Constructing accurate model-agnostic explanations for opaque machine learning models remains a challenging task. Classification models for high-dimensional data, like images, are often inherently complex. To reduce this complexity, individual predictions may be explained locally, either in terms of a simpler local surrogate model or by communicating how the predictions contrast with those of another class. However, existing approaches still fall short in the following ways: a) they measure locality using a (Euclidean) metric that is not meaningful for non-linear high-dimensional data; or b) they do not attempt to explain the decision boundary, which is the most relevant characteristic of classifiers that are optimized for classification accuracy; or c) they do not give the user any freedom in specifying attributes that are meaningful to them. We address these issues in a new procedure for local decision boundary approximation (DBA). To construct a meaningful metric, we train a variational autoencoder to learn a Euclidean latent space of encoded data representations. We impose interpretability by exploiting attribute annotations to map the latent space to attributes that are meaningful to the user. A difficulty in evaluating explainability approaches is the lack of a ground truth. We address this by introducing a new benchmark data set with artificially generated Iris images, and showing that we can recover the latent attributes that locally determine the class. We further evaluate our approach on tabular data and on the CelebA image data set.