In a variety of graph embedding methods, given association strength $w_{ij} \geq 0$ of given data pair $(\boldsymbol{x}_i,\boldsymbol{x}_j) \in \mathbb{R}^p \times \mathbb{R}^p$ is predicted by the kernel function $\mu_{\boldsymbol{\theta}}(\boldsymbol{x}_i,\boldsymbol{x}_j)$. Whereas the kernel function $\mu_{\boldsymbol{\theta}}$ can be determined by optimizing log-likelihood of some probabilistic models, the optimization relies on the quality of given association strength $\{w_{ij}\}$; their result strongly affected by noises in $\{w_{ij}\}$. To relieve the adverse effect of the noises, Okuno and Shimodaira (AISTATS2019, to appear) proposes β-graph embedding (β-GE) that employs a newly proposed robust moment matching β-score function (MMBSF). MMBSF includes negative log-likelihood of Poisson-based probabilistic model as β=0.

In the following, we plot feature vectors with noisy $\{w_{ij}\}$, obtained by (i) existing likelihood-based GE that corresponds to β=0, (ii) proposed GE with β=0.5, and (iii) proposed GE with β=1. Each point represents each vector whose color shows its cluster, and grey lines show associated data pairs $\{(i,j) \mid w_{ij}>0\}$. Proposed robust GE is able to distinguish colored clusters even if $\{w_{ij}\}$ is noisy, whereas the existing GE cannot.

MMBSF is related to unnormalized β-score function (UBSF; see Kanamori and Fujisawa, 2015) that robustly measures the discrepancy between two non-negative functions. Technically speaking, MMBSF has two advantages (i) also robust against distributional misspecification of $w_{ij} \mid \boldsymbol{x}_i,\boldsymbol{x}_j$ and (ii) computationally feasible, in contrast to naively applying UBSF to the probabilistic models for graph embedding.