In this work we consider the problem of learning a classifier from noisy labels when a few clean labeled examples are given. The structure of clean and noisy data is modeled by a graph per class and Graph Convolutional Networks (GCN) are used to predict class relevance of noisy examples. For each class, the GCN is treated as a binary classifier learning to discriminate clean from noisy examples using a weighted binary cross-entropy loss function, and then the GCN-inferred "clean" probability is exploited as a relevance measure. Each noisy example is weighted by its relevance when learning a classifier for the end task. We evaluate our method on an extended version of a few-shot learning problem, where the few clean examples of novel classes are supplemented with additional noisy data. Experimental results show that our GCN-based cleaning process significantly improves the classification accuracy over not cleaning the noisy data and standard few-shot classification where only few clean examples are used. The proposed GCN-based method outperforms the transductive approach (Douze et al., 2018) that is using the same additional data without labels.
The paper makes a significant attempt at solving one of the practical problems in machine learning -- learning from many noisy and limited number of clean labels. This setting is presumably more practical than the setting of few-shot learning. Noisy labels are often abundantly available and investing in methods that can take the noise into account for building a discriminative model is quite timely. 
To be honest, the theoretical contribution of the paper is limited.  The authors make use of the nearest neighbour graph obtained from a reduced-dimensional set of features to compute the weights of the noisy labels that must guide the predictive model. From this perspective, the paper seems like an application of existing tools (such as CNN, graph convolutional network and binary classification). However, that does not undermine the superior results the authors have received in the novel application they have targeted. I appreciate the effort that went validating these ideas with real-world datasets.
In future, I would like to see a joint approach to such training, where the function g(), the nearest neighbour graph loss and the classification loss are all tied in the same objective function and are optimized jointly. 
The paper has few really minor grammatical errors and typos. Please fix those before uploading the final draft.