一种高效的非参数密度估计器，在中等维度和样本量设置下，相对于一系列参数、非参数和深度学习基线表现出强劲性能。
英文摘要
Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.
