# Causal Matrix Completion

arXiv: https://arxiv.org/abs/2109.15154 · HTML: https://mlsanity.com/paper/2109.15154/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 66 references that are in the map.
- First posted on arXiv 30 Sep 2021.

## Builds on (most cited first, 21 of 21)

- [Recommendations as Treatments: Debiasing Learning and Evaluation](https://mlsanity.com/paper/1602.05352/) (2016)
- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Modeling User Exposure in Recommendation](https://mlsanity.com/paper/1510.07025/) (2015)
- [Synthetic Difference in Differences](https://mlsanity.com/paper/1812.09970/) (2018)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019)
- [Causal Inference with Noisy and Missing Covariates via Matrix Factorization](https://mlsanity.com/paper/1806.00811/) (2018)
- [Missing Not at Random in Matrix Completion: The Effectiveness of Estimating Missingness Probabilities Under a Low Nuclear Norm Assumption](https://mlsanity.com/paper/1910.12774/) (2019)
- [Rates of Convergence of Spectral Methods for Graphon Estimation](https://mlsanity.com/paper/1709.03183/) (2017)
- [On Robustness of Principal Component Regression](https://mlsanity.com/paper/1902.10920/) (2019)
- [Synthetic Interventions](https://mlsanity.com/paper/2006.07691/) (2020)
- [Robust Synthetic Control](https://mlsanity.com/paper/1711.06940/) (2017)
- [Imputation and low-rank estimation with Missing Not At Random data](https://mlsanity.com/paper/1812.11409/) (2018)
- [Matrix Completion via Max-Norm Constrained Optimization](https://mlsanity.com/paper/1303.0341/) (2013)
- [Causal Inference with Corrupted Data: Measurement Error, Missing Values, Discretization, and Differential Privacy](https://mlsanity.com/paper/2107.02780/) (2021)
- [High-dimensional principal component analysis with heterogeneous missingness](https://mlsanity.com/paper/1906.12125/) (2019)
- [Matrix completion with data-dependent missingness probabilities](https://mlsanity.com/paper/2106.02290/) (2021)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Collaborative Filtering with Social Exposure: A Modular Approach to Social Recommendation](https://mlsanity.com/paper/1711.11458/) (2017)
- [TenIPS: Inverse Propensity Sampling for Tensor Completion](https://mlsanity.com/paper/2101.00323/) (2021)
- [The Optimal Hard Threshold for Singular Values is 4/sqrt(3)](https://mlsanity.com/paper/1305.5870/) (2013)

## Built on by (most cited first, 30 of 45)

- [EdgeBench: Unveiling Scaling Laws of Learning from Real-World Environments](https://mlsanity.com/paper/2607.05155/) (2026)
- [Learning to Extrapolate: A Transductive Approach](https://mlsanity.com/paper/2304.14329/) (2023)
- [Counterfactual inference in sequential experiments](https://mlsanity.com/paper/2202.06891/) (2022)
- [Synthetic Combinations: A Causal Inference Framework for Combinatorial Interventions](https://mlsanity.com/paper/2303.14226/) (2023)
- [CausalSim: A Causal Framework for Unbiased Trace-Driven Simulation](https://mlsanity.com/paper/2201.01811/) (2022)
- [Doubly robust nearest neighbors in factor models](https://mlsanity.com/paper/2211.14297/) (2022)
- [Speed Up the Cold-Start Learning in Two-Sided Bandits with Many Arms](https://mlsanity.com/paper/2210.00340/) (2022)
- [Adaptive Principal Component Regression with Applications to Panel Data](https://mlsanity.com/paper/2307.01357/) (2023)
- [Tackling Combinatorial Distribution Shift: A Matrix Completion Perspective](https://mlsanity.com/paper/2307.06457/) (2023)
- [High-dimensional logistic regression with missing data: Imputation, regularization, and universality](https://mlsanity.com/paper/2410.01093/) (2024)
- [Strategyproof Decision-Making in Panel Data Settings and Beyond](https://mlsanity.com/paper/2211.14236/) (2022)
- [Double and Single Descent in Causal Inference with an Application to High-Dimensional Synthetic Control](https://mlsanity.com/paper/2305.00700/) (2023)
- [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)
- [One Pipeline, Many Transformers: Pattern-Specific Imputation Specialists for Tabular Missing Data](https://mlsanity.com/paper/2510.02625/) (2025)
- [Network Synthetic Interventions: A Causal Framework for Panel Data Under Network Interference](https://mlsanity.com/paper/2210.11355/) (2022)
- [Matrix Estimation for Offline Reinforcement Learning with Low-Rank Structure](https://mlsanity.com/paper/2305.15621/) (2023)
- [Synthetic Potential Outcomes and Causal Mixture Identifiability](https://mlsanity.com/paper/2405.19225/) (2024)
- [Distributional Matrix Completion via Nearest Neighbors in the Wasserstein Space](https://mlsanity.com/paper/2410.13112/) (2024)
- [Optimal Transfer Learning for Missing Not-at-Random Matrix Completion](https://mlsanity.com/paper/2503.00174/) (2025)
- [Causal Imputation for Counterfactual SCMs: Bridging Graphs and Latent Factor Models](https://mlsanity.com/paper/2402.14777/) (2024)
- [Causal Customer Churn Analysis with Low-rank Tensor Block Hazard Model](https://mlsanity.com/paper/2405.11377/) (2024)
- [Structured Learning of Compositional Sequential Interventions](https://mlsanity.com/paper/2406.05745/) (2024)
- [Learning Counterfactual Distributions via Kernel Nearest Neighbors](https://mlsanity.com/paper/2410.13381/) (2024)
- [A Causal Inference Framework for Data Rich Environments](https://mlsanity.com/paper/2504.01702/) (2025)
- [Covariate-Adjusted Deep Causal Learning for Heterogeneous Panel Data Models](https://mlsanity.com/paper/2505.20536/) (2025)
- [SYN-DIGITS: A Synthetic Control Framework for Calibrated Digital Twin Simulation](https://mlsanity.com/paper/2604.07513/) (2026)
- [Exploiting Observation Bias to Improve Matrix Completion](https://mlsanity.com/paper/2306.04775/) (2023)
- [Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling](https://mlsanity.com/paper/2403.00184/) (2024)
- [General Causal Imputation via Synthetic Interventions](https://mlsanity.com/paper/2410.20647/) (2024)

## Abstract (arXiv)

Matrix completion is the study of recovering an underlying matrix from a sparse subset of noisy observations. Traditionally, it is assumed that the entries of the matrix are "missing completely at random" (MCAR), i.e., each entry is revealed at random, independent of everything else, with uniform probability. This is likely unrealistic due to the presence of "latent confounders", i.e., unobserved factors that determine both the entries of the underlying matrix and the missingness pattern in the observed matrix. For example, in the context of movie recommender systems -- a canonical application for matrix completion -- a user who vehemently dislikes horror films is unlikely to ever watch horror films. In general, these confounders yield "missing not at random" (MNAR) data, which can severely impact any inference procedure that does not correct for this bias. We develop a formal causal model for matrix completion through the language of potential outcomes, and provide novel identification arguments for a variety of causal estimands of interest. We design a procedure, which we call "synthetic nearest neighbors" (SNN), to estimate these causal estimands. We prove finite-sample consistency and asymptotic normality of our estimator. Our analysis also leads to new theoretical results for the matrix completion literature. In particular, we establish entry-wise, i.e., max-norm, finite-sample consistency and asymptotic normality results for matrix completion with MNAR data. As a special case, this also provides entry-wise bounds for matrix completion with MCAR data. Across simulated and real data, we demonstrate the efficacy of our proposed estimator.