18 Background Kulkarni et al. (2015) propose DC-IGN, a model that learns to disentangle by using batches of data where only one single factor changes, providing a quite strong level of supervision. Siddharth et al. (2017) train a semi-supervised VAE that uses class labels to disentangle class information from other variation in the data, using only a small amount of labelled data points. Although this approach doesn’t really fit the perspective of disentangling multiple distinct generative factors into individual latent dimensions, it can be extended to handle such a situation as well. In an effort to reduce the amount of supervision needed to learn disentangled representations, Locatello et al. (2020) model pairs of data observations that share at least one generative factor. They then theoretically prove that only knowing how many factors have changed, but not which ones, suffices to learn a disentangled representation. Motivated by this, they propose AdaGVAE and AdaMLVAE, weakly-supervised disentanglement models that are trained on pairs of data where only a few factors are different. 2.3 Group Theory Group theory provides a useful language to formalise the notion of symmetry transformations and their effects. In this section, we summarise some concepts from group theory that are important to understand the contents of Chapter 4, where we examine symmetry-based disentanglement. For a more elaborate discussion on group theory, see e.g. the book by Hall (2015). Group A group is a non-empty set G together with a binary operation ◦ : G×G→Gthat satisfies three properties: 1. Associativity: For all f,g,h∈G, it holds that f ◦(g ◦h)=(f ◦g) ◦h. 2. Identity: There exists a unique element e ∈ Gsuch that for all g ∈ Git holds that e◦g =g ◦e =g. Here, e is called the identity element of G. 3. Inverse: For all g ∈Gthere exists an element g−1 ∈Gsuch that g−1 ◦g = g ◦g−1 =e. Here, g−1 is called the inverse of g. In a slight abuse of notation, it is common to write Gfor both the underlying set alone, as well as the group (G, ◦) consisting of the set Gand its associated binary operation◦.
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