  Detection of the outliers is pivotal for any machine learning model deployed
and operated in real-world. It is essential for the Deep Neural Networks that
were shown to be overconfident with such inputs. Moreover, even deep generative
models that allow estimation of the probability density of the input fail in
achieving this task. In this work, we concentrate on the specific type of these
models: Variational Autoencoders (VAEs). First, we unveil a significant
theoretical flaw in the assumption of the classical VAE model. Second, we
enforce an accommodating topological property to the image of the deep neural
mapping to the latent space: compactness to alleviate the flaw and obtain the
means to provably bound the image within the determined limits by squeezing
both inliers and outliers together. We enforce compactness using two
approaches: (i) Alexandroff extension and (ii) fixed Lipschitz continuity
constant on the mapping of the encoder of the VAEs. Finally and most
importantly, we discover that the anomalous inputs predominantly tend to land
on the vacant latent holes within the compact space, enabling their successful
identification. For that reason, we introduce a specifically devised score for
hole detection and evaluate the solution against several baseline benchmarks
achieving promising results.
