Qihao Cheng

Qihao Cheng, Qichen Liao, Hao Wu et al.

A linear-complexity algorithm for computing the Wasserstein-1 distance on non-uniform meshes is proposed. This work extends the fast Sinkhorn algorithms from [Q. Liao et al., Commun. Math. Sci., 20(2022)] and [Q. Liao et al., J. Sci. Comput., 98 (2024)] to non-uniform meshes. In those prior works, a distinctive collinear structure of the kernel matrix on uniform meshes was identified, enabling \(O(N)\) acceleration via dynamic programming. While non-uniform meshes are prevalent in practical applications like computational fluid dynamics and finance, their lack of collinearity has hindered direct acceleration. In this paper, we introduce the concept of a ``dividing index'', which partitions the kernel matrix into two blocks. We demonstrate that each block exhibits a quasi-collinear property, a generalization of the structure found in uniform meshes. Leveraging this insight, we develop \textbf{F}ast S\textbf{I}nkhorn algorithm on \textbf{NO}n-uniform \textbf{M}eshes (\textbf{FINOM}), a dynamic programming approach that reduces the per-iteration complexity of the Sinkhorn algorithm from \(O(N^2)\) to \(O(N)\). Extensive numerical experiments on 1D and 2D problems confirm these improvements, achieving speed-ups of several orders of magnitude while maintaining accuracy.

Qihao Cheng, Da Yan, Tianhao Wu et al.

Given a graph pair $(G^1, G^2)$, graph edit distance (GED) is defined as the minimum number of edit operations converting $G^1$ to $G^2$. GED is a fundamental operation widely used in many applications, but its exact computation is NP-hard, so the approximation of GED has gained a lot of attention. Data-driven learning-based methods have been found to provide superior results compared to classical approximate algorithms, but they directly fit the coupling relationship between a pair of vertices from their vertex features. We argue that while pairwise vertex features can capture the coupling cost (discrepancy) of a pair of vertices, the vertex coupling matrix should be derived from the vertex-pair cost matrix through a more well-established method that is aware of the global context of the graph pair, such as optimal transport. In this paper, we propose an ensemble approach that integrates a supervised learning-based method and an unsupervised method, both based on optimal transport. Our learning method, GEDIOT, is based on inverse optimal transport that leverages a learnable Sinkhorn algorithm to generate the coupling matrix. Our unsupervised method, GEDGW, models GED computation as a linear combination of optimal transport and its variant, Gromov-Wasserstein discrepancy, for node and edge operations, respectively, which can be solved efficiently without needing the ground truth. Our ensemble method, GEDHOT, combines GEDIOT and GEDGW to further boost the performance. Extensive experiments demonstrate that our methods significantly outperform the existing methods in terms of the performance of GED computation, edit path generation, and model generalizability.