90 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) In general, all models produce similar results consisting of objects with unclear shape or identity, with the exception of one model. The weakly supervised AdaGVAE model trained on COIL-100 appears to have a degenerate decoder producing only yellow objects. Such behaviour occurred for all ten trained instances of the AdaGVAE model. (a) Latent space structure. (b) Generated data. Figure 4.11: Image generation by traversing the circular latent variable for a sampled object identity. The high dimensional Euclidean space is depicted as a single dimension in a hyper-cylinder. (a) The latent variable corresponding to the object identity is sampled from the prior over the Euclidean latent space and combined with regularly spaced latent variables on S1. (b) Each row presents the decoded images for a fixed Euclidean latent variable while each column shows the images for a fixed latent variable onS1. The images are obtained from decoding the latent variables with LSBD-VAE/full trained on COIL-100. Even though the randomly generated images seem to have no clear identity or shape for COIL-100, LSBD-VAE allows to better determine the identity of such sampled objects, by showing multiple orientations thanks to the structure of its latent space. LSBD-VAE uses a latent space combining an S1 manifold encouraged to encode information about the SO(2) rotations and a Euclidean latent space encouraged to represent the information about the object’s identity. By first sampling a latent variable from the Euclidean latent space and combining it with a set of regularly spaced latent variables along S1 we can
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