74 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) Airplane The Airplane dataset consists of renders obtained using Blender v2.7 8 froma 3D model of an airplane within the ModelNet40 dataset (Wu et al., 2014) (this dataset is provided for the convenience of academic research only). We created each image by varying two properties: the airplane’s colour and its orientation with respect to the render camera. The orientation was changed via rotation with respect to a vertical axis (out-of-plane rotation). The colours of the model were selected from a predefined cyclic set of colours similar to the arrow rotation dataset. ModelNet40 The ModelNet40 dataset also consists of a dataset of renders obtained using Blender v2.7 from the 626 training 3D models within the airplane category of the ModelNet40 dataset (Wu et al., 2014). We created each image by varying each airplane’s orientation with respect to the render camera, via rotation with respect to a vertical axis (out-of-plane rotation). In this case we used 64 orientations for each object, i.e. |G| =64, for a total of 626 objects, thus the dataset consists of 40,064 images. COIL-100 The COIL-100 dataset (Nene et al., 1996) consists of images from 100 objects placed on a turntable against a black background. For each object, 72 views of the rotated object are provided. The original images have a resolution of 128×128 and were re-scaled to 64×64 to match our other datasets. In this case for each object |G| = 72, thus the total dataset consists of 7200 images. This dataset is intended for non-commercial research purposes only. This dataset was obtained using Tensorflow Datasets9. 4.6.2 LSBD-VAE with Semi-supervised Labelled Pairs For the Square, Arrow, and Airplane datasets, we test LSBD-VAE with transformation-labelled batches of size M=2. More specifically, for each experiment we randomly select Ldisjoint pairs of data points, and label the transformation between the data points in each pair. We vary the number of labelled pairs L 8http://www.blender.org 9https://www.tensorflow.org/datasets
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