126 Conclusion and Future Work not containing any anomalies. Filtering out anomalies to obtain a clean training set may require substantial effort, e.g. by human experts, and can thus be very costly. Moreover, in the presence of concept drift (Gama et al., 2014), these efforts may have to be repeated to maintain a practical anomaly detector. Moreover, the data sets on which we displayed successful anomaly detection are relatively simple and regular in nature, allowing us to formulate good representations and likelihood scores that are indeed capable of differentiating normal from anomalous data. More complicated data will need more sophisticated models to obtain good enough representations for anomaly detection, as demonstrated with the 3D NLST lung nodules dataset in Chapter 3. Traditional VAEs do not have the capacity to learn an accurate enough model for such data. Additionally, other works have recently questioned the suitability of model likelihood as a measure for anomaly detection. Lan and Dinh (2020) show that probability density may not actually be able to guarantee anomaly detection in principle, showing through reparameterisation that low probability density isn’t necessarily a good measure for anomaly. Nalisnick et al. (2018) show that models trained on one dataset may actually assign higher likelihood to another, unseen dataset. Wang et al. (2020) investigate this failure further, arguing that the problem may be related to typicality—the typical set doesn’t necessarily coincide with the highest probability density values. Havtorn et al. (2021) posit that the problem may be that low-level features are similar in both datasets and are the main component in determining the likelihood of the data, whereas high-level features are what we actually find relevant for anomaly detection but are not what models focus on to determine likelihood. They propose a solution with a hierarchical VAE, which can use likelihood ratios to focus on high-level features for anomaly detection. The DLSBD metric presented in Chapter 4 provides a well-formalised and general quantification of LSBD. However, while sound in theory, there is no fixed recipe to compute DLSBD in practice. We provide such a practical implementation for SO(2) subgroups, but this is technically an approximation of the true metric. Moreover, the implementation for other types of groups is not trivial. These limitations also affect LSBD-VAE, which relies on limiting assumptions to formulate a computable loss component that only works for the case of SO(2) subgroups. In particular, it requires knowledge on how to formulate suitable latent space topologies and group representations acting on them, given a particular subgroup decomposition. Moreover, for DLSBD and LSBD-VAE, we assume that the group action on the data space is regular, i.e. that it is transitive andfree. This somewhat limits the
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