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Matrix Completion with Hypergraphs:Sharp Thresholds and Efficient Algorithms

Zhongtian Ma, Qiaosheng Zhang, Zhen Wang

TL;DR

This work addresses exact matrix completion from a sub-sampled rating matrix augmented with observed social graphs and hypergraphs. It introduces MCH, a three-stage algorithm that uses spectral clustering on the hypergraph-augmented graph, majority-rule rating vector estimation, and iterative local refinement to achieve exact recovery. A key result is a sharp threshold for the sample probability $p$, expressed as $p^*= ext{max}igig rac{ ext{log terms}}{I_ heta ext{...}}, rac{K ext{log} m}{I_ heta n}igig$, below which exact recovery is information-theoretically impossible and above which MCH succeeds with high probability; this threshold decreases as hypergraph quality (and the combined information $I_d$) improves. The paper further quantifies the gain from hypergraphs and provides an information-theoretic lower bound that matches the algorithmic threshold in the symmetric setting, supported by synthetic and semi-real experiments. Overall, the results establish both the utility of hypergraphs in matrix completion and the near-optimal sample efficiency of MCH for exact recovery in structured social-data settings.

Abstract

This paper considers the problem of completing a rating matrix based on sub-sampled matrix entries as well as observed social graphs and hypergraphs. We show that there exists a \emph{sharp threshold} on the sample probability for the task of exactly completing the rating matrix -- the task is achievable when the sample probability is above the threshold, and is impossible otherwise -- demonstrating a phase transition phenomenon. The threshold can be expressed as a function of the ``quality'' of hypergraphs, enabling us to \emph{quantify} the amount of reduction in sample probability due to the exploitation of hypergraphs. This also highlights the usefulness of hypergraphs in the matrix completion problem. En route to discovering the sharp threshold, we develop a computationally efficient matrix completion algorithm that effectively exploits the observed graphs and hypergraphs. Theoretical analyses show that our algorithm succeeds with high probability as long as the sample probability exceeds the aforementioned threshold, and this theoretical result is further validated by synthetic experiments. Moreover, our experiments on a real social network dataset (with both graphs and hypergraphs) show that our algorithm outperforms other state-of-the-art matrix completion algorithms.

Matrix Completion with Hypergraphs:Sharp Thresholds and Efficient Algorithms

TL;DR

This work addresses exact matrix completion from a sub-sampled rating matrix augmented with observed social graphs and hypergraphs. It introduces MCH, a three-stage algorithm that uses spectral clustering on the hypergraph-augmented graph, majority-rule rating vector estimation, and iterative local refinement to achieve exact recovery. A key result is a sharp threshold for the sample probability , expressed as , below which exact recovery is information-theoretically impossible and above which MCH succeeds with high probability; this threshold decreases as hypergraph quality (and the combined information ) improves. The paper further quantifies the gain from hypergraphs and provides an information-theoretic lower bound that matches the algorithmic threshold in the symmetric setting, supported by synthetic and semi-real experiments. Overall, the results establish both the utility of hypergraphs in matrix completion and the near-optimal sample efficiency of MCH for exact recovery in structured social-data settings.

Abstract

This paper considers the problem of completing a rating matrix based on sub-sampled matrix entries as well as observed social graphs and hypergraphs. We show that there exists a \emph{sharp threshold} on the sample probability for the task of exactly completing the rating matrix -- the task is achievable when the sample probability is above the threshold, and is impossible otherwise -- demonstrating a phase transition phenomenon. The threshold can be expressed as a function of the ``quality'' of hypergraphs, enabling us to \emph{quantify} the amount of reduction in sample probability due to the exploitation of hypergraphs. This also highlights the usefulness of hypergraphs in the matrix completion problem. En route to discovering the sharp threshold, we develop a computationally efficient matrix completion algorithm that effectively exploits the observed graphs and hypergraphs. Theoretical analyses show that our algorithm succeeds with high probability as long as the sample probability exceeds the aforementioned threshold, and this theoretical result is further validated by synthetic experiments. Moreover, our experiments on a real social network dataset (with both graphs and hypergraphs) show that our algorithm outperforms other state-of-the-art matrix completion algorithms.
Paper Structure (35 sections, 11 theorems, 40 equations, 3 figures, 1 algorithm)

This paper contains 35 sections, 11 theorems, 40 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions.
  3. Related Works.
  4. Notations.
  5. Problem Statement
  6. Model.
  7. Observations.
  8. Objectives.
  9. MCH: An Efficient Matrix Completion Algorithm
  10. Algorithm Description.
  11. Stage 1 (Partial recovery of clusters)
  12. Stage 2 (Estimate rating vectors)
  13. Stage 3 (Local refinement of clusters)
  14. Computational Complexity.
  15. Theoretical Guarantees of MCH
  16. ...and 20 more sections

Key Result

Theorem 1

AssumeThe assumption that the sizes of users and items satisfy $m=\omega(\log n)$ and $m = o(e^n)$ avoids extreme cases wherein the rating matrix $R$ is excessively "tall" or "fat". This is only a mild assumption that arises from technical considerations, and is suitable for most practical scenarios then MCH ensures $\lim_{n \rightarrow \infty}P_{\mathrm{err}}^{(\gamma)} = 0$ (or equivalently, exa

Figures (3)

  • Figure 1: An illustration of the considered matrix completion problem. The goal is to exactly recover the nominal rating matrix by exploiting the sub-sampled matrix and the observed social graph and hypergraphs.
  • Figure 2: Consider a setting that contains $K = 4$ clusters, a social graph $HG_2$, and a $3$-uniform hypergraph $HG_3$. Let $\gamma=0.2$ and $\theta=0$. Figure \ref{['T1']} visualizes the gain due to $HG_2$ and $HG_3$ in terms of reducing the optimal sample probability $p^*$, where $g^*$ represents the maximum possible gain. Figure \ref{['T2']} shows the extra gain due to exploiting the hypergraph $HG_3$ for fixed values of the graph quality $I_2$. Note that $I_3/I_2$ represents the "relative quality" of hypergraphs, and $I_3/I_2=0$ means that no hypergraph information is available, corresponding to the setting considered in ahn2018binary.
  • Figure 3: Experimental results on synthetic and semi-real datasets show the superior performance of MCH.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 4 more