Corpus ID: 237491797

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

  title={On the Fundamental Limits of Matrix Completion: Leveraging Hierarchical Similarity Graphs},
  author={Junhyung Ahn and Adel M. Elmahdy and Soheil Mohajer and Changho Suh},
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… Expand


Matrix Completion with Hierarchical Graph Side Information
This work develops a universal, parameter-free, and computationally efficient algorithm that starts with hierarchical graph clustering and then iteratively refines estimates both on graph clusters and matrix ratings that achieves the information-theoretic limit on the number of observed matrix entries. Expand
Matrix Completion on Graphs
This work introduces a novel matrix completion model that makes use of proximity information about rows and columns by assuming they form communities, and borrows ideas from manifold learning to constrain the solution to be smooth on these graphs, in order to implicitly force row and column proximities. Expand
Binary Rating Estimation with Graph Side Information
To the best of the knowledge, this work is the first to reveal how much the graph side information reduces sample complexity, and proposes a computationally efficient algorithm that achieves the limit. Expand
The Power of Convex Relaxation: Near-Optimal Matrix Completion
  • E. Candès, T. Tao
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
  • 2010
This paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). Expand
SoRec: social recommendation using probabilistic matrix factorization
A factor analysis approach based on probabilistic matrix factorization to solve the data sparsity and poor prediction accuracy problems by employing both users' social network information and rating records is proposed. Expand
Query Complexity of Clustering with Side Information
The dramatic power of side information aka similarity matrix on reducing the query complexity of clustering is shown, and intriguing connection to popular community detection models such as the {\em stochastic block model}, significantly generalizes them, and opens up many venues for interesting future research. Expand
Collaborative Filtering with Graph Information: Consistency and Scalable Methods
This work formulate and derive a highly efficient, conjugate gradient based alternating minimization scheme that solves optimizations with over 55 million observations up to 2 orders of magnitude faster than state-of-the-art (stochastic) gradient-descent based methods. Expand
Collaborative Filtering with Social Local Models
SLOMA is the first work to incorporate social connections into the local low-rank framework by applying social regularization to submatrices factorization, denoted as SLOMA++ and experimental results from two real-world datasets demonstrate the superiority of the proposed models over LLORMA and MF. Expand
Information-theoretic bounds for exact recovery in weighted stochastic block models using the Renyi divergence
The main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order $\frac{1}{2}$ between the distributions of within-community and between-community edges. Expand
Discrete-valued Preference Estimation with Graph Side Information
This work proposes a new model in which the unknown preference matrix can have any discrete values, thereby relaxing the assumptions made in prior work and develops a computationally-efficient algorithm that matches the optimal sample complexity. Expand