# Matrix completion with data-dependent missingness probabilities

arXiv: https://arxiv.org/abs/2106.02290 · HTML: https://mlsanity.com/paper/2106.02290/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2021.

## Built on by (most cited first, 15 of 15)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Doubly Robust Inference in Causal Latent Factor Models](https://mlsanity.com/paper/2402.11652/) (2024)
- [Symmetric Matrix Completion with ReLU Sampling](https://mlsanity.com/paper/2406.05822/) (2024)
- [An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function](https://mlsanity.com/paper/2503.23832/) (2025)
- [Learning Counterfactual Distributions via Kernel Nearest Neighbors](https://mlsanity.com/paper/2410.13381/) (2024)
- [Optimal Transfer Learning for Missing Not-at-Random Matrix Completion](https://mlsanity.com/paper/2503.00174/) (2025)
- [Exploiting Observation Bias to Improve Matrix Completion](https://mlsanity.com/paper/2306.04775/) (2023)
- [Two-Sided Nearest Neighbors: An adaptive and minimax optimal procedure for matrix completion](https://mlsanity.com/paper/2411.12965/) (2024)
- [Adaptively-weighted Nearest Neighbors for Matrix Completion](https://mlsanity.com/paper/2505.09612/) (2025)
- [Learning to Recommend Using Non-Uniform Data](https://mlsanity.com/paper/2110.11248/) (2021)
- [Matrix Completion via Residual Spectral Matching](https://mlsanity.com/paper/2412.10005/) (2024)
- [Fast exact recovery of noisy matrix from few entries: the infinity norm approach](https://mlsanity.com/paper/2501.19224/) (2025)
- [Truncated Matrix Completion - An Empirical Study](https://mlsanity.com/paper/2504.09873/) (2025)
- [N2: A Unified Python Package and Test Bench for Nearest Neighbor-Based Matrix Completion](https://mlsanity.com/paper/2506.04166/) (2025)
- [Synthetic Nearest Neighbors: Extending Synthetic Controls for Matrix Completion with Missing Not at Random Data](https://mlsanity.com/paper/2609.13586/) (2026)

## Abstract (arXiv)

The problem of completing a large matrix with lots of missing entries has received widespread attention in the last couple of decades. Two popular approaches to the matrix completion problem are based on singular value thresholding and nuclear norm minimization. Most of the past works on this subject assume that there is a single number p such that each entry of the matrix is available independently with probability p and missing otherwise. This assumption may not be realistic for many applications. In this work, we replace it with the assumption that the probability that an entry is available is an unknown function f of the entry itself. For example, if the entry is the rating given to a movie by a viewer, then it seems plausible that high value entries have greater probability of being available than low value entries. We propose two new estimators, based on singular value thresholding and nuclear norm minimization, to recover the matrix under this assumption. The estimators involve no tuning parameters, and are shown to be consistent under a low rank assumption. We also provide a consistent estimator of the unknown function f.