# Entry-Specific Matrix Estimation under Arbitrary Sampling Patterns through the Lens of Network Flows

arXiv: https://arxiv.org/abs/2409.03980 · HTML: https://mlsanity.com/paper/2409.03980/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation](https://mlsanity.com/paper/1802.08397/) (2018)
- [Tightness of the maximum likelihood semidefinite relaxation for angular synchronization](https://mlsanity.com/paper/1411.3272/) (2014)
- [Adversarial Crowdsourcing Through Robust Rank-One Matrix Completion](https://mlsanity.com/paper/2010.12181/) (2020)
- [Design-based Analysis in Difference-In-Differences Settings with Staggered Adoption](https://mlsanity.com/paper/1808.05293/) (2018)
- [Consistency of weighted majority votes](https://mlsanity.com/paper/1312.0451/) (2013)
- [Gradient Descent for Sparse Rank-One Matrix Completion for Crowd-Sourced Aggregation of Sparsely Interacting Workers](https://mlsanity.com/paper/1904.11608/) (2019)
- [Computational Limits for Matrix Completion](https://mlsanity.com/paper/1402.2331/) (2014)
- [Fixed-Effect Regressions on Network Data](https://mlsanity.com/paper/1608.01532/) (2016)
- [Partial Matrix Completion](https://mlsanity.com/paper/2208.12063/) (2022)
- [Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling](https://mlsanity.com/paper/2403.00184/) (2024)

## Abstract (arXiv)

Matrix completion tackles the task of predicting missing values in a low-rank matrix based on a sparse set of observed entries. It is often assumed that the observation pattern is generated uniformly at random or has a very specific structure tuned to a given algorithm. There is still a gap in our understanding when it comes to arbitrary sampling patterns. Given an arbitrary sampling pattern, we introduce a matrix completion algorithm based on network flows in the bipartite graph induced by the observation pattern. For additive matrices, the particular flow we used is the electrical flow and we establish error upper bounds customized to each entry as a function of the observation set, along with matching minimax lower bounds. Our results show that the minimax squared error for recovery of a particular entry in the matrix is proportional to the effective resistance of the corresponding edge in the graph. Furthermore, we show that our estimator is equivalent to the least squares estimator. We apply our estimator to the two-way fixed effects model and show that it enables us to accurately infer individual causal effects and the unit-specific and time-specific confounders. For rank-1 matrices, we use edge-disjoint paths to form an estimator that achieves minimax optimal estimation when the sampling is sufficiently dense. Our discovery introduces a new family of estimators parametrized by network flows, which provide a fine-grained and intuitive understanding of the impact of the given sampling pattern on the relative difficulty of estimation at an entry-specific level. This graph-based approach allows us to quantify the inherent complexity of matrix completion for individual entries, rather than relying solely on global measures of performance.