| Dataset | Optimiser | Architecture |
| 2D shapes | Adagrad | Input | 4096 (flattened 64x64x1). |
| (VAE) | 1e-2 | Encoder | FC 1200,1200.ReLU activation. |
| | Latents Decoder | 10 |
| | | FC 1200,1200,1200,4096. Tanh activation.Bernoulli. |
| 2D shapes | rmsprop | Input | 64x64x1. |
| (DC-IGN) | (as in Kulkarni et al., 2015) | Encoder | Conv 96x3x3,48x3x3,48x3x3 (padding 1). ReLU activation and Max pooling 2x2. |
| | Latents | 10 |
| | Decoder | Unpooling,Conv 48x3x3,96x3x3,1x3x3. ReLU activation, Sigmoid. |
| 2D shapes (InfoGAN) | Adam 1e-3 (gen) | Generator | FC 256,256,Deconv 128x4x4, 64x4x4 (stride 2). Tanh. |
| 2e-4 (dis) | Discriminator | Conv and FC reverse of generator.Leaky ReLU activation. FC1. Sigmoid activation. |
| | Recognition Latents | Conv and FC shared with discriminator.FC 128,5.Gaussian 10: z1...5 ~ Unif(-1,1),c1...5 ~ Unif(-1,1) |
| Chairs | Adam | Input | 64x64x1. |
| (VAE) | 1e-4 | Encoder | Conv 32x4x4 (stride 2),32x4x4 (stride 2), 64x4x4 (stride 2), |
| | Latents | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Decoder | Deconv reverse of encoder. ReLU activation.Bernoulli. |
| CelebA | | | |
| (VAE) | Adam 1e-4 | Input Encoder | 64x64x3. Conv 32x4x4 (stride 2),32x4x4 (stride 2),64x4x4 (stride 2), |
| | Latents | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Decoder | Deconv reverse of encoder.ReLU activation.Gaussian. |
| 3DFaces | Adam | | 64x64x1. |
| (VAE) | 1e-4 | Input Encoder | Conv 32x4x4 (stride 2),32x4x4 (stride 2), 64x4x4 (stride 2), |
| | | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Latents Decoder | Deconv reverse of encoder.ReLU activation.Bernoulli. |
+
+# A.3 ICA AND PCA BASELINES
+
+In order to calculate the ICA benchmark, we applied fastICA (Pedregosa et al., 2011) algorithm to the whitened pixel data. Due to memory limitations we had to apply the algorithm to pairwise combinations of the subsets of the dataset corresponding to the transforms of each of the three 2D object identities. We calculated the disentangling metric for all three ICA models trained on each of the three pairwise combinations of 2D objects, before presenting the average of these scores in Fig. 6.
+
+We performed PCA on the raw and whitened pixel data. Both approaches resulted in similar disentangling metric scores. Fig. 6 reports the PCA results calculated using whitened pixel data for more direct comparison with the ICA score.
+
+# A.4 DISENTANGLEMENT METRIC DETAILS
+
+We used a linear classifier to learn the identity of the generative factor that produced $\mathbf { z } _ { \mathrm { d i f f } } ^ { b }$ (see Equations (5) for the process used to obtain samples of $\mathbf { z } _ { \mathrm { d i f f } } ^ { b } .$ ). We used a fully connected linear classifier to predict $p ( y | \mathbf { z } _ { \mathrm { d i f f } } ^ { b } )$ , where $y$ is one of four generative factors (position X, position Y, scale and rotation). We used softmax output nonlinearity and a negative log likelihood loss function. The classifier was trained using the Adagrad (Duchi et al., 2011) optimisation algorithm with learning rate of 1e-2 until convergence.
+
+$$
+\mathcal { D } = \{ V \in \mathbb { R } ^ { K } , W \in \mathbb { R } ^ { H } , X \in \mathbb { R } ^ { N } \} , y \sim U n i f [ 1 . . . K ]
+$$
+
+Repeat for $b = 1 \dots B$ :
+
+$$
+\begin{array} { l } { { \displaystyle { \bf v } _ { 1 , l } \sim p ( { \bf v } ) , ~ { \bf w } _ { 1 , l } \sim p ( { \bf w } ) , ~ { \bf w } _ { 2 , l } \sim p ( { \bf w } ) , ~ [ { \bf v } _ { 2 , l } ] _ { k } = \left\{ \begin{array} { l l } { { \displaystyle { \bf v } _ { 1 , l } ] _ { k } , } } & { { \mathrm { i f ~ } k = y } } \\ { { \displaystyle \sim p ( v _ { k } ) , } } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } \ ~ } \\ { { \displaystyle { \bf x } _ { 1 , l } \sim \mathrm { S i m } ( { \bf v } _ { 1 , 1 } , { \bf w } _ { 1 , 1 } ) , ~ { \bf x } _ { 2 , 1 } \sim \mathrm { S i m } ( { \bf v } _ { 2 , 1 } , { \bf w } _ { 2 , 1 } ) , } \ ~ } \\ { { \displaystyle q ( { \bf z } | { \bf x } ) \sim \mathcal N ( \mu ( { \bf x } ) , \sigma ( { \bf x } ) ) , ~ { \bf z } _ { 1 , l } = \mu ( { \bf x } _ { 1 , l } ) , ~ { \bf z } _ { 2 , l } = \mu ( { \bf x } _ { 2 , l } ) } \ ~ } \\ { \displaystyle { \bf z } _ { \mathrm { d i f f } } ^ { l } = | { \bf z } _ { 1 , l } - { \bf z } _ { 2 , l } | , ~ { \bf z } _ { \mathrm { d i f f } } ^ { b } = \frac { 1 } { L } \sum _ { l = 1 } ^ { L } { \bf z } _ { \mathrm { d i f f } } ^ { l } } \end{array}
+$$
+
+All disentanglement metric score results reported in the paper were calculated in the following manner. Ten replicas of each model with the same hyperparameters were trained using different random seeds to obtain disentangled representations. Each of the ten trained model replicas was evaluated three times using the disentanglement metric score algorithm, each time using a different random seed to initialise the linear classifier. We then discarded the bottom $50 \%$ of the thirty resulting scores and reported the remaining results. This was done to control for the outlier results from the few experiments that diverged during training.
+
+The results reported in table in Fig. 6 (left) were calculated using the following data. Ground truth uses independent data generating factors $\mathbf { v }$ (our dataset did not contain any correlated data generating factors w). PCA and ICA decompositions keep the first ten components (PCA components explain $6 0 . 8 \%$ of variance). $\beta$ -VAE $\beta = 4 ,$ ), VAE $\begin{array} { r } { \beta = 1 , } \end{array}$ ) and VAE untrained have the same fully connected architecture with ten latent units $\mathbf { z }$ . InfoGAN uses “inferred” values of the five continuous latents that were regularised with the mutual information objective during training.
+
+# A.5 CLASSIFYING THE GROUND TRUTH DATA GENERATIVE FACTORS VALUES
+
+In order to further verify the validity of our proposed disentanglement metric we ran an extra quantitative test: we trained a linear classifier to predict the ground truth value of each of the five data generative factors used to generate the 2D shapes dataset. While this test does not measure disentangling directly (since it does not measure independence of the latent representation), a disentangled representation should make such a classification trivial. It can be seen in Table 2 that the representation learnt by $\beta$ -VAE is on average the best representation for factor classification across all five factors. It is closely followed by DC-IGN. It is interesting to note that ICA does well only at encoding object identity, while PCA manages to learn a very good representation of object position.
+
+| Dataset | Optimiser | Architecture |
| 2D shapes | Adagrad | Input | 4096 (flattened 64x64x1). |
| (VAE) | 1e-2 | Encoder | FC 1200,1200.ReLU activation. |
| | Latents Decoder | 10 |
| | | FC 1200,1200,1200,4096. Tanh activation.Bernoulli. |
| 2D shapes | rmsprop | Input | 64x64x1. |
| (DC-IGN) | (as in Kulkarni et al., 2015) | Encoder | Conv 96x3x3,48x3x3,48x3x3 (padding 1). ReLU activation and Max pooling 2x2. |
| | Latents | 10 |
| | Decoder | Unpooling,Conv 48x3x3,96x3x3,1x3x3. ReLU activation, Sigmoid. |
| 2D shapes (InfoGAN) | Adam 1e-3 (gen) | Generator | FC 256,256,Deconv 128x4x4, 64x4x4 (stride 2). Tanh. |
| 2e-4 (dis) | Discriminator | Conv and FC reverse of generator.Leaky ReLU activation. FC1. Sigmoid activation. |
| | Recognition Latents | Conv and FC shared with discriminator.FC 128,5.Gaussian 10: z1...5 ~ Unif(-1,1),c1...5 ~ Unif(-1,1) |
| Chairs | Adam | Input | 64x64x1. |
| (VAE) | 1e-4 | Encoder | Conv 32x4x4 (stride 2),32x4x4 (stride 2), 64x4x4 (stride 2), |
| | Latents | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Decoder | Deconv reverse of encoder. ReLU activation.Bernoulli. |
| CelebA | | | |
| (VAE) | Adam 1e-4 | Input Encoder | 64x64x3. Conv 32x4x4 (stride 2),32x4x4 (stride 2),64x4x4 (stride 2), |
| | Latents | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Decoder | Deconv reverse of encoder.ReLU activation.Gaussian. |
| 3DFaces | Adam | | 64x64x1. |
| (VAE) | 1e-4 | Input Encoder | Conv 32x4x4 (stride 2),32x4x4 (stride 2), 64x4x4 (stride 2), |
| | | 64x4x4 (stride 2),FC 256.ReLU activation. 32 |
| | Latents Decoder | Deconv reverse of encoder.ReLU activation.Bernoulli. |
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| 5 | while Continue do |
| 6 | forK iterations do |
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