Original: We show four different plots for each dataset considered and include all available models.
Revised: We show four different plots for each dataset considered and include all available models.

Original: Each ﬁgure corresponds to a different dataset.
Revised: Each ﬁgure corresponds to a different dataset.

Revised: [Table]

Original:  In Figures 15- 21 we display the mean correlation coefﬁcient matrix and the latent representations for each ground-truth, as described in the main text for Fig. 5.
Revised: [Table] [Figure] [Table] In Figures 15- 21 we display the mean correlation coefﬁcient matrix and the latent representations for each ground-truth, as described in the main text for Fig. 5.

Original: The top row is the sorted absolute correlation coefﬁcient matrix between the latents (rows) and the ground truth generating factors (columns).
Revised: The top row is the sorted absolute correlation coefﬁcient matrix between the latents (rows) and the ground truth generating factors (columns).

Original: The latent dimensions are permuted such that the sum on the diagonal is maximal.
Revised: The latent dimensions are permuted such that the sum on the diagonal is maximal.

Original: This is achieved by an optimal, non-greedy matching process for each ground truth factor with its corresponding latent, as described in appendix D. As such, a more prevalent diagonal structure corresponds to a better mapping between the ground-truth factors and latent encoding.
Revised: This is achieved by an optimal, non-greedy matching process for each ground truth factor with its corresponding latent, as described in appendix C. As such, a more [Figure] [Table] [Table]

Revised: prevalent diagonal structure corresponds to a better mapping between the ground-truth factors and latent encoding.

Original: The middle set of plots are latent embeddings of random training data samples.
Revised: The middle set of plots are latent embeddings of random training data samples.

Original: The x-axis denotes the ground truth generating factor and the y-axis denotes the corresponding latent factor as matched according to the main diagonal of the correlation matrix.
Revised: The x-axis denotes the ground truth generating factor and the y-axis denotes the corresponding latent factor as matched according to the main diagonal of the correlation matrix.

Original: For each dataset, we further color-code the latents by a categorical variable as denoted in each ﬁgure.
Revised: For each dataset, we further color-code the latents by a categorical variable as denoted in each ﬁgure.

Original: The bottom set of plots show the ground truth encoding compared to the second best latent as opposed to the diagonally matched latent.
Revised: The bottom set of plots show the ground truth encoding compared to the second best latent as opposed to the diagonally matched latent.

Original: This plot can be used to judge how much the correspondence between latents is one-to-one or rather one-to-many.
Revised: This plot can be used to judge how much the correspondence between latents is one-to-one or rather one-to-many.

Original: To further investigate the latent representations, we show a scatter plot over the best and second best latents in ﬁgures 22-28.
Revised: To further investigate the latent representations, we show a scatter plot over the best and second best latents in ﬁgures 22-28.

Original: Here, the color-coding is matched by the ground truth factor denoted in each row.
Revised: Here, the color-coding is matched by the ground truth factor denoted in each row.

Original: [Table] [Table] When comparing the correlation matrix with the corresponding scatter plots, one can see that embeddings with sinusoidal curves have low correlation, which illustrates a shortcoming of the metric.
Revised: When comparing the correlation matrix with the corresponding scatter plots, one can see that embeddings with sinusoidal curves have low correlation, which illustrates a shortcoming of the metric.

Original: Another limitation is that categorical variables which have no natural ordering have an order-dependent MCC score, indicating the permutation variance of MCC.
Revised: Another limitation is that categorical variables which have no natural ordering have an order-dependent MCC score, indicating the permutation variance of MCC.

Original: With SlowVAE, we can infer three different types of embeddings.
Revised: With SlowVAE, we can infer three different types of embeddings.

Original: First, we have simple ordered ground truth factors with noncircular boundary conditions.
Revised: First, we have simple ordered ground truth factors with noncircular boundary conditions.

Original: Here, SlowVAE models often show a clear one-to-one correspondence  (e.g. Fig 22 scale, x-position and y-position; Fig 25 θ -rotation; Fig 26 Φ -rotation).
Revised: Here, SlowVAE models often show a clear one-to-one correspondence [Table] [Table] (e.g. Fig 22 scale, x-position and y-position; Fig 25 θ -rotation; Fig 26 Φ -rotation).

Original: Second, we observe circular embeddings due to boundary conditions for certain factors (e.g. Fig 15, 22 3rd row;
Revised: Second, we observe circular embeddings due to boundary conditions for certain factors (e.g. Fig 15, 22 3rd row;

Original: Fig 16, 23 2nd row).
Revised: Fig 16, 23 2nd row).

Original: Note that not all datasets with orientations exhibit full rotations and thus do not have circular boundary conditions, e.g. smallNORB.
Revised: Note that not all datasets with orientations exhibit full rotations and thus do not have circular boundary conditions, e.g. smallNORB.

Original: Finally, we have categorical variables, where no order exists (e.g. Fig.
Revised: Finally, we have categorical variables, where no order exists (e.g. Fig.

Original: 16, 23 top row, Fig 17, 24 top row, Fig 18, 25 top row) resulting in separated but not necessarily ordered clusters.
Revised: 16, 23 top row, Fig 17, 24 top row, Fig 18, 25 top row) resulting in separated but not necessarily ordered clusters.

Original: [Table]
Revised: [Table]

Revised: [Figure]

Original: [Table] 
Revised: [Table] [Table] [Figure] [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table] 
Revised: [Table] [Figure] [Table] [Figure]

Original: [Table] 
Revised: [Table] [Figure] [Table] [Figure]

Original: [Table] 
Revised: [Table] [Figure] [Table] [Figure]

Revised: [Table]

Original: [Table]
Revised: [Table]

Original: GT Factors
Revised: GT Factors

Original: L a t e n t s z
Revised: L a t e n t s z

Original: MCC = 74.8 86 141 5926 179 

Revised: MCC = 74.8 86 141 5926 179

Original: 45 9-VAE 0 

Revised: 45 9-VAE 0

Original: MCC = 76.0 

Revised: MCC = 76.0

Original: Ada-GVAE
Revised: Ada-GVAE

Original: MCC = 62.4 66 1530 6913 4812 503 263 3121 3334 1549 106 

Revised: MCC = 62.4 66 1530 6913 4812 503 263 3121 3334 1549 106

Original: PCL
Revised: PCL

Original: MCC = 90.6 95 210 817 550 179 1625 714 1110 711 159 

Revised: MCC = 90.6 95 210 817 550 179 1625 714 1110 711 159

Original: SlowVAE
Revised: SlowVAE

Original: [Table] [Figure]
Revised: [Table] [Figure]