Original: U NSUPERVISED M ODEL S ELECTION FOR V ARIATIONAL D ISENTANGLED R EPRESENTATION L EARNING
Revised: U NSUPERVISED M ODEL S ELECTION FOR V ARIATIONAL D ISENTANGLED R EPRESENTATION L EARNING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Disentangled representations have recently been shown to improve fairness, data efﬁciency and generalisation in simple supervised and reinforcement learning tasks.
Revised: Disentangled representations have recently been shown to improve fairness, data efﬁciency and generalisation in simple supervised and reinforcement learning tasks.

Original: To extend the beneﬁts of disentangled representations to more complex domains and practical applications, it is important to enable hyperparameter tuning and model selection of existing unsupervised approaches without requiring access to ground truth attribute labels, which are not available for most datasets.
Revised: To extend the beneﬁts of disentangled representations to more complex domains and practical applications, it is important to enable hyperparameter tuning and model selection of existing unsupervised approaches without requiring access to ground truth attribute labels, which are not available for most datasets.

Original: This paper addresses this problem by introducing a simple yet robust and reliable method for unsupervised disentangled model selection.
Revised: This paper addresses this problem by introducing a simple yet robust and reliable method for unsupervised disentangled model selection.

Original: Our approach, Unsupervised Disentanglement Ranking (UDR), leverages the recent theoretical results that explain why variational autoencoders disentangle (Rolinek et al., 2019), to quantify the quality of disentanglement by performing pairwise comparisons between trained model representations.
Revised: Our approach, Unsupervised Disentanglement Ranking (UDR), leverages the recent theoretical results that explain why variational autoencoders disentangle (Rolinek et al., 2019), to quantify the quality of disentanglement by performing pairwise comparisons between trained model representations.

Original: We show that our approach performs comparably to the existing supervised alternatives across 5400 models from six state of the art unsupervised disentangled representation learning model classes.
Revised: We show that our approach performs comparably to the existing supervised alternatives across 5400 models from six state of the art unsupervised disentangled representation learning model classes.

Original: Furthermore, we show that the ranking produced by our approach correlates well with the ﬁnal task performance on two different domains
Revised: Furthermore, we show that the ranking produced by our approach correlates well with the ﬁnal task performance on two different domains

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Happy families are all alike; every unhappy family is unhappy in its own way. —
Revised: Happy families are all alike; every unhappy family is unhappy in its own way. —

Original: Leo Tolstoy,
Revised: Leo Tolstoy,

Original: Despite the success of deep learning in the recent years (Hu et al., 2018; Espeholt et al., 2018;
Revised: Despite the success of deep learning in the recent years (Hu et al., 2018; Espeholt et al., 2018;

Original: Silver et al., 2018; Lample et al., 2018; Hessel et al., 2017; Oord et al., 2016), the majority of state of the art approaches are still missing many basic yet important properties, such as fairness, data efﬁcient learning, strong generalisation beyond the training data distribution, or the ability to transfer knowledge between tasks (Lake et al., 2016; Garnelo et al., 2016; Marcus, 2018).
Revised: Silver et al., 2018; Lample et al., 2018; Hessel et al., 2017; Oord et al., 2016), the majority of state of the art approaches are still missing many basic yet important properties, such as fairness, data efﬁcient learning, strong generalisation beyond the training data distribution, or the ability to transfer knowledge between tasks (Lake et al., 2016; Garnelo et al., 2016; Marcus, 2018).

Original: The idea that a good representation can help with such shortcomings is not new, and recently a number of papers have demonstrated that models with disentangled representations show improvements in terms of these shortcomings (Higgins et al., 2017b; 2018b; Achille et al., 2018; Steenbrugge et al., 2018; Nair et al., 2018;
Revised: The idea that a good representation can help with such shortcomings is not new, and recently a number of papers have demonstrated that models with disentangled representations show improvements in terms of these shortcomings (Higgins et al., 2017b; 2018b; Achille et al., 2018; Steenbrugge et al., 2018; Nair et al., 2018;

Original: Laversanne-Finot et al., 2018; van Steenkiste et al., 2019; Locatello et al., 2019).
Revised: Laversanne-Finot et al., 2018; van Steenkiste et al., 2019; Locatello et al., 2019).

Original: A common intuitive way to think about disentangled representations is that it should reﬂect the compositional structure of the world.
Revised: A common intuitive way to think about disentangled representations is that it should reﬂect the compositional structure of the world.

Original: For example, to describe an object we often use words pertaining to its colour, position, shape and size.
Revised: For example, to describe an object we often use words pertaining to its colour, position, shape and size.

Original: We can use different words to describe these properties because they relate to independent factors of variation in our world, i.e. properties which can be compositionally recombined.
Revised: We can use different words to describe these properties because they relate to independent factors of variation in our world, i.e. properties which can be compositionally recombined.

Original: Hence a disentangled representation of objects should reﬂect this by factorising into dimensions which correspond to those properties (Bengio et al., 2013; Higgins et al., 2018a).
Revised: Hence a disentangled representation of objects should reﬂect this by factorising into dimensions which correspond to those properties (Bengio et al., 2013; Higgins et al., 2018a).

Original: The ability to automatically discover the compositional factors of complex real datasets can be of great importance in many practical applications of machine learning and data science.
Revised: The ability to automatically discover the compositional factors of complex real datasets can be of great importance in many practical applications of machine learning and data science.

Original: However, it is important to be able to learn such representations in an unsupervised manner, since most interesting datasets do not have their generative factors fully labelled.
Revised: However, it is important to be able to learn such representations in an unsupervised manner, since most interesting datasets do not have their generative factors fully labelled.

Original: For a long time scalable unsupervised disentangled representation learning was impossible, until recently a new class of models based on
Revised: For a long time scalable unsupervised disentangled representation learning was impossible, until recently a new class of models based on

Original: Variational Autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014) was developed.
Revised: Variational Autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014) was developed.

Original: These approaches (Higgins et al., 2017a; Burgess et al., 2017; Chen et al., 2018; Kumar et al., 2017; Kim
Revised: These approaches (Higgins et al., 2017a; Burgess et al., 2017; Chen et al., 2018; Kumar et al., 2017; Kim

Original: [Figure]
Revised: [Figure]

Original: Seed
Revised: Seed

Original: [Figure]
Revised: [Figure]

Original: Seed
Revised: Seed

Original: Seed 3 UDR = 0.
Revised: Seed 3 UDR = 0.

Original: Seed 1 UDR = 0.
Revised: Seed 1 UDR = 0.

Original: zzzzz 5 -  -  -  -  zzzzzzzzzz
Revised: zzzzz 5 -  -  -  -  zzzzzzzzzz

Original: Mnih, 2018) scale reasonably well and are the current state of the art in unsupervised disentangled representation learning.
Revised: Mnih, 2018) scale reasonably well and are the current state of the art in unsupervised disentangled representation learning.

Original: However, so far the beneﬁts of these techniques have not been widely exploited because of two major shortcomings: First, the quality of the achieved disentangling is sensitive to the choice of hyperparameters, however, model selection is currently impossible without having access to the ground truth generative process and/or attribute labels, which are required by all the currently existing disentanglement metrics (Higgins et al., 2017a; Kim & Mnih, 2018; Chen et al., 2018;
Revised: However, so far the beneﬁts of these techniques have not been widely exploited because of two major shortcomings: First, the quality of the achieved disentangling is sensitive to the choice of hyperparameters, however, model selection is currently impossible without having access to the ground truth generative process and/or attribute labels, which are required by all the currently existing disentanglement metrics (Higgins et al., 2017a; Kim & Mnih, 2018; Chen et al., 2018;

Original: Eastwood & Williams, 2018; Ridgeway & Mozer, 2018).
Revised: Eastwood & Williams, 2018; Ridgeway & Mozer, 2018).

Original: Second, even if one could apply any of the existing disentanglement metrics for model selection, the scores produced by these metrics can vary a lot even for models with the same hyperparameters and trained on the same data (Locatello et al., 2018).
Revised: Second, even if one could apply any of the existing disentanglement metrics for model selection, the scores produced by these metrics can vary a lot even for models with the same hyperparameters and trained on the same data (Locatello et al., 2018).

Original: While a lot of this variance is explained by the actual quality of the learnt representations, some of it is introduced by the metrics themselves.
Revised: While a lot of this variance is explained by the actual quality of the learnt representations, some of it is introduced by the metrics themselves.

Original: In particular, all of the existing supervised disentanglement metrics assume a single “canonical” factorisation of the generative factors, any deviation from which is penalised.
Revised: In particular, all of the existing supervised disentanglement metrics assume a single “canonical” factorisation of the generative factors, any deviation from which is penalised.

Original: Such a “canonical” factorisation, however, is not chosen in a principled manner.
Revised: Such a “canonical” factorisation, however, is not chosen in a principled manner.

Original: Indeed, for the majority of datasets, apart from the simplest ones, multiple equally valid disentangled representations may be possible (see Higgins et al.
Revised: Indeed, for the majority of datasets, apart from the simplest ones, multiple equally valid disentangled representations may be possible (see Higgins et al.

Original: (2018a)
Revised: (2018a)

Original: for a discussion).
Revised: for a discussion).

Original: For example, the intuitive way that humans reason about colour is in terms of hue and saturation.
Revised: For example, the intuitive way that humans reason about colour is in terms of hue and saturation.

Original: However, colour may also be represented in RGB, YUV, HSV, HSL, CIELAB.
Revised: However, colour may also be represented in RGB, YUV, HSV, HSL, CIELAB.

Original: Any of the above representations are as valid as each other, yet only one of them is allowed to be “canonical” by the supervised metrics.
Revised: Any of the above representations are as valid as each other, yet only one of them is allowed to be “canonical” by the supervised metrics.

Original: Hence, a model that learns to represent colour in a subspace aligned with HSV will be penalised by a supervised metric which assumes that the canonical disentangled representation of colour should be in RGB.
Revised: Hence, a model that learns to represent colour in a subspace aligned with HSV will be penalised by a supervised metric which assumes that the canonical disentangled representation of colour should be in RGB.

Original: This is despite the fact that both representations are equal in terms of preserving the compositional property at the core of what makes disentangled representations useful (Higgins et al., 2018a).
Revised: This is despite the fact that both representations are equal in terms of preserving the compositional property at the core of what makes disentangled representations useful (Higgins et al., 2018a).

Original: Hence, the ﬁeld ﬁnds itself in a predicament.
Revised: Hence, the ﬁeld ﬁnds itself in a predicament.

Original: From one point of view, there exists a set of approaches capable of reasonably scalable unsupervised disentangled representation learning.
Revised: From one point of view, there exists a set of approaches capable of reasonably scalable unsupervised disentangled representation learning.

Original: On the other hand, these models are hard to use in practice, because there is no easy way to do a hyperparameter search and model selection.
Revised: On the other hand, these models are hard to use in practice, because there is no easy way to do a hyperparameter search and model selection.

Original: This paper attemptsto bridge this gap.
Revised: This paper attemptsto bridge this gap.

Original: We propose asimple yet effective method for unsupervised model selection for the class of current state-of-the-art VAE-based unsupervised disentangled representation learning methods.
Revised: We propose asimple yet effective method for unsupervised model selection for the class of current state-of-the-art VAE-based unsupervised disentangled representation learning methods.

Original: Our approach, Unsupervised Disentanglement Ranking (UDR), leverages the recent theoretical results that explain why variational autoencoders disentangle (Rolinek et al., 2019), to quantify the quality of disentanglement by performing pairwise comparisons between trained model representations.
Revised: Our approach, Unsupervised Disentanglement Ranking (UDR), leverages the recent theoretical results that explain why variational autoencoders disentangle (Rolinek et al., 2019), to quantify the quality of disentanglement by performing pairwise comparisons between trained model representations.

Original: We evaluate the validity of our unsupervised model selection metric against the four best existing supervised alternatives reported in the large scale study by Locatello et al.
Revised: We evaluate the validity of our unsupervised model selection metric against the four best existing supervised alternatives reported in the large scale study by Locatello et al.

Original: (2018): the β -VAE metric (Higgins et al., 2017a), the FactorVAE metric (Kim & Mnih, 2018), Mutual Information Gap (MIG) (Chen et al., 2018) and DCI Disentanglement scores (Eastwood & Williams, 2018).
Revised: (2018): the β -VAE metric (Higgins et al., 2017a), the FactorVAE metric (Kim & Mnih, 2018), Mutual Information Gap (MIG) (Chen et al., 2018) and DCI Disentanglement scores (Eastwood & Williams, 2018).

Original: We do so for all existing state of the art disentangled representation learning approaches: β -VAE (Higgins et al., 2017a),
Revised: We do so for all existing state of the art disentangled representation learning approaches: β -VAE (Higgins et al., 2017a),

Original: CCI-VAE (Burgess et al., 2017), FactorVAE (Kim & Mnih, 2018), TC-VAE (Chen et al., 2018) and two versions of DIP-VAE (Kumar et al., 2017).
Revised: CCI-VAE (Burgess et al., 2017), FactorVAE (Kim & Mnih, 2018), TC-VAE (Chen et al., 2018) and two versions of DIP-VAE (Kumar et al., 2017).

Original: We validate our proposed method on two datasets with fully known generative processes commonly used to evaluate the quality of disentangled representations: dSprites (Matthey et al., 2017) and 3D Shapes (Burgess & Kim, 2018), and show that our unsupervised model selection method is able to match the supervised baselines in terms of guiding a hyperparameter search and picking the most disentangled trained models both quantitatively and qualitatively.
Revised: We validate our proposed method on two datasets with fully known generative processes commonly used to evaluate the quality of disentangled representations: dSprites (Matthey et al., 2017) and 3D Shapes (Burgess & Kim, 2018), and show that our unsupervised model selection method is able to match the supervised baselines in terms of guiding a hyperparameter search and picking the most disentangled trained models both quantitatively and qualitatively.

Original: We also apply our approach to the 3D Cars dataset (Reed et al., 2014), where the full set of ground truth attribute labels is not available, and conﬁrm through visual inspection that the ranking produced by our method is meaningful (Fig. 1).
Revised: We also apply our approach to the 3D Cars dataset (Reed et al., 2014), where the full set of ground truth attribute labels is not available, and conﬁrm through visual inspection that the ranking produced by our method is meaningful (Fig. 1).

Original: Overall we evaluate 6 different model classes, with 6 separate hyperparameter settings and 50 seeds on 3 separate datasets, totalling 5400 models and show that our method is both accurate and consistent across models and datasets.
Revised: Overall we evaluate 6 different model classes, with 6 separate hyperparameter settings and 50 seeds on 3 separate datasets, totalling 5400 models and show that our method is both accurate and consistent across models and datasets.

Original: Finally, we validate that the model ranking produced by our approach correlates well with the ﬁnal task performance on two recently reported tasks: a classiﬁcation fairness task (Locatello et al., 2019) and a model-based reinforcement learning (RL) task (Watters et al., 2019).
Revised: Finally, we validate that the model ranking produced by our approach correlates well with the ﬁnal task performance on two recently reported tasks: a classiﬁcation fairness task (Locatello et al., 2019) and a model-based reinforcement learning (RL) task (Watters et al., 2019).

Original: Indeed, on the former our approach outperformed the reported supervised baseline scores.
Revised: Indeed, on the former our approach outperformed the reported supervised baseline scores.

Original:  O PERATIONAL DEFINITION OF DISENTANGLING
Revised:  O PERATIONAL DEFINITION OF DISENTANGLING

Original: Given a dataset of observations X
Revised: Given a dataset of observations X

Original: = { x 1 ,..., x N } , we assume that there exist a number of plausible generative processes g i that produce the observations from a small set of corresponding K
Revised: = { x 1 ,..., x N } , we assume that there exist a number of plausible generative processes g i that produce the observations from a small set of corresponding K

Original: i independent generative factors.
Revised: i independent generative factors.

Original: For each choice of i , g : c n (cid:55)→ x n , where p ( c n ) =
Revised: For each choice of i , g : c n (cid:55)→ x n , where p ( c n ) =

Original: (cid:81) Kj =1 p ( c jn ) .
Revised: (cid:81) Kj =1 p ( c jn ) .

Original: For example, a dataset containing images of an object, which can be of a particular shape and colour, and which can be in a particular vertical and horizontal positions, may be created by a generative process that assumes a ground truth disentangled factorisation into shape x colour x position , or shape x hue x
Revised: For example, a dataset containing images of an object, which can be of a particular shape and colour, and which can be in a particular vertical and horizontal positions, may be created by a generative process that assumes a ground truth disentangled factorisation into shape x colour x position , or shape x hue x

Original: saturation x position X x position Y .
Revised: saturation x position X x position Y .

Original: We operationalise a model as having learnt a disentangled representation, if it learns to invert of one of the generative processes g i and recover a latent representation z ∈ R L , so that it best explains the observed data p ( z , x ) ≈ p ( c i , x ) , and factorises the same way as the corresponding data generative factors c i .
Revised: We operationalise a model as having learnt a disentangled representation, if it learns to invert of one of the generative processes g i and recover a latent representation z ∈ R L , so that it best explains the observed data p ( z , x ) ≈ p ( c i , x ) , and factorises the same way as the corresponding data generative factors c i .

Original: The choice of the generative process can be determined by the interaction between the model class and the observed data distribution p ( x ) , as discussed next in Sec. 3.
Revised: The choice of the generative process can be determined by the interaction between the model class and the observed data distribution p ( x ) , as discussed next in Sec. 3.

Original: . V ARIATIONAL UNSUPERVISED DISENTANGLING
Revised: . V ARIATIONAL UNSUPERVISED DISENTANGLING

Original: The current state of the art approaches to unsupervised disentangled representation learning are based on the Variational Autoencoder (VAE) framework (Rezende et al., 2014; Kingma & Welling, 2014).
Revised: The current state of the art approaches to unsupervised disentangled representation learning are based on the Variational Autoencoder (VAE) framework (Rezende et al., 2014; Kingma & Welling, 2014).

Original: VAEs attempt to estimate the lower bound on the joint distribution of the data and the latent factors p ( x , z ) by optimising the following objective:
Revised: VAEs attempt to estimate the lower bound on the joint distribution of the data and the latent factors p ( x , z ) by optimising the following objective:

Original: [Equation]
Revised: [Equation]

Original: where, in the usual case, the prior p ( z ) is chosen to be an isotropic unit Gaussian.
Revised: where, in the usual case, the prior p ( z ) is chosen to be an isotropic unit Gaussian.

Original: In order to encourage disentangling, different approaches decompose the objective in Eq.
Revised: In order to encourage disentangling, different approaches decompose the objective in Eq.

Original: 1 into various terms and change their relative weighting.
Revised: 1 into various terms and change their relative weighting.

Original: In this paper we will consider six state of the art approaches to unsupervised disentangled representation learning that can be grouped into three broad classes based on how they modify the objective in Eq.
Revised: In this paper we will consider six state of the art approaches to unsupervised disentangled representation learning that can be grouped into three broad classes based on how they modify the objective in Eq.

Original: 1: 1) β -VAE (Higgins et al., 2017a)
Revised: 1: 1) β -VAE (Higgins et al., 2017a)

Original: and CCI-VAE (Burgess et al., 2017) upweight the KL term; 2) FactorVAE
Revised: and CCI-VAE (Burgess et al., 2017) upweight the KL term; 2) FactorVAE

Original: (Kim & Mnih, 2018) and TC-VAE (Chen et al., 2018) introduce a total correlation penalty; and 3) two different implementations of DIP-VAE (-I and -II)
Revised: (Kim & Mnih, 2018) and TC-VAE (Chen et al., 2018) introduce a total correlation penalty; and 3) two different implementations of DIP-VAE (-I and -II)

Original: (Kumar et al., 2017) penalise the deviation of the the marginal posterior from a factorised prior (see Sec.
Revised: (Kumar et al., 2017) penalise the deviation of the the marginal posterior from a factorised prior (see Sec.

Original: A.4.1 in
Revised: A.4.1 in

Original: Supplementary Material for details).
Revised: Supplementary Material for details).

Original: 3. W HY DO VAE S DISENTANGLE ?
Revised: 3. W HY DO VAE S DISENTANGLE ?

Original: In order to understand the reasoning behind our proposed unsupervised disentangled model selection method, it is ﬁrst important to understand why VAEs disentangle.
Revised: In order to understand the reasoning behind our proposed unsupervised disentangled model selection method, it is ﬁrst important to understand why VAEs disentangle.

Original: The objective in Eq.
Revised: The objective in Eq.

Original: 1 does not in itself encourage disentangling, as discussed in Rolinek et al.
Revised: 1 does not in itself encourage disentangling, as discussed in Rolinek et al.

Original: and Locatello et al.
Revised: and Locatello et al.

Original: Indeed, any rotationally invariant prior makes disentangled representations learnt in an unsupervised setting unidentiﬁable when optimising Eq.
Revised: Indeed, any rotationally invariant prior makes disentangled representations learnt in an unsupervised setting unidentiﬁable when optimising Eq.

Original: This theoretical result is not surprising and has been known for a while in the ICA literature (Comon, 1994), however what is surprising is that disentangling VAEs appear to work in practice.
Revised: This theoretical result is not surprising and has been known for a while in the ICA literature (Comon, 1994), however what is surprising is that disentangling VAEs appear to work in practice.

Original: The question of what makes VAEs disentangle was answered by Rolinek et al.
Revised: The question of what makes VAEs disentangle was answered by Rolinek et al.

Original: (2019), who were able to show that it is the peculiarities of the VAE implementation choices that allow disentanglingtoemerge(seealsodiscussioninBurgessetal.(2017);Mathieuetal.(2019)).
Revised: (2019), who were able to show that it is the peculiarities of the VAE implementation choices that allow disentanglingtoemerge(seealsodiscussioninBurgessetal.(2017);Mathieuetal.(2019)).

Original: Inparticular, the interactions between the reconstruction objective (the ﬁrst term in Eq. 1) and the enhanced pressure tomatchadiagonalpriorcreatedbythemodiﬁedobjectivesofthedisentanglingVAEs, forcethedecoder to approximate PCA-like behaviour locally around the data samples.
Revised: Inparticular, the interactions between the reconstruction objective (the ﬁrst term in Eq. 1) and the enhanced pressure tomatchadiagonalpriorcreatedbythemodiﬁedobjectivesofthedisentanglingVAEs, forcethedecoder to approximate PCA-like behaviour locally around the data samples.

Original: Rolinek et al.
Revised: Rolinek et al.

Original: (2019) demonstrated that during training VAEs often enter the so-called “polarised regime”, where many of the latent dimensions of the posterior are effectively switched off by being reduced to the prior q φ ( z j )
Revised: (2019) demonstrated that during training VAEs often enter the so-called “polarised regime”, where many of the latent dimensions of the posterior are effectively switched off by being reduced to the prior q φ ( z j )

Original: = p ( z j ) (this behaviour is often further encouraged by the extra disentangling terms added to the ELBO).
Revised: = p ( z j ) (this behaviour is often further encouraged by the extra disentangling terms added to the ELBO).

Original: When trained in such a regime, a linear approximation of the Jacobian of the decoder around a data sample x i ,
Revised: When trained in such a regime, a linear approximation of the Jacobian of the decoder around a data sample x i ,

Original: J i =
Revised: J i =

Original: ∂Dec θ ( µ φ ( x i ))
Revised: ∂Dec θ ( µ φ ( x i ))

Original: ∂µ φ ( x i ) , is forced to have orthogonal columns, and hence to separate the generative factors based on the amount of reconstruction variance they induce.
Revised: ∂µ φ ( x i ) , is forced to have orthogonal columns, and hence to separate the generative factors based on the amount of reconstruction variance they induce.

Original: Given that the transformations induced by different generative factors will typically have different effects on the pixel space (e.g. changing the position of a sprite will typically affect more pixels than changing its size), such local orthogonalisation of the decoder induces an identiﬁable disentangled latent space for each particular dataset.
Revised: Given that the transformations induced by different generative factors will typically have different effects on the pixel space (e.g. changing the position of a sprite will typically affect more pixels than changing its size), such local orthogonalisation of the decoder induces an identiﬁable disentangled latent space for each particular dataset.

Original: An equivalent statement is that for a well disentangled VAE, the SVD decomposition J = U Σ V (cid:62) of the
Revised: An equivalent statement is that for a well disentangled VAE, the SVD decomposition J = U Σ V (cid:62) of the

Original: Jacobian J calculated as above, results in a trivial V , which is a signed permutation matrix.
Revised: Jacobian J calculated as above, results in a trivial V , which is a signed permutation matrix.

Original:  U NSUPERVISED DISENTANGLED MODEL SELECTION
Revised:  U NSUPERVISED DISENTANGLED MODEL SELECTION

Original: We now describe how the insights from Sec.
Revised: We now describe how the insights from Sec.

Original: 3.1 motivate the development of our proposed
Revised: 3.1 motivate the development of our proposed

Original: Unsupervised Disentanglement Ranking (UDR) method.
Revised: Unsupervised Disentanglement Ranking (UDR) method.

Original: Our method relies on the assumption that for a particular dataset and a VAE-based unsupervised disentangled representation learning model class, disentangled representations are all alike, while every entangled representation is entangled in its own way, to rephrase Tolstoy.
Revised: Our method relies on the assumption that for a particular dataset and a VAE-based unsupervised disentangled representation learning model class, disentangled representations are all alike, while every entangled representation is entangled in its own way, to rephrase Tolstoy.

Original: We justify this assumption next.
Revised: We justify this assumption next and empirically verify them in Sec.

Revised: A.10 in Supplementary Materials.

Original: Disentangled representations are similar According to Rolinek et al.
Revised: Disentangled representations are similar According to Rolinek et al.

Original: for a given non-adversarial dataset a disentangling VAE will likely keep converging to the same disentangled representation (up to permutation and sign inverse).
Revised: for a given non-adversarial dataset a disentangling VAE will likely keep converging to the same disentangled representation (up to permutation and sign inverse).

Original: Note that this representation will correspond to a single plausible disentangled generative process g i using the notation we introduced in Sec. 2.
Revised: Note that this representation will correspond to a single plausible disentangled generative process g i using the notation we introduced in Sec. 2.

Original: This is because any two different disentangled representations z a and z b learnt by a VAE-based model will only differ in terms of the corresponding signed permutation matrices V a
Revised: This is because any two different disentangled representations z a and z b learnt by a VAE-based model will only differ in terms of the corresponding signed permutation matrices V a

Original: and V b of the SVD decompositions of the locally linear approximations of the Jacobians of their decoders.
Revised: and V b of the SVD decompositions of the locally linear approximations of the Jacobians of their decoders.

Original: Entangled representations are different Unfortunately the ﬁeld of machine learning has little theoretical understanding of the nature and learning dynamics of internal representations in neural networks.
Revised: Entangled representations are different Unfortunately the ﬁeld of machine learning has little theoretical understanding of the nature and learning dynamics of internal representations in neural networks.

Original: The few pieces of research that have looked into the nature of model representations (Raghu et al., 2017; Li et al., 2016; Wang et al., 2018; Morcos et al., 2018) have been empirical rather than theoretical in nature.
Revised: The few pieces of research that have looked into the nature of model representations (Raghu et al., 2017; Li et al., 2016; Wang et al., 2018; Morcos et al., 2018) have been empirical rather than theoretical in nature.

Original: All of them suggest that neural networks tend to converge to different hidden representations despite being trained on the same task with the same hyperparameters and architecture and reaching similar levels of task performance.
Revised: All of them suggest that neural networks tend to converge to different hidden representations despite being trained on the same task with the same hyperparameters and architecture and reaching similar levels of task performance.

Original: Furthermore, the theoretical analysis and the empirical demonstrations in Rolinek et al.
Revised: Furthermore, the theoretical analysis and the empirical demonstrations in Rolinek et al.

Original: (2019) suggest that the entangled VAEs learn representations that are different at least up to a rotation induced by a non-degenerate matrix V in the SVD decomposition of the local linear approximation of the decoder Jacobian J i .
Revised: (2019) suggest that the entangled VAEs learn representations that are different at least up to a rotation induced by a non-degenerate matrix V in the SVD decomposition of the local linear approximation of the decoder Jacobian J i .

Original: Unsupervised Disentanglement
Revised: Unsupervised Disentanglement

Original: Ranking Our proposed UDR method consists of four steps (illustrated in Fig.
Revised: Ranking Our proposed UDR method consists of four steps (illustrated in Fig.

Original: 4 in Supplementary Material):
Revised: 4 in Supplementary Material):

Original: Train M = H × S models, where H is the number of different hyperparameter settings, and S is the number of different initial model weight conﬁgurations (seeds).
Revised: Train M = H × S models, where H is the number of different hyperparameter settings, and S is the number of different initial model weight conﬁgurations (seeds).

Original: For each trained model i ∈ { 1 ,...,M } , sample without replacement P ≤ S other trained models with the same hyperparameters but different seeds.
Revised: For each trained model i ∈ { 1 ,...,M } , sample without replacement P ≤ S other trained models with the same hyperparameters but different seeds.

Original: Perform P pairwise comparisons per trained model and calculate the respective UDR ij scores, where i ∈{ 1 ,...,M } is the model index, and j ∈{ 1 ,...,P } is its unique pairwise match from Step 2. 4.
Revised: Perform P pairwise comparisons per trained model and calculate the respective UDR ij scores, where i ∈{ 1 ,...,M } is the model index, and j ∈{ 1 ,...,P } is its unique pairwise match from Step 2. 4.

Original: Aggregate UDR ij scores for each model i to report the ﬁnal UDR i
Revised: Aggregate UDR ij scores for each model i to report the ﬁnal UDR i

Original: = avg j ( UDR ij ) scores, where avg j ( · ) is the median over P scores.
Revised: = avg j ( UDR ij ) scores, where avg j ( · ) is the median over P scores.

Original: The key part of the UDR method is Step 3, where we calculate the UDR ij score that summarises how similar the representations of the two models i and j are.
Revised: The key part of the UDR method is Step 3, where we calculate the UDR ij score that summarises how similar the representations of the two models i and j are.

Original: As per the justiﬁcations above, two latent representations z i and z j should be scored as highly similar if they axis align with each other upto permutation (the same ground truth factor c k may be encoded by different latent dimensions within the two models,
Revised: As per the justiﬁcations above, two latent representations z i and z j should be scored as highly similar if they axis align with each other upto permutation (the same ground truth factor c k may be encoded by different latent dimensions within the two models,

Original: z i,a and z j,b where a (cid:54) =
Revised: z i,a and z j,b where a (cid:54) =

Original: b ), sign inverse (the two models may learn to encode the valuesof the generative factor in the opposite order compared to each other, z i,a = − z j,b ), and subsetting
Revised: b ), sign inverse (the two models may learn to encode the values

Original: [Table]
Revised: [Table]

Original:  (one model may learn a subset of the factors that the other model has learnt if the relevant disentangling hyperparameters encourage a different number of latents to be switched off in the two models).
Revised: of the generative factor in the opposite order compared to each other, z i,a = − z j,b ), and subsetting (one model may learn a subset of the factors that the other model has learnt if the relevant disentangling hyperparameters encourage a different number of latents to be switched off in the two models).

Original: In order for the UDR to be invariant to the ﬁrst scenario, we propose calculating a full L × L similarity matrix R ij between the individual dimensions of z i ∈ R L and z j ∈ R L (see Fig.
Revised: In order for the UDR to be invariant to the ﬁrst scenario, we propose calculating a full L × L similarity matrix R ij between the individual dimensions of z i ∈ R L and z j ∈ R L (see Fig.

Original: 5 in Supplementary
Revised: 5 in Supplementary

Original: Material).
Revised: Material).

Original: In order to address the second point, we take the absolute value of the similarity matrix | R ij | .
Revised: In order to address the second point, we take the absolute value of the similarity matrix | R ij | .

Original: Finally, to address the third point, we divide the UDR score by the average number of informative latents discovered by the two models.
Revised: Finally, to address the third point, we divide the UDR score by the average number of informative latents discovered by the two models.

Original: Note that even though disentangling often happens when the
Revised: Note that even though disentangling often happens when the

Original: VAEs enter the “polarised regime”, where many of the latent dimensions are switched off, the rankings produced by UDR are not affected by whether the model operates in such a regime or not.
Revised: VAEs enter the “polarised regime”, where many of the latent dimensions are switched off, the rankings produced by UDR are not affected by whether the model operates in such a regime or not.

Original: To populate the similarity matrix R ij we calculate each matrix element as the similarity between two vectors
Revised: To populate the similarity matrix R ij we calculate each matrix element as the similarity between two vectors

Original: z i,a and z j,b , where z i,a is a response of a single latent dimension z a of model i over the entire ordered dataset (see Sec.
Revised: z i,a and z j,b , where z i,a is a response of a single latent dimension z a of model i over the entire ordered dataset (see Sec.

Original: A.5 in Supplementary Material for details).
Revised: A.5 in Supplementary Material for details).

Original: We considered two versions of the
Revised: We considered two versions of the

Original: UDR score based on the method used for calculating the vector similarity: the non-parametric UDR S , using Spearman’s correlation; and the parametric UDR L , using Lasso regression following past work on evaluatingrepresentations(Eastwood&Williams,2018;Lietal.,2016).
Revised: UDR score based on the method used for calculating the vector similarity: the non-parametric UDR S , using Spearman’s correlation; and the parametric UDR L , using Lasso regression following past work on evaluatingrepresentations(Eastwood&Williams,2018;Lietal.,2016).

Original: InpracticetheLassoregression version worked slightly better, so the remainder of the paper is restricted to UDR L (we use UDR L and
Revised: InpracticetheLassoregression version worked slightly better, so the remainder of the paper is restricted to UDR L (we use UDR L and

Original: UDR interchangeably to refer to this version), while UDR S is discussed in the Supplementary Materials.
Revised: UDR interchangeably to refer to this version), while UDR S is discussed in the Supplementary Materials.

Original: Given a similarity matrix R ij , we want to ﬁnd one-to-one correspondence between all the informative latent dimensions within the chosen pair of models.
Revised: Given a similarity matrix R ij , we want to ﬁnd one-to-one correspondence between all the informative latent dimensions within the chosen pair of models.

Original: Hence, we want to see at most a single strong correlation in each row and column of the similarity matrix.
Revised: Hence, we want to see at most a single strong correlation in each row and column of the similarity matrix.

Original: To that accord, we step through the matrix
Revised: To that accord, we step through the matrix

Original: R = | R ij | , one column and row at a time, looking for the strongest correlation, and weighting it by the proportional weight it has within its respective column or row.
Revised: R = | R ij | , one column and row at a time, looking for the strongest correlation, and weighting it by the proportional weight it has within its respective column or row.

Original: We then average all such weighted scores over all the informative row and column latents to calculate the ﬁnal UDR ij score:
Revised: We then average all such weighted scores over all the informative row and column latents to calculate the ﬁnal UDR ij score:

Original: [Equation]
Revised: [Equation]

Original: where r a = max a R ( a,b ) and r b = max b R ( a,b ) .
Revised: where r a = max a R ( a,b ) and r b = max b R ( a,b ) .

Original: I KL indicates an “informative” latent within a model and d is the number of such latents: d a = (cid:80) a I KL ( a ) and d b = (cid:80) b I KL ( b ) .
Revised: I KL indicates an “informative” latent within a model and d is the number of such latents: d a = (cid:80) a I KL ( a ) and d b = (cid:80) b I KL ( b ) .

Original: We deﬁne a latent dimension as “informative” if it has learnt a latent posterior which diverges from the prior:
Revised: We deﬁne a latent dimension as “informative” if it has learnt a latent posterior which diverges from the prior:

Original: [Equation]
Revised: [Equation]

Original: UDR variations We explored whether doing all-to-all pairwise comparisons, with models in
Revised: UDR variations We explored whether doing all-to-all pairwise comparisons, with models in

Original: Step 2 sampled from the set of all M models rather than the subset of S models with the same hyperparameters, would produce more accurate results.
Revised: Step 2 sampled from the set of all M models rather than the subset of S models with the same hyperparameters, would produce more accurate results.

Original: Additionally we investigated the effect of choosing different numbers of models P for pairwise comparisons by sampling P ∼ U [5 , 45] .
Revised: Additionally we investigated the effect of choosing different numbers of models P for pairwise comparisons by sampling P ∼ U [5 , 45] .

Original: UDR assumptions and limitations Note that our approach is aimed at the current state of the art disentangling VAEs, for which the assumptions of our metric have been demonstrated to hold (Rolinek et al., 2019).
Revised: UDR assumptions and limitations Note that our approach is aimed at the current state of the art disentangling VAEs, for which the assumptions of our metric have been demonstrated to hold (Rolinek et al., 2019).

Original: It may be applied to other model classes, however the following assumptions and limitations need to be considered: dSprites
Revised: It may be applied to other model classes, however the following assumptions and limitations need to be considered: dSprites

Original: S c
Revised: S c

Original: o r e
Revised: o r e

Original: S c
Revised: S c

Original: o r e 3D Shapes
Revised: o r e 3D Shapes

Original: S c
Revised: S c

Original: o r e
Revised: o r e

Original: S c
Revised: S c

Original: o r e
Revised: o r e

Original: Disentangled representations produced by two models from the same class trained on the same dataset are likely to be more similar than entangled representations – this holds for disentangling VAEs (Rolinek et al., 2019), but may not hold more broadly.
Revised: Disentangled representations produced by two models from the same class trained on the same dataset are likely to be more similar than entangled representations – this holds for disentangling VAEs (Rolinek et al., 2019), but may not hold more broadly.

Original: Continuous, monotonic and scalar factors – UDR assumes that these properties hold for the data generative factors and their representations.
Revised: Continuous, monotonic and scalar factors – UDR assumes that these properties hold for the data generative factors and their representations.

Original: This is true for the disentangling approaches described in Sec. 3, but may not hold more generally.
Revised: This is true for the disentangling approaches described in Sec. 3, but may not hold more generally.

Original: It is likely that UDR can be adapted to work with other kinds of generative factors (e
Revised: It is likely that UDR can be adapted to work with other kinds of generative factors (e

Original: g. factors with special or no geometry) by exchanging the similarity calculations in Step 3 with an appropriate measure, however we leave this for future work.
Revised: g. factors with special or no geometry) by exchanging the similarity calculations in Step 3 with an appropriate measure, however we leave this for future work.

Original: Herd effect –
Revised: Herd effect –

Original: since UDR detects disentangled representations through pairwise comparisons, the score it assigns to each individual model will depend on the nature of the other models involved in these comparisons.
Revised: since UDR detects disentangled representations through pairwise comparisons, the score it assigns to each individual model will depend on the nature of the other models involved in these comparisons.

Original: This means that UDR is unable to detect a single disentangled model within a hyperparameter sweep.
Revised: This means that UDR is unable to detect a single disentangled model within a hyperparameter sweep.

Original: It also means that when models are only compared within a single hyperparameter setting, individual model scores may be over/under estimated as they tend to be drawn d = 3 CCI-VAE zz0.607 / 0.
Revised: It also means that when models are only compared within a single hyperparameter setting, individual model scores may be over/under estimated as they tend to be drawn d = 3 CCI-VAE zz0.607 / 0.

Original: d = 5 β-VAE zz0.444 / 0.774 TC-VAE d =zz0.212 / 0.692 DIP-VAE d =zz-  towards the mean of the scores of the other models within a hyperparameter group.
Revised: d = 5 β-VAE zz0.444 / 0.774 TC-VAE d =zz0.212 / 0.692 DIP-VAE d =zz-  towards the mean of the scores of the other models within a hyperparameter group.

Original: Thus, it is preferable to perform the UDR-A2A during model selection and UDR during hyperparameter selection.
Revised: Thus, it is preferable to perform the UDR-A2A during model selection and UDR during hyperparameter selection.

Original: Explicitness bias – UDR does not penalise models that learn a subset of the data generative factors.
Revised: Explicitness bias – UDR does not penalise models that learn a subset of the data generative factors.

Original: In fact, such models often score higher than those that learn the full set of generative factors, because the current state of the art disentangling approaches tend to trade-off the number of discovered factors for cleaner disentangling.
Revised: In fact, such models often score higher than those that learn the full set of generative factors, because the current state of the art disentangling approaches tend to trade-off the number of discovered factors for cleaner disentangling.

Original: As discussed in Sec. 2, we provide the practitioner with the ability to choose the most disentangled model per number of factors discovered by approximating this with the d score in Eq. 2.
Revised: As discussed in Sec. 2, we provide the practitioner with the ability to choose the most disentangled model per number of factors discovered by approximating this with the d score in Eq. 2.

Original: Computational cost – UDR requires training a number of seeds per hyperparameter setting and
Revised: Computational cost – UDR requires training a number of seeds per hyperparameter setting and

Original: M × P pairwise comparisons per hyperparameter search, which may be computationally expensive.
Revised: M × P pairwise comparisons per hyperparameter search, which may be computationally expensive.

Original: Saying this, training multiple seeds per hyperparameter setting is a good research practice to produce more robust results and UDR computations are highly parallelisable.
Revised: Saying this, training multiple seeds per hyperparameter setting is a good research practice to produce more robust results and UDR computations are highly parallelisable.

Original: To summarise, UDR relies on a number of assumptions and has certain limitations that we hope to relax in future work.
Revised: To summarise, UDR relies on a number of assumptions and has certain limitations that we hope to relax in future work.

Original: However, it offers improvements over the existing supervised metrics.
Revised: However, it offers improvements over the existing supervised metrics.

Original: Apart from being the only method that does not rely on supervised attribute labels, its scores are often more representative of the true disentanglement quality (e.g. see Fig.
Revised: Apart from being the only method that does not rely on supervised attribute labels, its scores are often more representative of the true disentanglement quality (e.g. see Fig.

Original: 3 and Fig.
Revised: 3 and Fig.

Original: 9 in Supplementary
Revised: 9 in Supplementary

Original: Materials), and it does not assume a single “canonical” disentangled factorisation per dataset.
Revised: Materials), and it does not assume a single “canonical” disentangled factorisation per dataset.

Original: Hence, we believe that UDR can be a useful method for unlocking the power of unsupervised disentangled representation learning to real-life practical applications, at least in the near future.
Revised: Hence, we believe that UDR can be a useful method for unlocking the power of unsupervised disentangled representation learning to real-life practical applications, at least in the near future.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Our hope was to develop a method for unsupervised disentangled model selection with the following properties: it should 1) help with hyperparameter tuning by producing an aggregate score that can be used to guide evolutionary or Bayesian methods (Jaderberg et al., 2018;
Revised: Our hope was to develop a method for unsupervised disentangled model selection with the following properties: it should 1) help with hyperparameter tuning by producing an aggregate score that can be used to guide evolutionary or Bayesian methods (Jaderberg et al., 2018;

Original: Snoek et al., 2012; Thornton et al., 2012; Bergstra et al., 2011; Hutter et al., 2011; Miikkulainen et al., 2017); 2) rank individual trained models based on their disentanglement quality; 3) correlate well with ﬁnal task performance.
Revised: Snoek et al., 2012; Thornton et al., 2012; Bergstra et al., 2011; Hutter et al., 2011; Miikkulainen et al., 2017); 2) rank individual trained models based on their disentanglement quality; 3) correlate well with ﬁnal task performance.

Original: In this section we evaluate our proposed UDR against these qualities.
Revised: In this section we evaluate our proposed UDR against these qualities.

Original: For the reported experiments we use the trained model checkpoints and supervised scores from Locatello et al.
Revised: For the reported experiments we use the trained model checkpoints and supervised scores from Locatello et al.

Original: (2018) to evaluate β -VAE, CCI-VAE, FactorVAE, TC-VAE, DIP-VAE-I and DIP-VAE-II on two benchmark datasets: dSprites (Matthey et al., 2017) and 3D Shapes (Burgess & Kim, 2018) (see Sec.
Revised: (2018) to evaluate β -VAE, CCI-VAE, FactorVAE, TC-VAE, DIP-VAE-I and DIP-VAE-II on two benchmark datasets: dSprites (Matthey et al., 2017) and 3D Shapes (Burgess & Kim, 2018) (see Sec.

Original: A.3 for details).
Revised: A.3 for details).

Original: Each model is trained with H =6 different hyperparameter settings (detailed in Sec.
Revised: Each model is trained with H =6 different hyperparameter settings (detailed in Sec.

Original: A.4.1 in Supplementary
Revised: A.4.1 in Supplementary

Original: Material), with S =50 seeds per setting, and P =50 pairwise comparisons.
Revised: Material), with S =50 seeds per setting, and P =50 pairwise comparisons.

Original: UDR correlates well with the supervised metrics.
Revised: UDR correlates well with the supervised metrics.

Original: To validate UDR, we calculate Spearman’s correlation between its model ranking and that produced by four existing supervised disentanglement metrics found to be the most meaningful in the large scale comparison study by Locatello et al.
Revised: To validate UDR, we calculate Spearman’s correlation between its model ranking and that produced by four existing supervised disentanglement metrics found to be the most meaningful in the large scale comparison study by Locatello et al.

Original: (2018): the original β -VAE metric (Higgins et al., 2017a), FactorVAE metric (Kim & Mnih, 2018),
Revised: (2018): the original β -VAE metric (Higgins et al., 2017a), FactorVAE metric (Kim & Mnih, 2018),

Original: Mutual Information Gap (MIG) (Chen et al., 2018) and DCI Disentanglement (Eastwood & Williams, 2018)
Revised: Mutual Information Gap (MIG) (Chen et al., 2018) and DCI Disentanglement (Eastwood & Williams, 2018)

Original: (see Sec.
Revised: (see Sec.

Original: A.6 in Supplementary Material for metric details).
Revised: A.6 in Supplementary Material for metric details).

Original: The average correlation for UDR is 0 .
Revised: The average correlation for UDR is 0 .

Original: 54 ± 0 .
Revised: 54 ± 0 .

Original: 06 and for UDR-A2A is 0 .
Revised: 06 and for UDR-A2A is 0 .

Original: 60 ± 0 .
Revised: 60 ± 0 .

Original: This is comparable to the average Spearman’s
Revised: This is comparable to the average Spearman’s

Original: [Table]
Revised: [Table]

Original: correlation between the model rankings produced by the different supervised metrics: 0 .
Revised: correlation between the model rankings produced by the different supervised metrics: 0 .

Original: 67 ± 0 . 2 .
Revised: 67 ± 0 . 2 .

Original: The variance in rankings produced by the different metrics is explained by the fact that the metrics capture different aspects of disentangling (see Sec.
Revised: The variance in rankings produced by the different metrics is explained by the fact that the metrics capture different aspects of disentangling (see Sec.

Original: A.2 in Supplementary Materials for a discussion of how UDR relates to other representation comparison methods).
Revised: A.2 in Supplementary Materials for a discussion of how UDR relates to other representation comparison methods).

Original: Tbl.
Revised: Tbl.

Original: 1 provides a breakdown of correlation scores between MIG and the different versions of UDR for different model classes and datasets.
Revised: 1 provides a breakdown of correlation scores between MIG and the different versions of UDR for different model classes and datasets.

Original: It is clear that the different versions of UDR perform similarly to each other, and this holds across datasets and model classes.
Revised: It is clear that the different versions of UDR perform similarly to each other, and this holds across datasets and model classes.

Original: Note that unlike the supervised metrics, UDR does not assume a “canonical” disentangled representation.
Revised: Note that unlike the supervised metrics, UDR does not assume a “canonical” disentangled representation.

Original: Instead, it allows any one of the many equivalent possible ground truth generative processes to become the “canonical” one for each particular dataset and model class, as per the theoretical results by Rolinek et al.
Revised: Instead, it allows any one of the many equivalent possible ground truth generative processes to become the “canonical” one for each particular dataset and model class, as per the theoretical results by Rolinek et al.

Original: (2019) summarised in Sec. 3.1.
Revised: (2019) summarised in Sec. 3.1.

Original: UDR is useful for hyperparameter selection.
Revised: UDR is useful for hyperparameter selection.

Original: Fig.
Revised: Fig.

Original: 2 compares the scores produced by UDR and the four supervised metrics for 3600 trained models, split over six model classes, two datasets and six hyperparametersettings.
Revised: 2 compares the scores produced by UDR and the four supervised metrics for 3600 trained models, split over six model classes, two datasets and six hyperparametersettings.

Original: Weconsiderthemedianscoreproﬁlesacrossthesixhyperparametersettingsto evaluatewhetheraparticularsettingisbetterthanothers.
Revised: Weconsiderthemedianscoreproﬁlesacrossthesixhyperparametersettingsto evaluatewhetheraparticularsettingisbetterthanothers.

Original: ItcanbeseenthatUDRbroadlyagreeswiththe supervised metrics on which hyperparameters are more promising for disentangling.
Revised: ItcanbeseenthatUDRbroadlyagreeswiththe supervised metrics on which hyperparameters are more promising for disentangling.

Original: This holds across datasets and model classes.
Revised: This holds across datasets and model classes.

Original: Hence, UDR may be useful for evaluating model ﬁtness for disentangled representation learning as part of an evolutionary algorithm or Bayesian hyperparameter tuning.
Revised: Hence, UDR may be useful for evaluating model ﬁtness for disentangled representation learning as part of an evolutionary algorithm or Bayesian hyperparameter tuning.

Original: UDR is useful for model selection.
Revised: UDR is useful for model selection.

Original: Fig.
Revised: Fig.

Original: 2 can also be used to examine whether a particular trained model has learnt a good disentangled representation.
Revised: 2 can also be used to examine whether a particular trained model has learnt a good disentangled representation.

Original: We see that some models reach high UDR scores.
Revised: We see that some models reach high UDR scores.

Original: For example, more models score highly as the value of the β hyperparameter is increased in the β -VAE model class.
Revised: For example, more models score highly as the value of the β hyperparameter is increased in the β -VAE model class.

Original: This is in line with the previously reported results (Higgins et al., 2017a).
Revised: This is in line with the previously reported results (Higgins et al., 2017a).

Original: Note that the 0th hyperparameter setting in this case corresponds to β =1 , which is equivalent to the standard VAE objective (Kingma & Welling, 2014; Rezende et al., 2014).
Revised: Note that the 0th hyperparameter setting in this case corresponds to β =1 , which is equivalent to the standard VAE objective (Kingma & Welling, 2014; Rezende et al., 2014).

Original: As expected, these models score low in terms of disentangling.
Revised: As expected, these models score low in terms of disentangling.

Original: We also see that for some model classes (e.g. DIP-VAE-I, DIP-VAE-II and 
Revised: We also see that for some model classes (e.g. DIP-VAE-I, DIP-VAE-II and FactorVAE on dSprites)

Original: FactorVAE on dSprites) no trained model scores highly according to UDR.
Revised: no trained model scores highly according to UDR.

Original: This suggests that none of the hyperparameter choices explored were good for this particular dataset, and that no instance of the model class learnt to disentangle well.
Revised: This suggests that none of the hyperparameter choices explored were good for this particular dataset, and that no instance of the model class learnt to disentangle well.

Original: To test this, we use latent traversals to qualitatively evaluate the level of disentanglement achieved by the models, ranked by their UDR scores.
Revised: To test this, we use latent traversals to qualitatively evaluate the level of disentanglement achieved by the models, ranked by their UDR scores.

Original: This is a common technique to qualitatively evaluate the level of disentanglement on simple visual datasets where no ground truth attribute labels are available.
Revised: This is a common technique to qualitatively evaluate the level of disentanglement on simple visual datasets where no ground truth attribute labels are available.

Original: Such traversals involve changing the value of one latent dimension at a time and evaluating its effect on the resulting reconstructions to understand whether the latent has learnt to represent anything semantically meaningful.
Revised: Such traversals involve changing the value of one latent dimension at a time and evaluating its effect on the resulting reconstructions to understand whether the latent has learnt to represent anything semantically meaningful.

Original: Fig.
Revised: Fig.

Original: 3 demonstrates that the qualitative disentanglement quality is reﬂected well in the UDR scores.
Revised: 3 demonstrates that the qualitative disentanglement quality is reﬂected well in the UDR scores.

Original: The ﬁgure also highlights that the supervised metric scores can sometimes be overoptimistic.
Revised: The ﬁgure also highlights that the supervised metric scores can sometimes be overoptimistic.

Original: For example, compare TC-VAE and β -VAE traversals in Fig.
Revised: For example, compare TC-VAE and β -VAE traversals in Fig.

Original: These are scored similarly by the supervised metric (0.774 and 0.751) but differently by UDR (0.444 and 0.607).
Revised: These are scored similarly by the supervised metric (0.774 and 0.751) but differently by UDR (0.444 and 0.607).

Original: Qualitative evaluation of the traversals clearly shows that β -VAE learnt a more disentangled representation than TC-VAE, which is captured by UDR but not by the supervised metric.
Revised: Qualitative evaluation of the traversals clearly shows that β -VAE learnt a more disentangled representation than TCVAE, which is captured by UDR but not by the supervised metric.

Original: Fig.
Revised: Fig.

Original: 9 in Supplementary Material provides more examples.
Revised: 9 in Supplementary Material provides more examples.

Revised: We also evaluated how well UDR ranks models trained on more complex datasets,

Revised: CelebA and ImageNet, and found that it performs well (see Sec.

Revised: A.9 in Supplementary Materials).

Original: UDR works well even with ﬁve pairwise comparisons.
Revised: UDR works well even with ﬁve pairwise comparisons.

Original: We test the effect of the number of pairwise comparisons P on the variance and accuracy of the UDR scores.
Revised: We test the effect of the number of pairwise comparisons P on the variance and accuracy of the UDR scores.

Original: Tbl.
Revised: Tbl.

Original: 2 reports the changes in the rank correlation with the β -VAE metric on the dSprites dataset as P is varied between 5 and 45.
Revised: 2 reports the changes in the rank correlation with the β -VAE metric on the dSprites dataset as P is varied between 5 and 45.

Original: We see that the correlation between the UDR and the β -VAE metric becomes higher and the variance decreases as the number of seeds is increased.
Revised: We see that the correlation between the UDR and the β -VAE metric becomes higher and the variance decreases as the number of seeds is increased.

Original: However, even with P =5 the correlation is reasonable.
Revised: However, even with P =5 the correlation is reasonable.

Original: UDR generalises to a dataset with no attribute labels.
Revised: UDR generalises to a dataset with no attribute labels.

Original: We investigate whether UDR can be useful for selecting well disentangled models trained on the 3D Cars (Reed et al., 2014) dataset with poorly labelled attributes, which makes it a bad ﬁt for supervised disentanglement metrics.
Revised: We investigate whether UDR can be useful for selecting well disentangled models trained on the 3D Cars (Reed et al., 2014) dataset with poorly labelled attributes, which makes it a bad ﬁt for supervised disentanglement metrics.

Original: Fig.
Revised: Fig.

Original: 1 shows that a highly ranked model according to UDR appears disentangled, while a poorly ranked one appears entangled.
Revised: 1 shows that a highly ranked model according to UDR appears disentangled, while a poorly ranked one appears entangled.

Original: Fig.
Revised: Fig.

Original: 10 in Supplementary Material provides more examples of high and low scoring models according to the UDR method.
Revised: 10 in Supplementary Material provides more examples of high and low scoring models according to the UDR method.

Original: UDR predicts ﬁnal task performance.
Revised: UDR predicts ﬁnal task performance.

Original: We developed UDR to help practitioners use disentangled representations to better solve subsequent tasks.
Revised: We developed UDR to help practitioners use disentangled representations to better solve subsequent tasks.

Original: Hence, we evaluate whether the model ranking produced by UDR correlates with task performance on two different domains: the fairness on a classiﬁcation task introduced by Locatello et al.
Revised: Hence, we evaluate whether the model ranking produced by UDR correlates with task performance on two different domains: the fairness on a classiﬁcation task introduced by Locatello et al.

Original: (2018), and data efﬁciency on a clustering task for a model-based reinforcement learning agent introduced by Watters et al.
Revised: (2018), and data efﬁciency on a clustering task for a model-based reinforcement learning agent introduced by Watters et al.

Original: (see Sec.
Revised: (see Sec.

Original: A.8 in Supplementary Materials for more details).
Revised: A.8 in Supplementary Materials for more details).

Original: We found that UDR had an average of 0.8 Spearman correlation with the fairness scores, which is higher than the average of 0.72 correlation between fairness and supervised scores reported by Locatello et al.
Revised: We found that UDR had an average of 0.8 Spearman correlation with the fairness scores, which is higher than the average of 0.72 correlation between fairness and supervised scores reported by Locatello et al.

Original: We also found that UDR scores had 0.56 Spearman correlation with data efﬁciency of the COBRA agent.
Revised: We also found that UDR scores had 0.56 Spearman correlation with data efﬁciency of the COBRA agent.

Original: The difference between the best and the worst models according to UDR amounted to around 66% reduction in the number of steps to 90% success rate on the task.
Revised: The difference between the best and the worst models according to UDR amounted to around 66% reduction in the number of steps to 90% success rate on the task.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We have introduced UDR, the ﬁrst method for unsupervised model selection for variational disentangled representation learning.
Revised: We have introduced UDR, the ﬁrst method for unsupervised model selection for variational disentangled representation learning.

Original: We have validated our approach on 5400 models covering all six state of the art VAE-based unsupervised disentangled representation learning model classes.
Revised: We have validated our approach on 5400 models covering all six state of the art VAE-based unsupervised disentangled representation learning model classes.

Original: We compared UDR to four existing supervised disentanglement metrics both quantitatively and qualitatively, and demonstrated that our approach works reliably well across three different datasets, often ranking models more accurately than the supervised alternatives.
Revised: We compared UDR to four existing supervised disentanglement metrics both quantitatively and qualitatively, and demonstrated that our approach works reliably well across three different datasets, often ranking models more accurately than the supervised alternatives.

Original: Moreover, UDR avoids one of the big shortcomings of the supervised disentangling metrics – the arbitrary choice of a “canonical” disentangled factorisation, instead allowing any of the equally valid disentangled generative processes to be accepted.
Revised: Moreover, UDR avoids one of the big shortcomings of the supervised disentangling metrics – the arbitrary choice of a “canonical” disentangled factorisation, instead allowing any of the equally valid disentangled generative processes to be accepted.

Original: Finally, we also demonstrated that UDR is useful for predicting ﬁnal task performance using two different domains.
Revised: Finally, we also demonstrated that UDR is useful for predicting ﬁnal task performance using two different domains.

Original: Hence, we hope that UDR can be a step towards unlocking the power of unsupervised disentangled representation learning to real-life applications.
Revised: Hence, we hope that UDR can be a step towards unlocking the power of unsupervised disentangled representation learning to real-life applications.

Original: A CKNOWLEDGEMENTS
Revised: A CKNOWLEDGEMENTS

Original: We thank Olivier Bachem and Francesco Locatello for helping us re-use their code and model checkpoints, and Neil Rabinowitz, Avraham Ruderman and Tatjana Chavdarova for useful feedback
Revised: We thank Olivier Bachem and Francesco Locatello for helping us re-use their code and model checkpoints, and Neil Rabinowitz, Avraham Ruderman and Tatjana Chavdarova for useful feedback

Original: A S UPPLEMENTARY M ATERIAL A. U SEFUL PROPERTIES OF DISENTANGLED REPRESENTATIONS
Revised: A S UPPLEMENTARY M ATERIAL A. U SEFUL PROPERTIES OF DISENTANGLED REPRESENTATIONS

Original: Disentangled representations are particularly useful because they re-represent the information contained in the data in a way that enables semantically meaningful compositionality .
Revised: Disentangled representations are particularly useful because they re-represent the information contained in the data in a way that enables semantically meaningful compositionality .

Original: For example, having discovered that the data is generated using two factors, colour and shape, such a model would be able to support meaningful reasoning about ﬁctitious objects, like pink elephants, despite having never seen one during training (Higgins et al., 2017b; 2018b).
Revised: For example, having discovered that the data is generated using two factors, colour and shape, such a model would be able to support meaningful reasoning about ﬁctitious objects, like pink elephants, despite having never seen one during training (Higgins et al., 2017b; 2018b).

Original: This opens up opportunities for counterfactual reasoning, more robust and interpretable inference and model-based planning (Higgins et al., 2018a; Suter et al., 2018).
Revised: This opens up opportunities for counterfactual reasoning, more robust and interpretable inference and model-based planning (Higgins et al., 2018a; Suter et al., 2018).

Original: Furthermore, such a representation would support more data efﬁcient learning for subsequent tasks, like a classiﬁcation objective for differentiating elephants from cats.
Revised: Furthermore, such a representation would support more data efﬁcient learning for subsequent tasks, like a classiﬁcation objective for differentiating elephants from cats.

Original: This could be achieved by ignoring the nuisance variables irrelevant for the task, e.g. the colour variations, by simply masking out the disentangled subspaces that learnt to represent such nuisances, while only paying attention to the task-relevant subspaces, e.g. the units that learnt to represent shape (Cohen & Welling, 2016; Gens & Domingos, 2014; Soatto, 2010).
Revised: This could be achieved by ignoring the nuisance variables irrelevant for the task, e.g. the colour variations, by simply masking out the disentangled subspaces that learnt to represent such nuisances, while only paying attention to the task-relevant subspaces, e.g. the units that learnt to represent shape (Cohen & Welling, 2016; Gens & Domingos, 2014; Soatto, 2010).

Original: Hence, the semantically meaningful compositional nature of disentangled representations is perhaps the most sought after aspect of disentangling, due to its strong implications for generalisation, data efﬁciency and interpretability (Schmidhuber, 1992; Bengio et al., 2013; Higgins et al., 2018a).
Revised: Hence, the semantically meaningful compositional nature of disentangled representations is perhaps the most sought after aspect of disentangling, due to its strong implications for generalisation, data efﬁciency and interpretability (Schmidhuber, 1992; Bengio et al., 2013; Higgins et al., 2018a).

Original: A. A SPECTS OF DISENTANGLEMENT MEASURED BY DIFFERENT METRICS
Revised: A. A SPECTS OF DISENTANGLEMENT MEASURED BY DIFFERENT METRICS

Original: Methods for evaluating and comparing representations have been proposed in the past.
Revised: Methods for evaluating and comparing representations have been proposed in the past.

Original: The most similar approaches to ours are the DCI Disentanglement score from Eastwood & Williams (2018) and the axis alignment comparison of representations in trained classiﬁers proposed in Li et al.
Revised: The most similar approaches to ours are the DCI Disentanglement score from Eastwood & Williams (2018) and the axis alignment comparison of representations in trained classiﬁers proposed in Li et al.

Original: The former is not directly applicable for unsupervised disentangled model selection, since it requires access to the ground truth attribute labels.
Revised: The former is not directly applicable for unsupervised disentangled model selection, since it requires access to the ground truth attribute labels.

Original: Even when adapted to compare two latent representations, our preliminary experiments suggested that the entropy based aggregation proposed in Eastwood & Williams (2018) is inferior to our aggregation in Eq.when used in the UDR setup.
Revised: Even when adapted to compare two latent representations, our preliminary experiments suggested that the entropy based aggregation proposed in Eastwood & Williams (2018) is inferior to our aggregation in Eq.when used in the UDR setup.

Original: The approach by Li et al.
Revised: The approach by Li et al.

Original: (2016) shares the similarity matrix calculation step with us, however they never go beyond that quantitatively, opting for qualitative evaluations of model representations instead.
Revised: (2016) shares the similarity matrix calculation step with us, however they never go beyond that quantitatively, opting for qualitative evaluations of model representations instead.

Original: Hence, their approach is not directly applicable to quantitative unsupervised disentangled model ranking.
Revised: Hence, their approach is not directly applicable to quantitative unsupervised disentangled model ranking.

Original: Other related approaches worth mentioning are the Canonical Correlation Analysis (CCA) and its modiﬁcations (Hardoon et al., 2004; Raghu et al., 2017; Morcos et al., 2018).
Revised: Other related approaches worth mentioning are the Canonical Correlation Analysis (CCA) and its modiﬁcations (Hardoon et al., 2004; Raghu et al., 2017; Morcos et al., 2018).

Original: These approaches, however, tend to be invariant to invertible afﬁne transformations and therefore to the axis alignment of individual neurons, which makes them unsuitable for evaluating disentangling quality.
Revised: These approaches, however, tend to be invariant to invertible afﬁne transformations and therefore to the axis alignment of individual neurons, which makes them unsuitable for evaluating disentangling quality.

Original: Finally, Representation Similarity Matrix (RSM) (Kriegeskorte et al., 2008) is a commonly used method in Neuroscience for comparing the representations of different systems to the same set of stimuli.
Revised: Finally, Representation Similarity Matrix (RSM) (Kriegeskorte et al., 2008) is a commonly used method in Neuroscience for comparing the representations of different systems to the same set of stimuli.

Original: This technique, however, is not applicable for measuring disentangling, because it ignores dimension-wise response properties.
Revised: This technique, however, is not applicable for measuring disentangling, because it ignores dimension-wise response properties.

Original: When talking about disentangled representations, three properties are generally considered: modularity , compactness and explicitness 1 (Ridgeway & Mozer, 2018).
Revised: When talking about disentangled representations, three properties are generally considered: modularity , compactness and explicitness 1 (Ridgeway & Mozer, 2018).

Original: Modularity measures whether each latent dimension encodes only one data generative factor, compactness measures whether each data generative factor is encoded by a single latent dimension, and explicitness measures whether all the information about the data generative factors can be decoded from the latent representation.
Revised: Modularity measures whether each latent dimension encodes only one data generative factor, compactness measures whether each data generative factor is encoded by a single latent dimension, and explicitness measures whether all the information about the data generative factors can be decoded from the latent representation.

Original: We believe that modularity is the key aspect of disentangling, since it measures whether the representation is compositional, which gives disentangled representations the majority of their beneﬁcial properties (see Sec.
Revised: We believe that modularity is the key aspect of disentangling, since it measures whether the representation is compositional, which gives disentangled representations the majority of their beneﬁcial properties (see Sec.

Original: A.1 in Supplementary Materials for more details).
Revised: A.1 in Supplementary Materials for more details).

Original: Compactness , on the other hand, may not always be desirable.
Revised: Compactness , on the other hand, may not always be desirable.

Original: For example, according to a recent deﬁnition of disentangled representations (Higgins et al., 2018a), it is theoretically impossible to represent 3D rotation in a single dimension (see also Ridgeway & Mozer (2018)).
Revised: For example, according to a recent deﬁnition of disentangled representations (Higgins et al., 2018a), it is theoretically impossible to represent 3D rotation in a single dimension (see also Ridgeway & Mozer (2018)).

Original: Finally, while explicitness is clearly desirable for preserving information about the data that may be useful for subsequent tasks, in practice models often fail to discover and represent the full set of the data generative factors due to restrictions on both the observed data distribution and the model capacity (Mathieu et al., 2019).
Revised: Finally, while explicitness is clearly desirable for preserving information about the data that may be useful for subsequent tasks, in practice models often fail to discover and represent the full set of the data generative factors due to restrictions on both the observed data distribution and the model capacity (Mathieu et al., 2019).

Original: Hence, we suggest noting the explicitness of a representation, but not necessarily punishing its disentanglement ranking if it is not fully explicit.
Revised: Hence, we suggest noting the explicitness of a representation, but not necessarily punishing its disentanglement ranking if it is not fully explicit.

Original: Instead, we suggest that the practitioner should have the choice to select the most disentangled model given a particular number of discovered generative factors.
Revised: Instead, we suggest that the practitioner should have the choice to select the most disentangled model given a particular number of discovered generative factors.

Original: Hence, in the rest of the paper we will use the term “disentanglement” to refer to the compositional property of a representation related to the modularity measure.
Revised: Hence, in the rest of the paper we will use the term “disentanglement” to refer to the compositional property of a representation related to the modularity measure.

Original: Tbl.
Revised: Tbl.

Original: 3 provides a summary of how the different metrics considered in the paper compare in terms of modularity, compactness and explicitness.
Revised: 3 provides a summary of how the different metrics considered in the paper compare in terms of modularity, compactness and explicitness.

Original: A. D ATASET DETAILS
Revised: A. D ATASET DETAILS

Original: dSprites A commonly used unit test for evaluating disentangling is the dSprites dataset (Matthey et al., 2017)
Revised: dSprites A commonly used unit test for evaluating disentangling is the dSprites dataset (Matthey et al., 2017)

Original: This dataset consists of images of a single binary sprite pasted on a blank be fully described generative y ( disentanglement
Revised: This dataset consists of images of a single binary sprite pasted on a blank be fully described generative y ( disentanglement

Original: [Table]
Revised: [Table]

Original: 3D Shapes
Revised: 3D Shapes

Original: A more complex dataset for evaluating disentangling is the 3D Shapes dataset (Burgess & Kim, 2018).
Revised: A more complex dataset for evaluating disentangling is the 3D Shapes dataset (Burgess & Kim, 2018).

Original: This dataset consists of images of a single 3D object in a room and is fully speciﬁed by six generative factors: ﬂoor colour (10 values), wall colour (10 values), object colour (10 values), size (8 values), shape (4 values) and rotation (15 values).
Revised: This dataset consists of images of a single 3D object in a room and is fully speciﬁed by six generative factors: ﬂoor colour (10 values), wall colour (10 values), object colour (10 values), size (8 values), shape (4 values) and rotation (15 values).

Original: All the generative factors are sampled from a uniform distribution.
Revised: All the generative factors are sampled from a uniform distribution.

Original: Colours are sampled from the circular hue space.
Revised: Colours are sampled from the circular hue space.

Original: Rotation is sampled from the [-30, 30] degree angle range.
Revised: Rotation is sampled from the [-30, 30] degree angle range.

Original: 3D Cars
Revised: 3D Cars

Original: This dataset was adapted from Reed et al.
Revised: This dataset was adapted from Reed et al.

Original: The full data generative process for this dataset is unknown.
Revised: The full data generative process for this dataset is unknown.

Original: Thelabelledfactorsinclude199carmodelsand24rotationssampledfromthefull360degreeoutofplane rotation range.
Revised: Thelabelledfactorsinclude199carmodelsand24rotationssampledfromthefull360degreeoutofplane rotation range.

Original: An example of an unlabelled generative factor is the colour of the car – this varies across the dataset.
Revised: An example of an unlabelled generative factor is the colour of the car – this varies across the dataset.

Original: A. U NSUPERVISED DISENTANGLED REPRESENTATION LEARNING MODELS
Revised: A. U NSUPERVISED DISENTANGLED REPRESENTATION LEARNING MODELS

Original: As mentioned in Sec. 3, current state of the art approaches to unsupervised disentangled representation learning are based on the VAE (Kingma & Welling, 2014; Rezende et al., 2014) objective presented in Eq.
Revised: As mentioned in Sec. 3, current state of the art approaches to unsupervised disentangled representation learning are based on the VAE (Kingma & Welling, 2014; Rezende et al., 2014) objective presented in Eq.

Original: These approaches decompose the objective in Eq.
Revised: These approaches decompose the objective in Eq.

Original: 1 into various terms and change their relative weighting to exploit the trade-off between the capacity of the latent information bottleneck with independent sources of noise, and the quality of the resulting reconstruction in order to learn a disentangled representation.
Revised: 1 into various terms and change their relative weighting to exploit the trade-off between the capacity of the latent information bottleneck with independent sources of noise, and the quality of the resulting reconstruction in order to learn a disentangled representation.

Original: The ﬁrst such modiﬁcation was introduced by Higgins et al.
Revised: The ﬁrst such modiﬁcation was introduced by Higgins et al.

Original: (2017a)
Revised: (2017a)

Original: in their β -VAE framework:
Revised: in their β -VAE framework:

Original: [Equation]
Revised: [Equation]

Original: In order to achieve disentangling in β -VAE, the KL term in Eq. 4 is typically up-weighted by setting β > 1 .
Revised: In order to achieve disentangling in β -VAE, the KL term in Eq. 4 is typically up-weighted by setting β > 1 .

Original: This implicitly reduces the latent bottleneck capacity and, through the interaction with the reconstruction term, encourages the generative factors
Revised: This implicitly reduces the latent bottleneck capacity and, through the interaction with the reconstruction term, encourages the generative factors

Original: c k with different reconstruction proﬁles to be encoded by different independent noisy channels z l in the latent bottleneck.
Revised: c k with different reconstruction proﬁles to be encoded by different independent noisy channels z l in the latent bottleneck.

Original: Building on the β -VAE ideas, CCI-VAE (Burgess et al., 2017) suggested slowly increasing the bottleneck capacity during training, thus improving the ﬁnal disentanglement and reconstruction quality: E
Revised: Building on the β -VAE ideas, CCI-VAE (Burgess et al., 2017) suggested slowly increasing the bottleneck capacity during training, thus improving the ﬁnal disentanglement and reconstruction quality: E

Original: [Equation]
Revised: [Equation]

Original: Later approaches (Kim & Mnih, 2018; Chen et al., 2018; Kumar et al., 2017) showed that the KL term in Eq.
Revised: Later approaches (Kim & Mnih, 2018; Chen et al., 2018; Kumar et al., 2017) showed that the KL term in Eq.

Original: 1 can be further decomposed according to:
Revised: 1 can be further decomposed according to:

Original: [Equation]
Revised: [Equation]

Original: Hence, penalising the full KL term as in Eqs.
Revised: Hence, penalising the full KL term as in Eqs.

Original: 4-5 is not optimal, since it unnecessarily penalises the mutual infor- mation between the latents and the data.
Revised: 4-5 is not optimal, since it unnecessarily penalises the mutual infor- mation between the latents and the data.

Original: To remove this undesirable side effect, different authors suggested instead adding more targeted penalised terms to the VAE objective function.
Revised: To remove this undesirable side effect, different authors suggested instead adding more targeted penalised terms to the VAE objective function.

Original: These include different implementations of the total correlation penalty (FactorVAE by Kim & Mnih (2018) and TC-VAE by Chen et al.
Revised: These include different implementations of the total correlation penalty (FactorVAE by Kim & Mnih (2018) and TC-VAE by Chen et al.

Original: [Equation]
Revised: [Equation]

Original: and different implementations of the penalty that pushes the marginal posterior towards a factorised prior (DIP-VAE by Kumar et al.
Revised: and different implementations of the penalty that pushes the marginal posterior towards a factorised prior (DIP-VAE by Kumar et al.

Original: [Equation]
Revised: [Equation]

Original: A.4. M ODEL IMPLEMENTATION DETAILS
Revised: A.4. M ODEL IMPLEMENTATION DETAILS

Original: We re-used the trained checkpoints from Locatello et al.
Revised: We re-used the trained checkpoints from Locatello et al.

Original: (2018), hence we recommend the readers to check the original paper for model implementation details.
Revised: (2018), hence we recommend the readers to check the original paper for model implementation details.

Original: Brieﬂy, the following architecture and optimiser were used.
Revised: Brieﬂy, the following architecture and optimiser were used.

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

Original: For consistency, all the models were trained using the same architecture, optimiser, and hyperparameters.
Revised: For consistency, all the models were trained using the same architecture, optimiser, and hyperparameters.

Original: All of the methods use a deep neural network to encode and decode the latent embedding and the parameters of the latent factors are predicted using a Gaussian encoder whose architecture is speciﬁed in Table 4.
Revised: All of the methods use a deep neural network to encode and decode the latent embedding and the parameters of the latent factors are predicted using a Gaussian encoder whose architecture is speciﬁed in Table 4.

Original: All of the models predict a latent vector with 10 factors.
Revised: All of the models predict a latent vector with 10 factors.

Original: Each model was also trained with 6 different levels of regularisation strength speciﬁed in Table 5.
Revised: Each model was also trained with 6 different levels of regularisation strength speciﬁed in Table 5.

Original: The ranges of the hyperparameters used for the various levels of regularisation were speciﬁed to show a diversity of different performance on different datasets without relying on pre-existing intuition on good hyperparameters, however ranges were based on hyperparameters that were used previously in literature.
Revised: The ranges of the hyperparameters used for the various levels of regularisation were speciﬁed to show a diversity of different performance on different datasets without relying on pre-existing intuition on good hyperparameters, however ranges were based on hyperparameters that were used previously in literature.

Original: For each of the model classes outlined above, we tried 6 hyperparameter values with 50 seeds each.
Revised: For each of the model classes outlined above, we tried 6 hyperparameter values with 50 seeds each.

Original: β -VAE
Revised: β -VAE

Original: The β -VAE (Higgins et al., 2017a) model is similar to the vanilla VAE model but with an additional hyperparameter β to modify the strength of the KL regulariser.
Revised: The β -VAE (Higgins et al., 2017a) model is similar to the vanilla VAE model but with an additional hyperparameter β to modify the strength of the KL regulariser.

Original: [Equation]
Revised: [Equation]

Original: where a β value of 1 corresponds to the vanilla VAE model.
Revised: where a β value of 1 corresponds to the vanilla VAE model.

Original: Increasing β enforces a stronger prior on the latent distribution and encourages the representation to be independent.
Revised: Increasing β enforces a stronger prior on the latent distribution and encourages the representation to be independent.

Original: CCI-VAE The CCI-VAE model (Burgess et al., 2017) is a variant of the β -VAE where the KL divergence is encouraged to match a controlled value C which is increased gradually throughout training.
Revised: CCI-VAE The CCI-VAE model (Burgess et al., 2017) is a variant of the β -VAE where the KL divergence is encouraged to match a controlled value C which is increased gradually throughout training.

Original: This yields us the objective function for CCI-VAE.
Revised: This yields us the objective function for CCI-VAE.

Original: [Equation]
Revised: [Equation]

Original: FactorVAE FactorVAE (Kim & Mnih, 2018) speciﬁcally penalises the dependencies between the latent dimensions such that the “Total Correlation” term is targeted yielding a modiﬁed version of the β -VAE objective.
Revised: FactorVAE FactorVAE (Kim & Mnih, 2018) speciﬁcally penalises the dependencies between the latent dimensions such that the “Total Correlation” term is targeted yielding a modiﬁed version of the β -VAE objective.

Original: [Equation]
Revised: [Equation]

Original: The “Total Correlation” term is intractable in this case so for FactorVAE, samples are used from both q ( z | x ) and q ( z ) as well as the density-ratio trick to compute an estimate of the “Total Correlation” term.
Revised: The “Total Correlation” term is intractable in this case so for FactorVAE, samples are used from both q ( z | x ) and q ( z ) as well as the density-ratio trick to compute an estimate of the “Total Correlation” term.

Original: FactorVAE uses an additional discriminator network to approximate the density ratio in the KL divergence.
Revised: FactorVAE uses an additional discriminator network to approximate the density ratio in the KL divergence.

Original: The implementation details for the discriminator network and its hyperparameters can be found in Table 5(b) and 5(c).
Revised: The implementation details for the discriminator network and its hyperparameters can be found in Table 5(b) and 5(c).

Original: TC-VAE
Revised: TC-VAE

Original: The TC-VAE model (Chen et al., 2018) which independently from FactorVAE has a similar objective
Revised: The TC-VAE model (Chen et al., 2018) which independently from FactorVAE has a similar objective

Original: KL regulariser which contains a “Total Correlation” term.
Revised: KL regulariser which contains a “Total Correlation” term.

Original: In the case of TC-VAE the “Total Correlation” term is estimated using a biased Monte-Carlo estimate.
Revised: In the case of TC-VAE the “Total Correlation” term is estimated using a biased Monte-Carlo estimate.

Original: Create MxP similarity matrices, for each calculate UDR ij score
Revised: Create MxP similarity matrices, for each calculate UDR ij score

Original: UDR 1 = 0.
Revised: UDR 1 = 0.

Original: Aggregate UDR ij scores for each of M models
Revised: Aggregate UDR ij scores for each of M models

Original: UDR 2 = 0.
Revised: UDR 2 = 0.

Original: UDR M = 0.
Revised: UDR M = 0.

Original: Train M=HxS models
Revised: Train M=HxS models

Original: Hyperseed 1 seedseed S
Revised: Hyperseed 1 seedseed S

Original: Hyper
Revised: Hyper

Original: Hyper H (2)
Revised: Hyper H (2)

Original: Select P models for pairwise comparisons with each of M models 0.
Revised: Select P models for pairwise comparisons with each of M models 0.

Original: Model i=1 of M j =j =j = P seed 1 seedseed S seed 1 seedseed S
Revised: Model i=1 of M j =j =j = P seed 1 seedseed S seed 1 seedseed S

Original: DIP-VAE
Revised: DIP-VAE

Original: The DIP-VAE model also adds regularisation to the aggregated posterior but instead an additional loss term is added to encourage it to match the factorised prior.
Revised: The DIP-VAE model also adds regularisation to the aggregated posterior but instead an additional loss term is added to encourage it to match the factorised prior.

Original: Since the KL divergence is intractable, other measures of divergence are used instead.
Revised: Since the KL divergence is intractable, other measures of divergence are used instead.

Original: Cov p
Revised: Cov p

Original: ( x ) [ µ φ ( x )] can be used, yielding the DIP-VAE-I objective
Revised: ( x ) [ µ φ ( x )] can be used, yielding the DIP-VAE-I objective

Original: [Equation]
Revised: [Equation]

Original: or Cov q φ [ z ] is used instead yielding the DIP-VAE-II objective.
Revised: or Cov q φ [ z ] is used instead yielding the DIP-VAE-II objective.

Original: [Equation]
Revised: [Equation]

Original: A. UDR IMPLEMENTATION DETAILS
Revised: A. UDR IMPLEMENTATION DETAILS

Original: Similarity matrix To compute the similarity matrix R ij we follow the approach of Li et al.
Revised: Similarity matrix To compute the similarity matrix R ij we follow the approach of Li et al.

Original: (2016) and Morcos et al.
Revised: (2016) and Morcos et al.

Original: For a given dataset X = { x 1 , x 2 ,...,, x N } and a neuron a ∈ { 1 ,...,L } of model i (denoted as z i,a ), we deﬁne z
Revised: For a given dataset X = { x 1 , x 2 ,...,, x N } and a neuron a ∈ { 1 ,...,L } of model i (denoted as z i,a ), we deﬁne z

Original: i,a to be the vector of mean inferred posteriors q i
Revised: i,a to be the vector of mean inferred posteriors q i

Original: ( z i | x i ) across the full dataset: z i,a = ( z i,a ( x 1 ) ,...,z i,a ( x N ))
Revised: ( z i | x i ) across the full dataset: z i,a = ( z i,a ( x 1 ) ,...,z i,a ( x N ))

Original: ∈ R N .
Revised: ∈ R N .

Original: Note that this is different from the often considered notion of a “latent representation vector”.
Revised: Note that this is different from the often considered notion of a “latent representation vector”.

Original: Here z i,a is a response of a single latent dimension over the entire dataset, not an entire latent response for a single input.
Revised: Here z i,a is a response of a single latent dimension over the entire dataset, not an entire latent response for a single input.

Original: We then calculate the similarity between each two of such vectors
Revised: We then calculate the similarity between each two of such vectors

Original: z i,a and z j,b using either Lasso regression or Spearman’s correlation.
Revised: z i,a and z j,b using either Lasso regression or Spearman’s correlation.

Original: Lasso regression (UDR L )
Revised: Lasso regression (UDR L )

Original: We trained L lasso regressors to predict each of the latent responses z i,a from z j using the dataset of latent encodings Z i,a = { ( z j, 1 ,z i,a, 1 ) ,..., ( z j,N ,z i,a,N ) } .
Revised: We trained L lasso regressors to predict each of the latent responses z i,a from z j using the dataset of latent encodings Z i,a = { ( z j, 1 ,z i,a, 1 ) ,..., ( z j,N ,z i,a,N ) } .

Original: Each row in R ij ( a ) is then ﬁlled in using the weights of the trained Lasso regressor for z
Revised: Each row in R ij ( a ) is then ﬁlled in using the weights of the trained Lasso regressor for z

Original: i,a .
Revised: i,a .

Original: The lasso regressors were trained using the default
Revised: The lasso regressors were trained using the default

Original: Scikit-learn
Revised: Scikit-learn

Original: [Table]
Revised: [Table]

Original: w 1 2 n samples || XW − Y || 

Revised: w 1 2 n samples || XW − Y ||

Original: Fro + λ || W
Revised: Fro + λ || W

Original: || 21 where Fro is the Frobenius norm:
Revised: || 21 where Fro is the Frobenius norm:

Original: || A || Fro = (cid:113)(cid:80) ij a 2 ij and the l 1 l 2 loss is computed as || A || 

Revised: || A || Fro = (cid:113)(cid:80) ij a 2 ij and the l 1 l 2 loss is computed as || A ||

Original: = (cid:80) i
Revised: = (cid:80) i

Original: (cid:113)(cid:80)
Revised: (cid:113)(cid:80)

Original: j a 2 ij .
Revised: j a 2 ij .

Original: λ is chosen using cross validation and the lasso is trained until convergence until either 1000 iterations have been run or our updates are below a tolerance of 0.0001.
Revised: λ is chosen using cross validation and the lasso is trained until convergence until either 1000 iterations have been run or our updates are below a tolerance of 0.0001.

Original: Lasso regressors were trained on a dataset of 10000 latents from each model and training was performed using coordinate descent over the entire dataset.
Revised: Lasso regressors were trained on a dataset of 10000 latents from each model and training was performed using coordinate descent over the entire dataset.

Original: R nm is then computed by extracting the weights in the trained lasso regressor and computing their absolute value (Eastwood & Williams, 2018).
Revised: R nm is then computed by extracting the weights in the trained lasso regressor and computing their absolute value (Eastwood & Williams, 2018).

Original: It is important that the representations are normalised per-latent such that the relative importances computed per latent are scaled to reﬂect their contribution to the output.
Revised: It is important that the representations are normalised per-latent such that the relative importances computed per latent are scaled to reﬂect their contribution to the output.

Original: Normalising our latents also ensures that the weights that are computed roughly lie in the interval [ − 1 , 1] .
Revised: Normalising our latents also ensures that the weights that are computed roughly lie in the interval [ − 1 , 1] .

Original: Spearman’s based similarity matrix (UDR S )
Revised: Spearman’s based similarity matrix (UDR S )

Original: We calculate each entry in the similarity matrix according to R ij ( a,b )=
Revised: We calculate each entry in the similarity matrix according to R ij ( a,b )=

Original: Corr ( z i,a , z j,b ) , where Corr stands for Spearman’s correlation.
Revised: Corr ( z i,a , z j,b ) , where Corr stands for Spearman’s correlation.

Original: We use Spearman’s correlation to measure the similarity between z i,a and z j,b , because we do not want to necessarily assume a linear relationship between the two latent encodings, since the geometry of the representational space is not crucial for measuring whether a representation is disentangled (see Sec. 2), but we do hope to ﬁnd a monotonic dependence between them.
Revised: We use Spearman’s correlation to measure the similarity between z i,a and z j,b , because we do not want to necessarily assume a linear relationship between the two latent encodings, since the geometry of the representational space is not crucial for measuring whether a representation is disentangled (see Sec. 2), but we do hope to ﬁnd a monotonic dependence between them.

Original: Spearman correlation coefﬁcients were computed by extracting 1000 samples from each model and computing the Spearman correlation over all the samples on a per-latent basis.
Revised: Spearman correlation coefﬁcients were computed by extracting 1000 samples from each model and computing the Spearman correlation over all the samples on a per-latent basis.

Original: All-to-all calculations To make all-to-all comparisons, we picked 10 random seeds per hyperparameter setting and limited all the calculations to those models.
Revised: All-to-all calculations To make all-to-all comparisons, we picked 10 random seeds per hyperparameter setting and limited all the calculations to those models.

Original: Hence we made the maximum of (60 choose 2) pairwise model comparisons when calculating UDR-A2A.
Revised: Hence we made the maximum of (60 choose 2) pairwise model comparisons when calculating UDR-A2A.

Original: Informative latent thresholding Uninformative latents typically have KL (cid:28) 0 .
Revised: Informative latent thresholding Uninformative latents typically have KL (cid:28) 0 .

Original: 01 while informative latents haveKL (cid:29) 0 .
Revised: 01 while informative latents haveKL (cid:29) 0 .

Original: 01 ,soKL =0 .
Revised: 01 ,soKL =0 .

Original: 01 thresholdinEq.3issomewhatarbitrarilychosentopickouttheinformativelatents z .
Revised: 01 thresholdinEq.3issomewhatarbitrarilychosentopickouttheinformativelatents z .

Original: Sample reduction experiments We randomly sampled without replacement 20 different sets of P models for pairwise comparison from the original set of 50 models with the same hyperparameter setting for UDR or 60 models with different seeds and hyperparameters for UDR-A2A.
Revised: Sample reduction experiments We randomly sampled without replacement 20 different sets of P models for pairwise comparison from the original set of 50 models with the same hyperparameter setting for UDR or 60 models with different seeds and hyperparameters for UDR-A2A.

Original: A. S UPERVISED METRIC IMPLEMENTATION DETAILS
Revised: A. S UPERVISED METRIC IMPLEMENTATION DETAILS

Original: Original β -VAE metric.
Revised: Original β -VAE metric.

Original: First proposed in Higgins et al.
Revised: First proposed in Higgins et al.

Original: (2017a)
Revised: (2017a)

Original: , this metric suggests sampling two batches of observations x where in both batches the same single data generative factor is ﬁxed to a particular value, while the other factors are sampled randomly from the underlying distribution.
Revised: , this metric suggests sampling two batches of observations x where in both batches the same single data generative factor is ﬁxed to a particular value, while the other factors are sampled randomly from the underlying distribution.

Original: These two batches are encoded into the corresponding latent representations q φ ( z | x ) and the pairwise differences between the corresponding mean latent values from the two batches are taken.
Revised: These two batches are encoded into the corresponding latent representations q φ ( z | x ) and the pairwise differences between the corresponding mean latent values from the two batches are taken.

Original: Disentanglement is measured as the ability of a linear classiﬁer to predict the index of the data generative factor that was ﬁxed when generating x .
Revised: Disentanglement is measured as the ability of a linear classiﬁer to predict the index of the data generative factor that was ﬁxed when generating x .

Original: We compute the β -VAE score by ﬁrst randomly picking a single factor of variation and ﬁxing the value of that factor to a randomly sampled value.
Revised: We compute the β -VAE score by ﬁrst randomly picking a single factor of variation and ﬁxing the value of that factor to a randomly sampled value.

Original: We then generate two batches of 64 where all the other factors are sampled randomly and take the mean of the differences between the latent mean responses in the two batches to generate a training point.
Revised: We then generate two batches of 64 where all the other factors are sampled randomly and take the mean of the differences between the latent mean responses in the two batches to generate a training point.

Original: This process is repeated 10000 times to generate a training set by using the ﬁxed factor of variation as the label.
Revised: This process is repeated 10000 times to generate a training set by using the ﬁxed factor of variation as the label.

Original: We then train a logistic regression on the data using Scikit-learn and report the evaluation accuracy on a test set of 5000 as the disentanglement score.
Revised: We then train a logistic regression on the data using Scikit-learn and report the evaluation accuracy on a test set of 5000 as the disentanglement score.

Original: FactorVAE metric.
Revised: FactorVAE metric.

Original: Kim & Mnih (2018) proposed a modiﬁcation on the β -VAE metric which made the classiﬁer non-parametric (majority vote based on the index of the latent dimension with the least variance after the pairwise difference step).
Revised: Kim & Mnih (2018) proposed a modiﬁcation on the β -VAE metric which made the classiﬁer non-parametric (majority vote based on the index of the latent dimension with the least variance after the pairwise difference step).

Original: This made the FactorVAE metric more robust, since the classiﬁer did not need to be optimised.
Revised: This made the FactorVAE metric more robust, since the classiﬁer did not need to be optimised.

Original: Furthermore, the FactorVAE metric is more accurate than the β -VAE one, since the β -VAE metric often over-estimates the level of disentanglement by reporting 100% disentanglement even when only K − 1 factors were disentangled.
Revised: Furthermore, the FactorVAE metric is more accurate than the β -VAE one, since the β -VAE metric often over-estimates the level of disentanglement by reporting 100% disentanglement even when only K − 1 factors were disentangled.

Original: The Factor VAE score is computed similarly to the β -VAE metric but with a few modiﬁcations.
Revised: The Factor VAE score is computed similarly to the β -VAE metric but with a few modiﬁcations.

Original: First we draw a set of 10000 random samples from the dataset and we estimate the variance of the mean latent responses in the model.
Revised: First we draw a set of 10000 random samples from the dataset and we estimate the variance of the mean latent responses in the model.

Original: Latents with a variance of less than 0.05 are discarded.
Revised: Latents with a variance of less than 0.05 are discarded.

Original: Then batches of 64 samples are generated by a random set of generative factors with a single ﬁxed generative factor.
Revised: Then batches of 64 samples are generated by a random set of generative factors with a single ﬁxed generative factor.

Original: The variances of all the latent responses over the 64 samples are computed and divided by the latent variance computed in the ﬁrst step.
Revised: The variances of all the latent responses over the 64 samples are computed and divided by the latent variance computed in the ﬁrst step.

Original: The variances are averaged to generate a single training point using the ﬁxed factor of variation as the label.
Revised: The variances are averaged to generate a single training point using the ﬁxed factor of variation as the label.

Original: 10000 such training points are generated as the training set.
Revised: 10000 such training points are generated as the training set.

Original: A majority vote classiﬁer is trained to pick out the ﬁxed generative factor and the evaluation accuracy is computed on test set of 5000 and reported as the disentanglement score.
Revised: A majority vote classiﬁer is trained to pick out the ﬁxed generative factor and the evaluation accuracy is computed on test set of 5000 and reported as the disentanglement score.

Original: VAE
Revised: VAE

Original: VAE
Revised: VAE

Original: [Figure]
Revised: [Figure]

Original: Trained model
Revised: Trained model

Original: [Figure]
Revised: [Figure]

Original: Spearman
Revised: Spearman

Original: Lasso
Revised: Lasso

Original: Entangled
Revised: Entangled

Original: Disentangled
Revised: Disentangled

Original: [Figure]
Revised: [Figure]

Original: the pairwise model comparison.
Revised: the pairwise model comparison.

Original: Two trained models i and j are sampled for pairwise comparison.
Revised: Two trained models i and j are sampled for pairwise comparison.

Original: Both models learnt a perfectly disentangled representation, learning to represent two (positions x/y) and three (positions x/y, and size) generative factors respectively.
Revised: Both models learnt a perfectly disentangled representation, learning to represent two (positions x/y) and three (positions x/y, and size) generative factors respectively.

Original: The latter is often cleaner
Revised: The latter is often cleaner

Original: Mutual Information Gap (MIG).
Revised: Mutual Information Gap (MIG).

Original: The MIG metric proposed in Chen et al.
Revised: The MIG metric proposed in Chen et al.

Original: (2018) proposes estimating the mutual information (MI) between each data generative factor and each latent dimension.
Revised: (2018) proposes estimating the mutual information (MI) between each data generative factor and each latent dimension.

Original: For each factor, they consider two latent dimensions with the highest MI scores.
Revised: For each factor, they consider two latent dimensions with the highest MI scores.

Original: It is assumed that in a disentangled representation only one latent dimension will have high MI with a single data generative factor, and hence the difference between these two MI scores will be large.
Revised: It is assumed that in a disentangled representation only one latent dimension will have high MI with a single data generative factor, and hence the difference between these two MI scores will be large.

Original: Hence, the MIG score is calculated as the average normalised difference between such pairs of MI scores per each data generative factor.
Revised: Hence, the MIG score is calculated as the average normalised difference between such pairs of MI scores per each data generative factor.

Original: Chen et al.
Revised: Chen et al.

Original: (2018) suggest that the MIG score is more general and unbiased than the β -VAE and FactorVAE metrics.
Revised: (2018) suggest that the MIG score is more general and unbiased than the β -VAE and FactorVAE metrics.

Original: We compute the Mutual Information Gap by taking the discretising the mean representation of 10000 samples into 20 bins.
Revised: We compute the Mutual Information Gap by taking the discretising the mean representation of 10000 samples into 20 bins.

Original: The disentanglement score is then derived by computing, per generative factor, the difference between the top two latents with the greatest mutual information with the generative factor and taking the mean.
Revised: The disentanglement score is then derived by computing, per generative factor, the difference between the top two latents with the greatest mutual information with the generative factor and taking the mean.

Original: [Table]
Revised: [Table]

Original: where K is the number of generative factors, from which v k is a single generative factor z j is the mean representation and j ( k ) =
Revised: where K is the number of generative factors, from which v k is a single generative factor z j is the mean representation and j ( k ) =

Original: argmax j I n ( z j ; v k ) is the latent representation with the greatest mutual information with the generative factor.
Revised: argmax j I n ( z j ; v k ) is the latent representation with the greatest mutual information with the generative factor.

Original: H v k is the computed entropy of the generative factor.
Revised: H v k is the computed entropy of the generative factor.

Original: DCI Disentanglement.
Revised: DCI Disentanglement.

Original: This is the disentanglement part of the three-part metric proposed by Eastwood &
Revised: This is the disentanglement part of the three-part metric proposed by Eastwood &

Original: Williams (2018).
Revised: Williams (2018).

Original: The DCI disentanglement metric is somewhat similar to our unsupervised metric, whereby the authors train a random forest classiﬁer to predict the ground truth factors from the corresponding latent encodings q ( z | x ) .
Revised: The DCI disentanglement metric is somewhat similar to our unsupervised metric, whereby the authors train a random forest classiﬁer to predict the ground truth factors from the corresponding latent encodings q ( z | x ) .

Original: They then use the resulting M × N matrix of feature importance weights to calculate the difference between the entropy of the probability that a latent dimension is important for predicting a particular ground truth factor weighted by the relative importance of each dimension.
Revised: They then use the resulting M × N matrix of feature importance weights to calculate the difference between the entropy of the probability that a latent dimension is important for predicting a particular ground truth factor weighted by the relative importance of each dimension.

Original: The DCI disentanglement metric is an implementation of the disentanglement metric as described in Eastwood & Williams (2018) using a gradient boosted tree.
Revised: The DCI disentanglement metric is an implementation of the disentanglement metric as described in Eastwood & Williams (2018) using a gradient boosted tree.

Original: It was computed by ﬁrst extracting the relative importance of each latent mean representation as a predictor for each generative factor by training a gradient boosted tree using the de- fault Scikit-learn model on 10000 training and 1000 test points and extracting the importance weights.
Revised: It was computed by ﬁrst extracting the relative importance of each latent mean representation as a predictor for each generative factor by training a gradient boosted tree using the de- fault Scikit-learn model on 10000 training and 1000 test points and extracting the importance weights.

Original: The weights are summarised into an importance matrix R ij with the number of rows equal to
Revised: The weights are summarised into an importance matrix R ij with the number of rows equal to

Original: A. A DDITIONAL RESULTS
Revised: A. A DDITIONAL RESULTS

Original: We evaluated four UDR versions, which differed in terms of whether Spearman- and Lasso-based similarity matrices R ij were used (subscripts S and L respectively), and whether the models for pairwise similarity comparison are picked from the pool of different seeds trained with the same hyperparameters or from the pool of all models (the latter indicated by the A2A sufﬁx).
Revised: We evaluated four UDR versions, which differed in terms of whether Spearman- and Lasso-based similarity matrices R ij were used (subscripts S and L respectively), and whether the models for pairwise similarity comparison are picked from the pool of different seeds trained with the same hyperparameters or from the pool of all models (the latter indicated by the A2A sufﬁx).

Original: The A2A correlations in Tbl.
Revised: The A2A correlations in Tbl.

Original: 7 are on average slightly higher, however these scores are more computationally expensive to compute due to the higher number of total pairwise similarity calculations.
Revised: 7 are on average slightly higher, however these scores are more computationally expensive to compute due to the higher number of total pairwise similarity calculations.

Original: For that reason, the scores presented in the table are calculated using only 20% of all the trained models.
Revised: For that reason, the scores presented in the table are calculated using only 20% of all the trained models.

Original: Hence, the results presented in the main text of the paper are computed using the UDR L score, which allowed us to evaluate all 5400 models and performed slightly better than the UDR S score.
Revised: Hence, the results presented in the main text of the paper are computed using the UDR L score, which allowed us to evaluate all 5400 models and performed slightly better than the UDR S score.

Original: Figs.
Revised: Figs.

Original: 6-8 provide more details on the performance of the different UDR versions.
Revised: 6-8 provide more details on the performance of the different UDR versions.

Original: To qualitatively validate that the UDR method is ranking models well, we look into more detail into the β -VAE model ranking when evaluated with the DCI disentanglement metric on the dSprites dataset.
Revised: To qualitatively validate that the UDR method is ranking models well, we look into more detail into the β -VAE model ranking when evaluated with the DCI disentanglement metric on the dSprites dataset.

Original: This scenario resulted in the worst disagreement between UDR and the supervised metric as shown in Fig.
Revised: This scenario resulted in the worst disagreement between UDR and the supervised metric as shown in Fig.

Original: We consider the UDR L version of our method, since it appears to give the best trade off between overall correlations with the supervised metrics and hyperparameter selection accuracy.
Revised: We consider the UDR L version of our method, since it appears to give the best trade off between overall correlations with the supervised metrics and hyperparameter selection accuracy.

Original: Fig.
Revised: Fig.

Original: 9 demonstrates that the poor correlation between UDR L and DCI Disentanglement is due to the supervised metric.
Revised: 9 demonstrates that the poor correlation between UDR L and DCI Disentanglement is due to the supervised metric.

Original: Models ranked highly by UDR L but poorly by DCI
Revised: Models ranked highly by UDR L but poorly by DCI

Original: Disentanglement appear to be qualitatively disentangled through visual inspection of latent traversals.
Revised: Disentanglement appear to be qualitatively disentangled through visual inspection of latent traversals.

Original: Conversely, models scored highly by DCI Disentanglement but poorly by UDR L appear entangled.
Revised: Conversely, models scored highly by DCI Disentanglement but poorly by UDR L appear entangled.

Original: A. UDR CORRELATION WITH FINAL TASK PERFORMANCE
Revised: A. UDR CORRELATION WITH FINAL TASK PERFORMANCE

Original: To illustrate the usefulness of UDR to select disentangled models, we ran two experiments.
Revised: To illustrate the usefulness of UDR to select disentangled models, we ran two experiments.

Original: We computed the UDR correlation with fairness scores and with data efﬁciency on a model-based RL task.
Revised: We computed the UDR correlation with fairness scores and with data efﬁciency on a model-based RL task.

Original: Fairness scores.
Revised: Fairness scores.

Original: Fig.
Revised: Fig.

Original: 11 (left) demonstrates that UDR correlates well with the classiﬁcation fairness scores introducedbyLocatelloetal.(2019).
Revised: 11 (left) demonstrates that UDR correlates well with the classiﬁcation fairness scores introducedbyLocatelloetal.(2019).

Original: WeadoptedasimilarsetupdescribedinLocatelloetal.(2019)tocomputefair- ness, using a gradient booting classiﬁer over 10000 labelled examples.
Revised: WeadoptedasimilarsetupdescribedinLocatelloetal.(2019)tocomputefair- ness, using a gradient booting classiﬁer over 10000 labelled examples.

Original: The fairness score was computed by taking
Revised: The fairness score was computed by taking

Original: [Figure]
Revised: [Figure]

Original: the mean of the fairness scores across all targets and all sensitive variables where the fairness scores are computed by measuring the total variation after intervening on the sensitive variable.
Revised: the mean of the fairness scores across all targets and all sensitive variables where the fairness scores are computed by measuring the total variation after intervening on the sensitive variable.

Original: The fairness scores were compared against the Lasso regression version of UDR where models were paired only within the same hyperparameters.
Revised: The fairness scores were compared against the Lasso regression version of UDR where models were paired only within the same hyperparameters.

Original: Model-based RL data efﬁciency.
Revised: Model-based RL data efﬁciency.

Original: We reproduced the results from the COBRA agent (Watters et al., 2019), toobserveifUDRwouldcorrelatewiththeﬁnaltasksperformancewhenusingVAEsasstaterepresentations.
Revised: We reproduced the results from the COBRA agent (Watters et al., 2019), toobserveifUDRwouldcorrelatewiththeﬁnaltasksperformancewhenusingVAEsasstaterepresentations.

Original: More precisely, we will look at the training data efﬁciency, reported as the number of steps needed to achieve 90% performance on the Clustering tasks (see Watters et al.
Revised: More precisely, we will look at the training data efﬁciency, reported as the number of steps needed to achieve 90% performance on the Clustering tasks (see Watters et al.

Original: (2019) for details), while using differently disentangled models.
Revised: (2019) for details), while using differently disentangled models.

Original: The agent is provided with a pre-trained MONet (Burgess et al., 2019), an exploration policy and a transition model and has to learn a good reward predictor for the task in a dense reward setting.
Revised: The agent is provided with a pre-trained MONet (Burgess et al., 2019), an exploration policy and a transition model and has to learn a good reward predictor for the task in a dense reward setting.

Original: It uses Model Predictive
Revised: It uses Model Predictive

Original: Control in order to plan and solve the task, where sprites have to be clustered by color (e.g. two blue sprites and two red sprites).
Revised: Control in order to plan and solve the task, where sprites have to be clustered by color (e.g. two blue sprites and two red sprites).

Original: In COBRA, the authors use a MONet with disentangled representation by using a high β =1 .
Revised: In COBRA, the authors use a MONet with disentangled representation by using a high β =1 .

Original: When pre-training MONet, we used β ∈{ 0 . 01 , 0 . 1 , 1 } in order to introduce entanglement in the representations without compromising reconstruction accuracy and pre-trained 10 seeds for each value of β .
Revised: When pre-training MONet, we used β ∈{ 0 . 01 , 0 . 1 , 1 } in order to introduce entanglement in the representations without compromising reconstruction accuracy and pre-trained 10 seeds for each value of β .

Original: We use 5 random initialisations of the reward predictor for each possible MONet model, and train them to perform the clustering task as explained in Watters et al.
Revised: We use 5 random initialisations of the reward predictor for each possible MONet model, and train them to perform the clustering task as explained in Watters et al.

Original: We report the number of steps to reach 90% success, averaged across the initialisations.
Revised: We report the number of steps to reach 90% success, averaged across the initialisations.

Original: The UDR score is computed by feeding images with a single sprite to obtain an associated unique representation and proceeding as described in the 
Revised: The UDR score is computed by feeding images with a single sprite to obtain an associated unique representation and proceeding as described in the main text.

Original: As can be seen in Figure 11 (right), we ﬁnd that the UDR scores correlate with this ﬁnal data efﬁciency (linear regression shown, Spearman correlation ρ =0 . 56 ).
Revised: As can be seen in Figure 11 (right), we ﬁnd that the UDR scores correlate with this ﬁnal data efﬁciency (linear regression shown, Spearman correlation ρ =0 . 56 ).

Original: This indicates that one could leverage the UDR score as a metric to select representations for further tasks.
Revised: This indicates that one could leverage the UDR score as a metric to select representations for further tasks.

Original: In this analysis we used the version of UDR that uses Spearman correlations and within-hyperparameter model comparisons.
Revised: In this analysis we used the version of UDR that uses Spearman correlations and within-hyperparameter model comparisons.

Revised: A. E VALUATING UDR ON MORE COMPLEX DATASETS

Revised: We evaluated whether UDR is useful for model selection on more complex datasets.

Revised: In particular, we chose CelebA and ImageNet.

Original: dSprites
Revised: While disentangling VAEs have been shown to perform well on CelebA in the past (e.g. Higgins et al. (2018b)) , ImageNet is notoriously too complex for even vanilla VAEs to model. However, we still wanted to verify whether the coarse representations of VAEs on ImageNet could be disentangled, and if so, whether UDR would be useful for model selection.

Revised: To this end, we ran a hyperparameter sweep for the β -VAE and ranked its representations using UDR.

Revised: Fig.

Revised: 12 shows that UDR scores are clearly different for the different values of the β hyperparameter.

Revised: It is also clear that the models were able to learn about CelebA and produce reasonable reconstructions, but on ImageNet even the vanilla VAEs struggled to represent anything but the coarsest information.

Revised: Figs.

Revised: 13-14 plot latent traversals for three randomly chosen models with high (>0.6) and low (<0.3)

Revised: UDR scores.

Revised: The latents are sorted by their informativeness, as approximated by their batch-wise per dimension KL with the prior as per Eq.

Revised: It is clear that for both datasets those models that are ranked high by the UDR have both more interpretable and more similar representations than those models that are ranked low.

Revised: dSprites

Original: Shapes 3D
Revised: Shapes 3D

Original: S c
Revised: S c

Original: o r e β -VAE
Revised: o r e β -VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Original: DIP-VAE
Revised: DIP-VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Original: TC-VAE
Revised: TC-VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Original: S c
Revised: S c

Original: o r e β -VAE
Revised: o r e β -VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Original: DIP-VAE
Revised: DIP-VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Original: TC-VAE
Revised: TC-VAE

Original: U D R L
Revised: U D R L

Original: U D R S U D R L - A 2 A U D
Revised: U D R S U D R L - A 2 A U D

Original: R S - A 2 A
Revised: R S - A 2 A

Revised: A. Q UALITATIVE EVALUATION OF MODEL REPRESENTATIONS RANKED BY UDR SCORES

Revised: In this section we attempt to qualitatively verify our assumption that “for a particular dataset and a VAE-based un- supervised disentangled representation learning model class, disentangled representations are all alike, while every entangled representation is entangled in its own way, to rephrase Tolstoy”.

Revised: The theoretical justiﬁcation of the pro- posed UDR hinges on the work by Rolinek et al.

Revised: However, that work only empirically evaluated their analysis on the β -VAE model class.

Original: [Figure]
Revised: Even though we have reasons to believe that their theoretical results would also hold for the other disentangling VAEs evaluated in this paper, in this section we empirically evaluate whether this is true. First, we check if all model classes operate in the so called “polarised regime”, which was highlighted by Rolinek et al. as being important for pushing VAEs towards disentanglement. It is known that even vanilla VAEs (Kingma & Welling, 2014; Rezende et al., 2014) enter the “polarised regime”, which is often cited as one of their shortcomings (e.g. see Rezende & Viola (2018)).

Revised: All of the disentangling VAEs considered in this paper augment the original ELBO objective with extra terms.

Revised: None of these extra terms penalise entering the “polarised regime”, apart from that of DIP-VAE-I. We tested empirically whether different model classes entered the “polarised regime” during our hyperparameter sweeps.

Revised: We did this by counting the number of latents that were “switched off” in each of the 5400 models considered in our paper by using Eq. 3.

Revised: We found that all models apart from

Revised: DIP-VAE-I entered the polarised regime during the hyperparameter sweep, having on average 2.95/10 latents “switched off” (with a standard deviation of 1.97).

Revised: Second, we check if the models scored highly by the UDR do indeed have similar representations, and models that are scored low have dissimilar representations.

Revised: We do this qualitatively by plotting latent traversals for three randomly chosen models within each of the six disentangling VAE model classes considered in this paper.

Revised: Lasso

Revised: Spearman

Original: H y p
Revised: H y p

Original: e r
Revised: e r

Original: A ll - t o - a ll
Revised: A ll - t o - a ll

Original: [Figure]
Revised: [Figure]

Revised: We groups these plots by UDR scores into three bands: high (UDR>0.4), medium (0.3<UDR<0.4) and low (UDR<0.3).

Revised: Fig.

Revised: 15 shows latent traversals for all model classes that were able to achieve the high range of

Revised: UDR scores (note that some model classes were not able to achieve high UDR values with the hyperparameter settings evaluated in this paper).

Revised: We present the latent traversals for all ten latents per model without sorting them in any particular way.

Revised: We also colour code the latents by their semantic meaning (if the meaning is apparent from the latent traversal).

Revised: Fig.

Revised: 15 shows that the representations learnt by the highly ranked models all appear to be very similar (up to subsetting, sign inverse and permutation).

Revised: Note that the models also include many latents that are “switched off”.

Revised: Fig.

Revised: 16 shows latent traversals for two model classes that did not achieve high UDR scores.

Revised: We see that these models now have fewer semantically meaningful latent dimensions, and fewer latents that are uninformative or “switched off”.

Revised: Finally, Fig.

Revised: 17 shows latent traversals for all model classes that had models which scored low on UDR.

Revised: We see that many of these models do not have any uninformative latents, and their representations are hard to interpret.

Revised: Furthermore, it is hard to ﬁnd similarity between the representations learnt by the different models.

Revised: Together, Figs.

Revised: 15-17 empirically verify that our assumption holds for the model classes considered in this paper.

Revised: However, we recommend that any practitioner using UDR on new disentangling model classes developed in the future ﬁrst verify that the assumptions of the UDR hold for those models.

Revised: We suggest training the new models on a number of toy but well studied datasets (like dSprites or Shapes3D) and checking if the ranks produced by UDR correlate with those produced by the supervised metrics.

Revised: Furthermore, we suggest a qualitative evaluation of the traversals plots for the high and low scoring models.

Original: [Figure]
Revised: [Figure]

Original: by informative latents #
Revised: by informative latents #

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

Original: -  zz-  -
Revised: -  zz-  -

Original: [Figure]
Revised: [Figure]

Original: UDR = 0.
Revised: UDR = 0.

Original: d =
Revised: d =

Original: UDR = 0.
Revised: UDR = 0.

Original: d =
Revised: d =

Original: UDR = 0.
Revised: UDR = 0.

Original: d =
Revised: d =

Original: UDR = 0.
Revised: UDR = 0.

Original: d =zzUDR = 0. 
Revised: d =zzUDR = 0. d =zzUDR = 0. d =zzzzzzzz-

Original: d =zzUDR = 0.
Revised: [Figure] La t en t Traversal La t en t

Original: d =zzzzzzzz-
Revised: Traversal [Figure] UDR [Figure]

Revised: La t en t

Revised: Traversal

Revised: La t en t

Revised: Traversal

Revised: [Figure]

Revised: [Figure]

Revised: β-VAE

Revised: TC-VAE

Revised: [Figure]

Revised: UDR > 0.

Revised: [Figure]

Revised: T r a v e r s a l

Revised: [Figure]

Revised: T r a v e r s a l

Revised: Factor VAE

Revised: DIP-VAE-II

Revised: [Figure]

Revised: 0.3 < UDR < 0.

Revised: Latent

Revised: T r a v e r s a l

Revised: Latent

Revised: T r a v e r s a l

Revised: Latent

Revised: T r a v e r s a l β-VAE

Revised: TC-VAE

Revised: Factor VAE

Revised: Latent

Revised: T r a v e r s a l

Revised: DIP-VAE-

Revised: I

Revised: Latent

Revised: T r a v e r s a l

Revised: DIP-VAE-II

Revised: Position x Position y

Revised: Size Rotation

Revised: Entangled Uninformative

Revised: UDR < 0.