arXiv：cs.LG· Robert A. Vandermeulen·· 4 小时前AI 评分30
轮廓算子：从一维投影识别低秩测度
The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections
AI 导读
研究者提出"轮廓算子"框架，将低秩符号测度映射为一维线性前推测度的有限集合，证明2k个投影边缘分布即可唯一识别紧支撑的秩≤k符号测度，且该数量最优、投影方向不能任意选取。基于此框架提出轮廓混合估计（SME），通过Wasserstein距离匹配一维投影边缘分布来构造低秩经验测度，结合一维密度估计器后，在中等维度和样本量下优于多种参数、非参数及深度学习基线。
正文
Abstract: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.
| Subjects: | Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Functional Analysis (math.FA); Machine Learning (stat.ML) |
| MSC classes: | 62H05 (Primary), 62H30, 62G07, 28A33 (Secondary) |
| Cite as: | arXiv:2610.09687 [math.ST] |
| (or arXiv:2610.09687v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2610.09687 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Robert Vandermeulen [view email]
[v1]
Wed, 7 Oct 2026 08:49:43 UTC (258 KB)
来源：arXiv：cs.LG · arxiv.org