Conformalized Tensor Completion with Riemannian Optimization

Abstract

Tensor data, or multidimensional arrays, 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 that contain missing values, and tensor completion aims to estimate the missing values given the partially observed tensor. Efforts have been spared in devising scalable tensor completion algorithms, but few on quantifying the uncertainty of the estimator. In this article, 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. 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. Supplemental materials for the article are available online.