Local Distance Preserving Auto-encoders using Continuous k-Nearest Neighbours Graphs

Auto-encoder models that preserve similarities in the data are a popular tool in representation learning. In this paper we introduce several auto-encoder models that preserve local distances when mapping from the data space to the latent space. We use a local distance-preserving loss that is based on the continuous k-nearest neighbours graph which is known to capture topological features at all scales simultaneously. To improve training performance, we formulate learning as a constraint optimisation problem with local distance preservation as the main objective and reconstruction accuracy as a constraint. Our method provides state-ofthe-art or comparable performance across several standard datasets and evaluation metrics. Auto-encoders and variational auto-encoders (Kingma & Welling, 2014; Rezende et al., 2014) are often used in machine learning to find meaningful latent representations of the data. What constitutes meaningful usually depends on the application and on the downstream tasks, for example, finding representations that have important factors of variations in the data (disentanglement) (Higgins et al., 2017; Chen et al., 2018), have high mutual information with the data (Chen et al., 2016), or show clustering behaviour w.r.t. These representations are usually incentivised by regularisers or architectural/structural choices. One criterion for finding a meaningful latent representation is geometric faithfulness to the data. This is important for data visualisation or further downstream tasks that involve geometric algorithms such as clustering or kNN classification. The data often lies in a small, sparse, low-dimensional manifold in the space it inhabits and finding a lower dimensional projection that is geometrically faithful to it can help not only in visualisation and interpretability but also in predictive performance and robustness (e.g.