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 the log-likelihood for user-specified probabilistic models, the optimization relies on the quality of given association strength $\{w_{ij}\}$; the optimization is strongly affected by noises in $\{w_{ij}\}$. To relieve the adverse effect of the noise, Okuno and Shimodaira (AISTATS2019) proposes β-graph embedding (β-GE) that employs a newly proposed robust empirical moment β-score (EMBS). EMBS 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.

existing
Existing GE (β=0)
proposed05
Proposed robust GE (β=0.5)
proposed1
Proposed robust GE (β=1)

EMBS is related to density-power score (see Kanamori and Fujisawa, 2014, 2015) that robustly measures the discrepancy between two non-negative functions. EMBS is called β-cross entropy, if the two non-negative functions are restricted to probability density functions. Similarly to Ghosh et al., (2013) that proposes regression using β-cross entropy, β-cross entropy based graph embedding can be easily formulated. Compared with simply using β-cross entropy, using EMBS has two advantages (i) free from specifying the distribution for $(w_{ij} \mid \boldsymbol{x}_i,\boldsymbol{x}_j)$ and (ii) computationally tractable.

Applying EMBS to PMvGE (Okuno, Hada, and Shimodaira, ICML2018), that is a multi-view extension of graph embeddding, leads to robust PMvGE. This robust PMvGE achieves the same goal as Iteratively-Reweighted Cross-Domain Matching Correlation Analysis (IR-CDMCA), that we presented at ICoRS2018 [slides; not published], though IR-CDMCA only considers linear setting.