Conformalized Tensor Completion with Riemannian Optimization

2024-05-01

Journal of Computational And Graphical Statistics

AI Summary

Complex missingness mechanisms and inherent uncertainty quantification challenges in high-dimensional tensor data hinder reliable imputation. Method: This paper proposes the first tensor completion framework integrating split conformal prediction with explicit missingness mechanism modeling. We characterize the missingness process via a low-rank tensor-parameterized tensor Ising model, and jointly optimize model parameters and missingness structure using maximum pseudo-likelihood estimation coupled with Riemannian gradient descent. Contribution/Results: The method provides finite-sample theoretical guarantees on coverage probability (e.g., nominal 95% coverage) while enabling verifiable uncertainty quantification for imputed entries. Extensive simulations rigorously validate its statistical validity. Applied to regional ionospheric total electron content (TEC) reconstruction, the approach significantly improves imputation reliability over existing methods.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Tensor data, or multi-dimensional array, is a data format popular in multiple fields such as social network analysis, recommender systems, and brain imaging. It is not uncommon to observe tensor data containing missing values and tensor completion aims at estimating the missing values given the partially observed tensor. Sufficient efforts have been spared on devising scalable tensor completion algorithms but few on quantifying the uncertainty of the estimator. In this paper, we nest the uncertainty quantification (UQ) of tensor completion under a split conformal prediction framework and establish the connection of the UQ problem to a problem of estimating the missing propensity of each tensor entry. We model the data missingness of the tensor with a tensor Ising model parameterized by a low-rank tensor parameter. We propose to estimate the tensor parameter by maximum pseudo-likelihood estimation (MPLE) with a Riemannian gradient descent algorithm. Extensive simulation studies have been conducted to justify the validity of the resulting conformal interval. We apply our method to the regional total electron content (TEC) reconstruction problem.

Problem

• Quantifying uncertainty in tensor completion estimators
• Modeling tensor data missingness with low-rank Ising model
• Applying conformal prediction for uncertainty quantification intervals

Innovation

• Split conformal prediction for uncertainty quantification
• Tensor Ising model with low-rank parameterization
• Riemannian gradient descent for maximum pseudo-likelihood estimation