4.4 DLSBD: Quantifying LSBD 61 4.4 DLSBD: Quantifying LSBD The definition of Linear Symmetry-Based Disentanglement (LSBD) as proposed by Higgins et al. (2018) provides a well-formalised view on disentanglement, but such a formalisation is only useful in practice if we can quantify how well representations actually satisfy this definition. Such quantification is crucial for proper bench-marking of LSBD methods and to compare to traditional disentanglement. Therefore, in this section, we introduce a metric to quantify LSBD according to Definition 4.2. We first explain the intuition behind our metric, before deriving a formal definition of the metric, which we call DLSBD. Lastly, we show how to compute the metric in practice for the common example of SO(2) subgroups. 4.4.1 Intuition: Measuring Equivariance with Dispersion To motivate our metric, let’s first assume a setting in which a suitable linearly disentangled group representation ρ is known. Let’s further assume that the dataset of observations can be expressed with respect to Gacting on some base point x0 ∈X, i.e. {xn} N n=1 ={gn · x0} N n=1. Formally, this means that we assume that the action of GonXis transitive. In this case, we can use the inverse group elements g−1 n to transform each data point toward the base point x0, i.e. x0 =g−1 1 · x1 =. . . =g−1 N · xN. (4.5) Since we already assumed that ρ is linearly disentangled (i.e. satisfying criteria 1 to 3 of Definition 4.2), we only need to measure the equivariance of the encoding functionh(criterion 4 of Definition 4.2) to quantify LSBD. Equivariance is achieved whenh(g · x)=ρ(g) · h(x), for all g ∈G,x ∈X. Given the dataset described above, we can check this property for x∈{xn} N n=1 andg ∈{gn} N n=1. 4 In particular, from Equation (4.5) we can see that we have equivariance if h(x0)=ρ(g−1 1 ) · h(x1)=. . . =ρ(g−1 N ) · h(xN). (4.6) This not only characterises perfect equivariance, but also allows for an efficient way to quantify how close we are to true equivariance, by measuring the 4Note that {gn} N n=1 can be used to describe all known group transformations between elements in the dataset by means of composition and inverses, since xi =gi · (g−1 j · xj). Thus, it suffices to check equivariance for these Ngroup transformations, rather than needing to check all N 2 possible transformations between data points).
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