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

2026-10-07

AI Summary

This study addresses the challenges of identifiability and efficient estimation in recovering low-rank signed measures from one-dimensional projections. To this end, it establishes a theoretical framework based on profile operators, proving that only 2k projections are required to uniquely identify a measure of rank k. Furthermore, this work proposes a sliced measure estimation (SME) algorithm grounded in the Wasserstein distance, which leverages linear pushforward maps to achieve computationally efficient empirical measure reconstruction. The theoretical analysis is subsequently extended to high-dimensional product measures and nonparametric density estimation settings. By establishing optimal conditions for both the number and directions of projections, the proposed approach demonstrates significantly superior performance over existing baseline models in moderate-dimensional regimes.

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.

Problem

• low-rank signed measures
• one-dimensional projections
• identifiability
• pushforward marginals
• structured recovery

Innovation

• Silhouette Operator
• Low-Rank Signed Measures
• One-Dimensional Projections
• Silhouette Mixture Estimation
• Nonparametric Density Estimation