# Synthetic Nearest Neighbors: Extending Synthetic Controls for Matrix Completion with Missing Not at Random Data

arXiv: https://arxiv.org/abs/2609.13586 · HTML: https://mlsanity.com/paper/2609.13586/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 60 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (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)
- [Robust Synthetic Control](https://mlsanity.com/paper/1711.06940/) (2017)
- [The Augmented Synthetic Control Method](https://mlsanity.com/paper/1811.04170/) (2018)
- [Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data](https://mlsanity.com/paper/2001.05484/) (2020)
- [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)
- [Matrix Completion When Missing Is Not at Random and Its Applications in Causal Panel Data Models](https://mlsanity.com/paper/2308.02364/) (2023)
- [Entrywise Inference for Missing Panel Data: A Simple and Instance-Optimal Approach](https://mlsanity.com/paper/2401.13665/) (2024)
- [Learning Treatment Effects in Panels with General Intervention Patterns](https://mlsanity.com/paper/2106.02780/) (2021)
- [Same Root Different Leaves: Time Series and Cross-Sectional Methods in Panel Data](https://mlsanity.com/paper/2207.14481/) (2022)
- [From Control to Treatment: Causal Forecasting Beyond the Observed Panel](https://mlsanity.com/paper/2606.18512/) (2026)

## Abstract (arXiv)

We develop a causal framework for matrix completion under missing not at random (MNAR) data. Drawing on synthetic controls from the econometric panel data literature, our approach relaxes two assumptions common in MNAR matrix completion: positivity and independence of observation indicators. Unlike traditional panel data models, which often require prescribed block-sparse geometries, our framework accommodates flexible, heterogeneous observation patterns through target-specific local information structures. We propose synthetic nearest neighbors (SNN), a local synthetic-controls-inspired estimator, and establish finite-sample entrywise error bounds and consistency for mean recovery under suitable conditions. We further derive asymptotic normality under heteroskedastic noise and develop feasible entrywise inference. To estimate entry-specific noise variances, we apply the same local principle to squared outcomes, obtaining consistency under bounded noise and asymptotic unbiasedness under general subgaussian noise. Simulation studies corroborate the theoretical findings across a range of missingness designs and observation patterns.