Fast and Provably Convergent Algorithms for Gromov-Wasserstein in Graph Data

Abstract

In this paper, we study the design and analysis of a class of efficient algorithms for computing the Gromov-Wasserstein (GW) distance tailored to large-scale graph learning tasks. Armed with the Luo-Tseng error bound condition (Luo & Tseng, 1992), two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) are proven to be (linearly) convergent. Upon taskspecific properties, our analysis further provides novel theoretical insights to guide how to select the best fit method. As a result, we are able to provide comprehensive experiments to validate the effectiveness of our methods on a host of tasks, including graph alignment, graph partition, and shape matching. In terms of both wall-clock time and modeling performance, the proposed methods achieve state-of-the-art results.

Robust Attributed Graph Alignment via Joint Structure Learning and Optimal Transport

“…Notably, the GW distance only requires modeling the topological or relational aspects of the distributions within each domain. In view of these nice properties, the GW distance has attracted intense research over the last decade, especially for structured data analysis, e.g., molecule analysis [48], [50], 3D shape matching [30], [43], graph embedding and classification [51], [53], generative models [3], [59], to name a few.…”

“…We first consider the case of dense graphs. The complexity of candidate bases construction in (6) t ), which is the same order as other optimal transport based alignment methods [6], [30], [48], [60].…”

“…Besides the "embed-then-cross-compare" paradigm, another line of research is to reformulate the graph alignment problem as finding the optimal probabilistic correspondence between two probability measures on graphs. Specifically, Gromov-Wasserstein (GW) distance serves as an effective tool in modeling the correspondence problems between two graphs on unaligned metric spaces [30], [43]. We show the procedure of the resulting optimal transport based alignment in Figure 2(b).…”

“…However, previous optimal transport based methods mainly consider the alignment between plain graphs without attributes and rely on manually designed cost matrices (e.g., the original graph adjacency matrix [30], [58], [60] or the heat kernel of graph Laplacian matrix [1], [6], [27]). Thus, these methods are potentially fragile to structure inconsistency.…”

Outlier-Robust Gromov Wasserstein for Graph Data

“…In the subgraph alignment task, the nodes in the large target graph, except for the nodes in the source graph, can be viewed as outliers in the target graph, which allows us to perform the RGW model on this task. Here, we compare the proposed RGW with unbalanced GW, partial GW, and also methods for computing the balanced GW: FW [32], BPG [35], SpecGW [13], eBPG [31] and BAPG [22].…”