Jianming Huang

Zhongxi Fang, Jianming Huang, Xun Su et al.

The Weisfeiler-Lehman (WL) test is a widely used algorithm in graph machine learning, including graph kernels, graph metrics, and graph neural networks. However, it focuses only on the consistency of the graph, which means that it is unable to detect slight structural differences. Consequently, this limits its ability to capture structural information, which also limits the performance of existing models that rely on the WL test. This limitation is particularly severe for traditional metrics defined by the WL test, which cannot precisely capture slight structural differences. In this paper, we propose a novel graph metric called the Wasserstein WL Subtree (WWLS) distance to address this problem. Our approach leverages the WL subtree as structural information for node neighborhoods and defines node metrics using the $L_1$-approximated tree edit distance ($L_1$-TED) between WL subtrees of nodes. Subsequently, we combine the Wasserstein distance and the $L_1$-TED to define the WWLS distance, which can capture slight structural differences that may be difficult to detect using conventional metrics. We demonstrate that the proposed WWLS distance outperforms baselines in both metric validation and graph classification experiments.

Jianming Huang, Xun Su, Zhongxi Fang et al.

In machine learning, Optimal Transport (OT) theory is extensively utilized to compare probability distributions across various applications, such as graph data represented by node distributions and image data represented by pixel distributions. In practical scenarios, it is often necessary to solve multiple OT problems. Traditionally, these problems are treated independently, with each OT problem being solved sequentially. However, the computational complexity required to solve a single OT problem is already substantial, making the resolution of multiple OT problems even more challenging. Although many applications of fast solutions to OT are based on the premise of a single OT problem with arbitrary distributions, few efforts handle such multiple OT problems with multiple distributions. Therefore, we propose the anchor space optimal transport (ASOT) problem: an approximate OT problem designed for multiple OT problems. This proposal stems from our finding that in many tasks the mass transport tends to be concentrated in a reduced space from the original feature space. By restricting the mass transport to a learned anchor point space, ASOT avoids pairwise instantiations of cost matrices for multiple OT problems and simplifies the problems by canceling insignificant transports. This simplification greatly reduces its computational costs. We then prove the upper bounds of its $1$-Wasserstein distance error between the proposed ASOT and the original OT problem under different conditions. Building upon this accomplishment, we propose three methods to learn anchor spaces for reducing the approximation error. Furthermore, our proposed methods present great advantages for handling distributions of different sizes with GPU parallelization.