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abstract indexed · arXiv 2610.09687 · updated 2026-10-18

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

First page of The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections
From the original paper · arXiv 2610.09687Page 1. Select the image to read the PDF.

SME replaces general multivariate nonparametric or deep-learning density estimation with a low-rank empirical measure built by matching 2k one-dimensional projected marginals in Wasserstein distance. It costs an identifiability constraint (the rank-k structure and optimally chosen, non-arbitrary projection directions) plus the complexity of Wasserstein marginal matching, while remaining cheap in compute. It fails when the true distribution is not well approximated by a finite sum of product measures, when projection directions are poorly chosen, or in high-dimensional regimes beyond the moderate settings evaluated.

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Silhouette mixture estimation for low-rank density recovery
published
2026-10-07

This is a theoretical statistics and density-estimation result outside production LLM inference or training practice, so it is worth monitoring only if the reader's workloads include moderate-dimensional nonparametric density estimation.

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  1. claim 01Paperraft inference
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    SME replaces general multivariate nonparametric or deep-learning density estimation with a low-rank empirical measure built by matching 2k one-dimensional projected marginals in Wasserstein distance. It costs an identifiability constraint (the rank-k structure and optimally chosen, non-arbitrary projection directions) plus the complexity of Wasserstein marginal matching, while remaining cheap in compute. It fails when the true distribution is not well approximated by a finite sum of product measures, when projection directions are poorly chosen, or in high-dimensional regimes beyond the moderate settings evaluated.

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  2. claim 02Paperraft inference

    The authors report that SME, combined with one-dimensional density estimators, performs strongly relative to parametric, nonparametric, and deep-learning baselines in moderate dimension and sample size; no quantitative factor is given in the abstract.

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arXiv 2610.09687

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