# Topological Obstructions to Autoencoding. (arXiv:2102.08380v1 [hep-ph])

Autoencoders have been proposed as a powerful tool for model-independent
anomaly detection in high-energy physics. The operating principle is that
events which do not belong to the space of training data will be reconstructed
poorly, thus flagging them as anomalies. We point out that in a variety of
examples of interest, the connection between large reconstruction error and
anomalies is not so clear. In particular, for data sets with nontrivial
topology, there will always be points that erroneously seem anomalous due to
global issues. Conversely, neural networks typically have an inductive bias or
prior to locally interpolate such that undersampled or rare events may be
reconstructed with small error, despite actually being the desired anomalies.
Taken together, these facts are in tension with the simple picture of the
autoencoder as an anomaly detector. Using a series of illustrative
low-dimensional examples, we show explicitly how the intrinsic and extrinsic
topology of the dataset affects the behavior of an autoencoder and how this
topology is manifested in the latent space representation during training. We
ground this analysis in the discussion of a mock “bump hunt” in which the
autoencoder fails to identify an anomalous “signal” for reasons tied to the
intrinsic topology of $n$-particle phase space.