Counterfactual explanations can be obtained by identifying the smallest change made to a feature vector to qualitatively influence a prediction; for example, from 'loan rejected' to 'awarded' or from 'high risk of cardiovascular disease' to 'low risk'. 
Previous approaches often emphasized that counterfactuals should be easily interpretable to humans, motivating sparse solutions with few changes to the feature vectors. 
However, these approaches would not ensure that the produced counterfactuals be proximate (i.e., not local outliers) and connected to regions with substantial data density (i.e., close to correctly classified observations), two requirements known as counterfactual faithfulness. 
These requirements are fundamental when making suggestions to individuals that are indeed attainable. Our contribution is twofold. 
On one hand, we suggest to complement the catalogue of counterfactual quality measures [1] using a criterion to quantify the degree of difficulty for a certain counterfactual suggestion. 
On the other hand, drawing ideas from the manifold learning literature, we develop a framework that generates attainable counterfactuals. 
We suggest the counterfactual conditional heterogeneous variational autoencoder (C-CHVAE) to identify attainable counterfactuals that lie within regions of high data density.