Adaptive Softassign via Hadamard-Equipped Sinkhorn

TL;DR

This work tackles the sensitivity of softassign-based graph matching to the inflation parameter by proposing an adaptive softassign framework that automatic ally tunes $\beta$ under a prescribed error bound. Central to the approach are Hadamard-Equipped Sinkhorn formulas, which enable fast, stable computation of the entropic projection and smooth transitions between $\beta$ values, with connections to entropic optimal transport and proximal point methods. The adaptive softassign matching (ASM) algorithm combines this adaptive softassign with a projected fixed-point scheme, achieving notable gains in accuracy on large graphs while maintaining competitive efficiency. Experimental results on protein networks, social networks, and real-image graphs demonstrate substantial accuracy improvements over state-of-the-art large-graph matching methods. The paper also highlights the broader applicability of the Hadamard-Equipped Sinkhorn framework to OT problems, suggesting potential future extensions to plain graph matching and related optimization tasks.

Abstract

Softassign is a pivotal method in graph matching and other learning tasks. Many softassign-based algorithms exhibit performance sensitivity to a parameter in the softassign. However, tuning the parameter is challenging and almost done empirically. This paper proposes an adaptive softassign method for graph matching by analyzing the relationship between the objective score and the parameter. This method can automatically tune the parameter based on a given error bound to guarantee accuracy. The Hadamard-Equipped Sinkhorn formulas introduced in this study significantly enhance the efficiency and stability of the adaptive softassign. Moreover, these formulas can also be used in optimal transport problems. The resulting adaptive softassign graph matching algorithm enjoys significantly higher accuracy than previous state-of-the-art large graph matching algorithms while maintaining comparable efficiency.

Figures (7)

• Figure 1: Mean matching accuracy and running time of different algorithms on protein network matching.
• Figure 2: The heights of histograms represent values of corresponding elements in ${S}^{\beta}_{ {X}}$. As $\beta$ increases, ${S}^{\beta}_{{X}}$ gradually converges towards the solution of the assignment problem, namely, the identity matrix.
• Figure 3: Softassign and adaptive softassign process.
• Figure 4: The orange solid line represents the performance of adaptive softassign; the blue dashed line represents the performance of adaptive softassign* (adaptive softassign with the softassign transition). These two methods are evaluated on random matrices over 20 runs.
• Figure 5: The change of $\beta_\epsilon$ in ASM when $\beta_0$ is $\ln n$ in adaptive softassign. PPI and image are two kinds of graph matching tasks introduced in experiments.

Theorems & Definitions (21)

• Proposition 1
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• Proposition 3
• Lemma 1
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• Lemma 3
• Theorem
• Example
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• Proposition 2