Optimal Estimation of Discrete Multiview Distributions under Heteroskedastic Multinomial Sampling

2026-05-24

AI Summary

This work addresses the challenging problem of discrete multi-view density tensor estimation under heteroscedastic and negatively correlated noise. The authors propose a general scaling framework that integrates spectral methods with ℓ₁ loss to construct both oracle and data-driven low-rank nonnegative tensor estimators. For the first time, they establish minimax upper and lower bounds in Frobenius norm that explicitly depend on fiber-wise quality, revealing how the distribution of sample quality fundamentally influences estimation difficulty. They further prove that the proposed oracle estimator achieves near-optimal performance under fixed rank and slice normalization constraints. Both theoretical analysis and simulation experiments confirm the near-optimality, robustness, and broad applicability of the proposed approach.

Abstract

Multiview latent-variable models provide a fundamental framework for discrete data analysis, with applications to latent structure models, topic models, and mixtures of product distributions. In the discrete setting, the joint distribution of the observed views can be represented as a nonnegative low-rank tensor, which we call a multiview density tensor. We study the problem of estimating this tensor from multinomial count data. A key challenge is that multinomial sampling induces heteroskedastic and dependent noise, so the difficulty of estimation depends not only on the ambient dimensions and rank, but also on how the probability mass is distributed across different locations of sample space. We propose a general scaling framework for density tensor estimation under multinomial sampling. This framework leads to a spectral estimator for which we prove a Frobenius-norm upper bound that directly handles heteroskedasticity and negative dependence. For the original multiview model, we obtain fiber-mass-dependent Frobenius upper bounds and minimax lower bounds showing that this dependence is unavoidable. Under $\ell_1$ loss, we develop both oracle and feasible data-driven estimators based on the same scaling principle, establish minimax lower bounds, and show near-optimality for the oracle rule at fixed rank and for slice normalization under bounded slice-to-fiber imbalance. Simulations support the theory and demonstrate the robustness of the proposed methods.

Problem

• multiview distributions
• heteroskedastic multinomial sampling
• density tensor estimation
• discrete data analysis
• latent-variable models

Innovation

• multiview latent-variable models
• heteroskedastic multinomial sampling
• density tensor estimation
• spectral estimator
• minimax optimality