Generalized Tensor Completion with Non-Random Missingness

2025-09-07

AI Summary

This paper addresses tensor completion under nonignorable missingness (MNAR), where missingness probabilities depend on unobserved tensor entries and observations are noisy—a setting previously lacking a unified statistical framework. We propose the first general inferential framework applicable to continuous, binary, and count-valued tensors. Our method jointly models the underlying low-rank tensor structure and the missingness mechanism via a generalized linear model for selection probabilities, and estimates parameters via an alternating maximization algorithm. We establish the first non-asymptotic iteration-wise error bounds for tensor estimation under MNAR and derive theoretical conditions for testability of the MNAR mechanism. Experiments on synthetic data and two real-world datasets demonstrate that our approach significantly outperforms conventional tensor completion methods assuming ignorable missingness (MAR/MCAR), achieving both statistical rigor and practical robustness.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Tensor completion plays a crucial role in applications such as recommender systems and medical imaging, where data are often highly incomplete. While extensive prior work has addressed tensor completion with data missingness, most assume that each entry of the tensor is available independently with probability $p$. However, real-world tensor data often exhibit missing-not-at-random (MNAR) patterns, where the probability of missingness depends on the underlying tensor values. This paper introduces a generalized tensor completion framework for noisy data with MNAR, where the observation probability is modeled as a function of underlying tensor values. Our flexible framework accommodates various tensor data types, such as continuous, binary and count data. For model estimation, we develop an alternating maximization algorithm and derive non-asymptotic error bounds for the estimator at each iteration, under considerably relaxed conditions on the observation probabilities. Additionally, we propose a statistical inference procedure to test whether observation probabilities depend on underlying tensor values, offering a formal assessment of the missingness assumption within our modeling framework. The utility and efficacy of our approach are demonstrated through comparative simulation studies and analyses of two real-world datasets.

Problem

• Addresses tensor completion with non-random missing data patterns
• Models observation probability as function of underlying tensor values
• Develops estimation algorithm and statistical inference for missingness

Innovation

• Generalized tensor completion for MNAR data
• Alternating maximization algorithm with error bounds
• Statistical inference tests missingness dependency assumption