math.STOct 7, 2026

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

Authors: Robert A. Vandermeulen

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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\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 2k2k projected marginals suffices to identify every compactly supported rank-≤k\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.

Figures & tables

Figure 1: Illustration of a silhouette operator for k=2k=2 , corresponding to a=−2a=-2 and b1=b2=1b_{1}=b_{2}=1 in Theorem 1 . A measure μ\mu on R2\mathbb{R}^{2} is mapped to its one-dimensional pushforwards along the resulting four projection directions. When μ\mu has rank at most 22 , these four projected measures uniquely determine μ\mu .
Figure 2: Ground-truth component assignments and partitions recovered by SME using 11 , 22 , 44 , and 66 non-axis projections. Colors and markers indicate component labels. Every fit also matches the empirical coordinate marginals exactly by construction.
Dataset (d,n)(d,n)HEPMASS (21,250)(21,250)kin8nm (9,1000)(9,1000)
SME (proposed)−27.36±0.13\mathbf{-27.36\pm 0.13} (3.1±0.8)(3.1\pm 0.8)−11.495±0.051\mathbf{-11.495\pm 0.051} (2.2±0.4)(2.2\pm 0.4)
KDE−30.71±0.09-30.71\pm 0.09 (2×10−6)(2\times 10^{-6})−12.386±0.016-12.386\pm 0.016 (2×10−6)(2\times 10^{-6})
NB-KDE−29.24±0.70-29.24\pm 0.70 (2×10−6)(2\times 10^{-6})−11.563±0.012-11.563\pm 0.012 (4×10−5)(4\times 10^{-5})
GMM−28.19±0.25-28.19\pm 0.25 (2×10−6)(2\times 10^{-6})−12.176±0.059-12.176\pm 0.059 (2×10−6)(2\times 10^{-6})
GMM-AA−27.73±0.12-27.73\pm 0.12 (2×10−6)(2\times 10^{-6})−12.293±0.037-12.293\pm 0.037 (2×10−6)(2\times 10^{-6})
MAF−28.57±0.55-28.57\pm 0.55 (2×10−6)(2\times 10^{-6})−12.216±0.107-12.216\pm 0.107 (2×10−6)(2\times 10^{-6})
Table 1: Test log-likelihood (mean ±\pm standard deviation across 2020 random splits). For SME, parentheses report the validation-selected number of components kk (mean ±\pm standard deviation); for competing methods, parentheses report two-sided Wilcoxon signed-rank pp -values from paired comparisons with SME. Bold indicates the best mean in each column.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Silhouette objective and estimator log-likelihood during SME optimization on HEPMASS.

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