62 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) dispersionof the points {ρ(g−1 n ) · h(xn)} N n=1. 5 Given a suitable norm∥ · ∥Z inZ, we can thus quantify LSBD in this setting as 1 N NX n=1 ρ(g−1 n ) · h(xn) − 1 N NX n′=1 ρ(g−1 n′ ) · h(xn′) 2 Z , (4.7) i.e. we compute the mean of {ρ(g−1 n ) · h(xn)} N n=1 and use the average squared norm to this mean for points in {ρ(g−1 n ) · h(xn)} N n=1 as our LSBD metric, see Figure 4.1. Figure 4.1: A dataset of images from a rotating object expressed in terms of the group G=SO(2) acting on a base image x0. It is possible to quantify the level of LSBD of an encoding functionhby measuring its equivariance with respect to a group representationρ. Since all data is generated fromx0, equivariance can be measured as the dispersion of the points {ρ(g−1 n ) · h(xn)} N n=1. However, this formulation requires knowing the right linearly disentangled group representation and a suitable norm inZ. Moreover, it implicitly assumes a uniform probability measure over the group elements {gn} N n=1. In the following section we formulate our metric for a more general setting. 4.4.2 DLSBD: A Metric for LSBD Generalising the ideas from the previous section with concepts frommeasure theory6 and metric spaces, we propose a metric to measure the level of LSBD 5Note that we do not actually need to knowx0 nor h(x0). In practice, any of the data points xn could serve as the base point x0. 6See Section 2.4 for an overview of some key concepts from measure theory.
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