# On the Fundamental Limits of Matrix Completion: Leveraging Hierarchical Similarity Graphs

arXiv: https://arxiv.org/abs/2109.05408 · HTML: https://mlsanity.com/paper/2109.05408/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 70 references that are in the map.
- First posted on arXiv 12 Sep 2021.

## Builds on (most cited first, 16 of 16)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks](https://mlsanity.com/paper/1704.06803/) (2017)
- [Exact Recovery in the Stochastic Block Model](https://mlsanity.com/paper/1405.3267/) (2014)
- [Community Detection in Networks with Node Attributes](https://mlsanity.com/paper/1401.7267/) (2014)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (2019)
- [Hierarchical Block Structures and High-resolution Model Selection in Large Networks](https://mlsanity.com/paper/1310.4377/) (2013)
- [Clustering with Same-Cluster Queries](https://mlsanity.com/paper/1606.02404/) (2016)
- [Matrix Completion on Graphs](https://mlsanity.com/paper/1408.1717/) (2014)
- [Community Detection and Classification in Hierarchical Stochastic Blockmodels](https://mlsanity.com/paper/1503.02115/) (2015)
- [Query Complexity of Clustering with Side Information](https://mlsanity.com/paper/1706.07719/) (2017)
- [Information Limits for Recovering a Hidden Community](https://mlsanity.com/paper/1509.07859/) (2015)
- [Information-theoretic bounds for exact recovery in weighted stochastic block models using the Renyi divergence](https://mlsanity.com/paper/1509.06418/) (2015)
- [Adapting the Stochastic Block Model to Edge-Weighted Networks](https://mlsanity.com/paper/1305.5782/) (2013)
- [Recovering a Single Community with Side Information](https://mlsanity.com/paper/1809.01738/) (2018)
- [MC2G: An Efficient Algorithm for Matrix Completion with Social and Item Similarity Graphs](https://mlsanity.com/paper/2006.04373/) (2020)
- [Collaborative Filtering with Social Local Models](https://mlsanity.com/paper/1704.05735/) (2017)

## Built on by (most cited first, 2 of 2)

- [Online Low Rank Matrix Completion](https://mlsanity.com/paper/2209.03997/) (2022)
- [Online Matrix Completion: A Collaborative Approach with Hott Items](https://mlsanity.com/paper/2408.05843/) (2024)

## Abstract (arXiv)

We study the matrix completion problem that leverages hierarchical similarity graphs as side information in the context of recommender systems. Under a hierarchical stochastic block model that well respects practically-relevant social graphs and a low-rank rating matrix model, we characterize the exact information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) by proving sharp upper and lower bounds on the sample complexity. In the achievability proof, we demonstrate that probability of error of the maximum likelihood estimator vanishes for sufficiently large number of users and items, if all sufficient conditions are satisfied. On the other hand, the converse (impossibility) proof is based on the genie-aided maximum likelihood estimator. Under each necessary condition, we present examples of a genie-aided estimator to prove that the probability of error does not vanish for sufficiently large number of users and items. One important consequence of this result is that exploiting the hierarchical structure of social graphs yields a substantial gain in sample complexity relative to the one that simply identifies different groups without resorting to the relational structure across them. More specifically, we analyze the optimal sample complexity and identify different regimes whose characteristics rely on quality metrics of side information of the hierarchical similarity graph. Finally, we present simulation results to corroborate our theoretical findings and show that the characterized information-theoretic limit can be asymptotically achieved.