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Graph Matching via convex relaxation to the simplex

Ernesto Araya Valdivia, Hemant Tyagi

TL;DR

This work presents a convex relaxation of graph matching on the unit simplex and solves it via a mirror descent method with closed-form updates. Under the Correlated Gaussian Wigner model in the noiseless regime, the simplex relaxation has a unique solution, enabling exact recovery of the ground-truth permutation after a single MD step, and the authors establish a weaker sufficiency condition for exact recovery that improves upon diagonal dominance. They also strengthen GRAMPA-related guarantees and validate the approach with extensive synthetic and real-data experiments, showing favorable accuracy and competitive efficiency against existing convex-relaxation methods. The methods and results advance theoretical understanding and practical performance of seedless graph matching, with potential extensions to other correlated graph models and accelerated MD variants that can benefit applications in computer vision and network analysis.

Abstract

This paper addresses the Graph Matching problem, which consists of finding the best possible alignment between two input graphs, and has many applications in computer vision, network deanonymization and protein alignment. A common approach to tackle this problem is through convex relaxations of the NP-hard \emph{Quadratic Assignment Problem} (QAP). Here, we introduce a new convex relaxation onto the unit simplex and develop an efficient mirror descent scheme with closed-form iterations for solving this problem. Under the correlated Gaussian Wigner model, we show that the simplex relaxation admits a unique solution with high probability. In the noiseless case, this is shown to imply exact recovery of the ground truth permutation. Additionally, we establish a novel sufficiency condition for the input matrix in standard greedy rounding methods, which is less restrictive than the commonly used `diagonal dominance' condition. We use this condition to show exact one-step recovery of the ground truth (holding almost surely) via the mirror descent scheme, in the noiseless setting. We also use this condition to obtain significantly improved conditions for the GRAMPA algorithm [Fan et al. 2019] in the noiseless setting.

Graph Matching via convex relaxation to the simplex

TL;DR

This work presents a convex relaxation of graph matching on the unit simplex and solves it via a mirror descent method with closed-form updates. Under the Correlated Gaussian Wigner model in the noiseless regime, the simplex relaxation has a unique solution, enabling exact recovery of the ground-truth permutation after a single MD step, and the authors establish a weaker sufficiency condition for exact recovery that improves upon diagonal dominance. They also strengthen GRAMPA-related guarantees and validate the approach with extensive synthetic and real-data experiments, showing favorable accuracy and competitive efficiency against existing convex-relaxation methods. The methods and results advance theoretical understanding and practical performance of seedless graph matching, with potential extensions to other correlated graph models and accelerated MD variants that can benefit applications in computer vision and network analysis.

Abstract

This paper addresses the Graph Matching problem, which consists of finding the best possible alignment between two input graphs, and has many applications in computer vision, network deanonymization and protein alignment. A common approach to tackle this problem is through convex relaxations of the NP-hard \emph{Quadratic Assignment Problem} (QAP). Here, we introduce a new convex relaxation onto the unit simplex and develop an efficient mirror descent scheme with closed-form iterations for solving this problem. Under the correlated Gaussian Wigner model, we show that the simplex relaxation admits a unique solution with high probability. In the noiseless case, this is shown to imply exact recovery of the ground truth permutation. Additionally, we establish a novel sufficiency condition for the input matrix in standard greedy rounding methods, which is less restrictive than the commonly used `diagonal dominance' condition. We use this condition to show exact one-step recovery of the ground truth (holding almost surely) via the mirror descent scheme, in the noiseless setting. We also use this condition to obtain significantly improved conditions for the GRAMPA algorithm [Fan et al. 2019] in the noiseless setting.
Paper Structure (51 sections, 19 theorems, 91 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 51 sections, 19 theorems, 91 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A general algorithmic strategy
  3. Convex relaxations of graph matching.
  4. Graph matching on the simplex and mirror descent.
  5. Contributions.
  6. Related work
  7. Convex relaxations.
  8. Information-theoretic limits of graph matching.
  9. MD algorithm and Multiplicative Weights Updates.
  10. Other algorithms for seedless graph matching.
  11. Notation.
  12. Algorithms
  13. A greedy rounding procedure and a sufficiency lemma
  14. Mirror descent algorithm for graph matching
  15. EMD and PGD updates.
  16. ...and 36 more sections

Key Result

Lemma 1

Let $\mathcal{K}$ be a convex subset of $\mathbb{R}^{n\times n}$ such that $\mathcal{K}\Pi^T=\mathcal{K}$, for any $\Pi\in\mathcal{P}_n$, and for $A,B\in\mathbb{R}^{n\times n}$ define $\mathcal{S}(A,B)$ to be the set of solutions of eq:convex_relax_general. Assume that the set defined by is a singleton. Then the algorithm that outputs $\hat{X}_\mathcal{P}=\texttt{GMWM}(\hat{X})$ for any $\hat{X}\

Figures (6)

  • Figure 1: Comparison of the performance of seedless convex methods for CGW (Fig. \ref{['fig:comp_convex_a']}) and CER (Fig. \ref{['fig:comp_convex_b']}) models. We plot the average overlap with the ground truth (recovery fraction) over $15$ Monte Carlo runs for graphs of size $n=300$. We used $N=125$ iterations in $\texttt{EMDGM}$ and in $\texttt{Grampa}$ we use the regularization parameter $\eta=0.2$. For $\texttt{EMDGM}$ we used the dynamic step-size rule \ref{['eq:dynamic_step_md']}.
  • Figure 2: We compare $\texttt{EMDGM}$, $\texttt{Grampa}$ and PGD under the CGW and CER models for $n=500$. For $\texttt{EMDGM}$ we used the dynamic step-size rule \ref{['eq:dynamic_step_md']} and PGD we used the heuristic rule \ref{['eq:heuristic_pgd_1']} with $\theta=1$. In both cases, we fix the number of iterations to $N=25$ (Figs.\ref{['fig:pgd_md_c']} and \ref{['fig:pgd_md_d']}) and $N=125$ (Figs. \ref{['fig:pgd_md_a']} and \ref{['fig:pgd_md_b']}). The $\texttt{Grampa}$ regularization parameter $\gamma$ is set to $0.2$ as recommended in Grampa. We present the average overlap (recovery fraction) for $25$ Monte Carlo runs. The shaded area indicates where $90\%$ of the data falls (excluding the top and bottom $5\%$).
  • Figure 3: Heatmap for the metrics \ref{['eq:metric_1']} and \ref{['eq:metric_2']} for the similarity matrices of $\texttt{Grampa}$ (Figs. \ref{['fig:comp_prop_a']} and \ref{['fig:comp_prop_c']}) and $\texttt{EMDGM}$ with $N=125$ (Figs. \ref{['fig:comp_prop_b']} and \ref{['fig:comp_prop_d']}). The input graphs were generated using the CGW model with $n=500$ and the noise level corresponds to the parameter $\sigma$. Recall that here a value of $0$ means perfect recovery on that particular realization.
  • Figure 4: Performance comparison of $\texttt{EMDGM}$, $\texttt{PGDGM}$, and $\texttt{Grampa}$ on the SHREC'16 dataset, based on the error cumulative distribution function (CDF) defined in \ref{['eq:error_cdf']}. We conducted 100 iterations each for $\texttt{EMDGM}$ and $\texttt{PGDGM}$.
  • Figure 5: Comparison of $\texttt{EMDGM}$, $\texttt{Grampa}$, and $\texttt{PGDGM}$ using a high-degree subsampled graph with $1,000$ vertices for each time instant. The input graph $A$ remains fixed as the graph from March 31, while the input graph $B$ varies (with random permutations of its rows and columns), as indicated on the x-axis. We used the dynamic step rule for $\texttt{EMDGM}$, derived from \ref{['eq:dynamic_step_md']} and the heuristic rule \ref{['eq:heuristic_pgd_1']} for $\texttt{PGDGM}$, with $125$ iterations in both cases.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Lemma 1
  • Lemma 2: Sufficient conditions for $\texttt{GMWM}$ returning the identity
  • proof
  • Lemma 3
  • Remark 1: Complexity of Algorithm \ref{['alg:mirror_simplex']}
  • Remark 2: Another CGW model
  • Lemma 4
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:uniqueness']}
  • Lemma 5
  • ...and 36 more