# The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections

arXiv: https://arxiv.org/abs/2610.09687 · HTML: https://mlsanity.com/paper/2610.09687/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 90 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 4 of 4)

- [Estimation and inference for the Wasserstein distance between mixing measures in topic models](https://mlsanity.com/paper/2206.12768/) (2022)
- [Sample Complexity Using Infinite Multiview Models](https://mlsanity.com/paper/2302.04292/) (2023)
- [PMODE: Theoretically Grounded and Modular Mixture Modeling](https://mlsanity.com/paper/2508.21396/) (2025)
- [Optimal Estimation of Discrete Multiview Distributions under Heteroskedastic Multinomial Sampling](https://mlsanity.com/paper/2605.24858/) (2026)

## Abstract (arXiv)

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.