  Anomalies are samples that significantly deviate from the rest of the data
and their detection plays a major role in building machine learning models that
can be reliably used in applications such as data-driven design and novelty
detection. The majority of existing anomaly detection methods either are
exclusively developed for (semi) supervised settings, or provide poor
performance in unsupervised applications where there is no training data with
labeled anomalous samples. To bridge this research gap, we introduce a robust,
efficient, and interpretable methodology based on nonlinear manifold learning
to detect anomalies in unsupervised settings. The essence of our approach is to
learn a low-dimensional and interpretable latent representation (aka manifold)
for all the data points such that normal samples are automatically clustered
together and hence can be easily and robustly identified. We learn this
low-dimensional manifold by designing a learning algorithm that leverages
either a latent map Gaussian process (LMGP) or a deep autoencoder (AE). Our
LMGP-based approach, in particular, provides a probabilistic perspective on the
learning task and is ideal for high-dimensional applications with scarce data.
We demonstrate the superior performance of our approach over existing
technologies via multiple analytic examples and real-world datasets.
