The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections
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 ℝ2, 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-≤ 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.
Related
- Ergodic × p-invariant measures on 𝕋2 with no dimension dropping projections
- Absolutely Continuous Convolutions and Projections of Fractal Measures
- Absolute continuity of projections and dimension conservation for self-similar sets and measures
- Deep operator learning for efficient sampling from invariant measures of stochastic differential equations
- Finite-dimensional normed spaces without small-norm hyperplane projections
BibTeX
@misc{vandermeulen2026silhouette,
title = {{The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections}},
author = {Robert A. Vandermeulen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09687},
eprint = {2610.09687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09687},
}
RIS
TY - UNPB
TI - The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections
AU - Robert A. Vandermeulen
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09687
UR - https://arxiv.org/abs/2610.09687
AB - 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 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-≤ 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.
ER -