vi no consensus on how disentanglement should be defined exactly and whether statistical independence is the right measure to reason about disentanglement. Therefore, we focus on Linear Symmetry-Based Disentanglement (LSBD); a formal group-theoretic definition of disentanglement inspired by physics, which describes how symmetries from the real world should be reflected in learned representations. The second question this thesis addresses is how LSBD can be properly quantified, how LSBD representations can be learned, and how LSBD compares to other notions of disentanglement. Lastly, this thesis addresses how disentanglement can help in situations where regular VAE-based anomaly detection is not well-aligned with the empirical data distribution. In particular, we focus on cases where certain combinations of generative factors are not observed, and thus empirically out-of-distribution (OOD), but where we still wish to develop a model that can generalise to such factor combinations, i.e. they should not be detected as anomalous. LSBD provides a sensible framework to deal with suchOOD generalisation, since the generative factors are modelled by symmetry groups that need not depend on the observed data distribution. The contributions in this thesis can be summarised as follows: • With a VAE trained on normal data samples, we can detect anomalous samples if their assigned probability density is lower than for normal samples. We test this anomaly detection framework on applications for visual quality control and lung cancer detection. Results show that anomaly detection is possible in certain cases, confirming the validity of this approach, but in more complicated settings the models fail to represent the data well enough for reliable anomaly detection. • The LSBD definition gives an explicit formalisation of disentanglement, but it does not provide a metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to compare to previous notions of disentanglement. Therefore, we propose DLSBD, a formal metric based on group theory to quantify LSBD for arbitrary representations; as well as a practical implementation to compute this metric for common group decompositions. Furthermore, from this metric we derive LSBD-VAE, a weakly-supervised model to learn LSBD representations. We also demonstrate how this model can be used for a particular image retrieval challenge. We demonstrate the utility of the DLSBD metric by showing that (1) common VAE-based disentanglement methods don’t learn LSBD representations, (2) LSBD-VAE, as well as other recent methods, canlearn LSBD representations needing
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