124 Conclusion and Future Work learn a good enough representation of the normal data, leading to likelihood scores that are uninformative for anomaly detection. In conclusion, we find that the performance of anomaly detection with VAEs relies a lot on the representative capabilities of the models. For accurate anomaly detection, a model should learn to represent the normal data and its particularities well, without becoming overly general as to include anomalous behaviour as well. Thus, further developments in learning better representations are important for this type of anomaly detection. As a particular example of learning better representations, our second set of research questions focused onLinear Symmetry-Based Disentanglement (LSBD) (Higgins et al., 2018), a definition that formalises the idea of disentangling underlying real-world symmetries in learned representations. In particular, our questions focus on the lack of a general metric to quantify LSBD, and the need for methods that can learn LSBD representations. In Chapter 4, we present a well-formalised metric to quantify LSBD, which we name DLSBD, and show how to practically compute this metric in common settings involving cyclic factors that correspond to 2D rotations. Furthermore, using a simplified formulation of this metric, we derive LSBD-VAE; a model that can learn LSBD representations using weakly-supervised information on the transformations between data points. A metric for quantifying LSBD was lacking from the literature, but is crucial to be able to properly analyse LSBD and to relate it to previous notions of disentanglement. We demonstrate the utility of the DLSBD metric by evaluating traditional disentanglement methods as well as recent methods that specifically target LSBD, including our own LSBD-VAE. We observe that traditional methods don’t learn LSBD representations, whereas LSBD-oriented methods can learn them in certain cases. Furthermore, when comparingDLSBDscores to traditional quantification metrics for disentanglement, we see that LSBD representations also satisfy the desirable properties that traditional disentanglement metrics aim to capture. Overall, we conclude that LSBD is a better-formalised and more specific requirement than traditional disentanglement. This shows that it is a desirable goal that matches the objectives of traditional disentanglement, while adding additional aims in terms of representing symmetries. However, this also makes the applications of LSBD more specific and limited compared to traditional disentanglement. Motivated by the need for better representations that properly describe the data—e.g. in downstream tasks such as anomaly detection—as well as by the
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