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

Robert A. Vandermeulen

math.STOct 7, 2026

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

Authors: Robert A. Vandermeulen

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.

Figures & tables

Figure 1: Illustration of a silhouette operator for

k=2

, corresponding to

a=-2

and

b_{1}=b_{2}=1

in Theorem 1 . A measure

\mu

\mathbb{R}^{2}

is mapped to its one-dimensional pushforwards along the resulting four projection directions. When

\mu

has rank at most

2

, these four projected measures uniquely determine

\mu

Panel of Figure 2 — Figure 2: Ground-truth component assignments and partitions recovered by SME using $1$ , $2$ , $4$ , and $6$ non-axis projections. Colors and markers indicate component labels. Every fit also matches the empirical coordinate marginals exactly by construction.

Dataset $(d,n)$	HEPMASS $(21,250)$	kin8nm $(9,1000)$
SME (proposed)	$\mathbf{-27.36\pm 0.13}$ $(3.1\pm 0.8)$	$\mathbf{-11.495\pm 0.051}$ $(2.2\pm 0.4)$
KDE	$-30.71\pm 0.09$ $(2\times 10^{-6})$	$-12.386\pm 0.016$ $(2\times 10^{-6})$
NB-KDE	$-29.24\pm 0.70$ $(2\times 10^{-6})$	$-11.563\pm 0.012$ $(4\times 10^{-5})$
GMM	$-28.19\pm 0.25$ $(2\times 10^{-6})$	$-12.176\pm 0.059$ $(2\times 10^{-6})$
GMM-AA	$-27.73\pm 0.12$ $(2\times 10^{-6})$	$-12.293\pm 0.037$ $(2\times 10^{-6})$
MAF	$-28.57\pm 0.55$ $(2\times 10^{-6})$	$-12.216\pm 0.107$ $(2\times 10^{-6})$

Table 1: Test log-likelihood (mean

\pm

standard deviation across

20

random splits). For SME, parentheses report the validation-selected number of components

k

(mean

\pm

standard deviation); for competing methods, parentheses report two-sided Wilcoxon signed-rank

p

-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

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

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