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In this section, we analyze the impact of the encoder/decoder architecture on the generation quality of considered models. The generation quality experiment of section 5 is repeated on the fMNIST and MNIST data set, where the architecture and hyperparameters are adapted from Dupont (2018). From Table 5 and Figure 9, we see that the overall FID scores and generation quality have improved; however, the relative scores among the models did not change significantly.

Table 5:

FID Scores Computed on Randomly Generated 8000 Images When Trained with Architecture and Hyperparameters.

	St-RKM	VAE	$β$ -VAE	FactorVAE	InfoGAN
MNIST	24.63 (0.22)	36.11 (1.01)	42.81 (2.01)	35.48 (0.07)	45.74 (2.93)
fMNIST	61.44 (1.02)	73.47 (0.73)	75.21 (1.11)	69.73 (1.54)	84.11 (2.58)

	St-RKM	VAE	$β$ -VAE	FactorVAE	InfoGAN
MNIST	24.63 (0.22)	36.11 (1.01)	42.81 (2.01)	35.48 (0.07)	45.74 (2.93)
fMNIST	61.44 (1.02)	73.47 (0.73)	75.21 (1.11)	69.73 (1.54)	84.11 (2.58)

Notes: Lower is better with standard deviations. Adapted from Dupont (2018).

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Table 6:

Computing the Diagonalization Scores (see Figure 3).

Models	dSprites	3DShapes	3D cars
St-RKM-sl (⁠ $σ = 10^{- 3}$ ⁠, $U_{★}$ ⁠)	0.17 (0.05)	0.23 (0.03)	0.21 (0.04)
St-RKM (⁠ $σ = 10^{- 3}$ ⁠, $U_{★}$ ⁠)	0.26 (0.05)	0.30 (0.10)	0.31 (0.09)
St-RKM (⁠ $σ = 10^{- 3}$ ⁠, random $U$ ⁠)	0.61 (0.02)	0.72 (0.01)	0.69 (0.03)

Models	dSprites	3DShapes	3D cars
St-RKM-sl (⁠ $σ = 10^{- 3}$ ⁠, $U_{★}$ ⁠)	0.17 (0.05)	0.23 (0.03)	0.21 (0.04)
St-RKM (⁠ $σ = 10^{- 3}$ ⁠, $U_{★}$ ⁠)	0.26 (0.05)	0.30 (0.10)	0.31 (0.09)
St-RKM (⁠ $σ = 10^{- 3}$ ⁠, random $U$ ⁠)	0.61 (0.02)	0.72 (0.01)	0.69 (0.03)

Notes: Denote $M = \frac{1}{| C |} \sum_{i \in C} U_{★}^{⊤} \nabla ψ (y_{i}) \nabla ψ {(y_{i})}^{⊤} U_{★}, with y_{i} = P_{U} ϕ_{θ} (x_{i})$ (cf. equation 3.6). Then we compute the score as ${∥M - diag (M)∥}_{F} / {∥M∥}_{F}$ ⁠, where $diag : R^{m \times m} \mapsto R^{m \times m}$ sets the off-diagonal elements of matrix to zero. The scores are computed for each model over 10 random seeds and show the mean (standard deviation). Lower scores indicate better diagonalization.

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Figure 8:

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Samples of randomly generated batch of images used to compute FID scores and SWD scores (see Figure 4).

Figure 9:

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Samples of randomly generated images used to compute the FID scores. See Table 5.

Figure 10:

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(a) Loss evolution (⁠ $log$ plot) during the training of equation A.2 over 1000 epochs with $ɛ = 10^{- 5}$ once with Cayley ADAM optimizer (green curve) and then without (blue curve). (b) Traversals along the principal components when the model was trained with a fixed $U$ ⁠, that is, with the objective given by equation A.2 and $ɛ = 10^{- 5}$ ⁠. There is no clear isolation of a feature along any of the principal components, indicating further that optimizing over $U$ is key to better disentanglement.

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