Original: L EARNING L IKELIHOODS WITH C ONDITIONAL N ORMALIZING F LOWS
Revised: L EARNING L IKELIHOODS WITH C ONDITIONAL N ORMALIZING F LOWS

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: Normalizing Flows (NFs) are able to model complicated distributions p Y ( y ) with strong inter-dimensional correlations and high multimodality by transforming a simple base density p Z ( z ) through an invertible neural network under the change of variables formula.
Revised: Normalizing Flows (NFs) are able to model complicated distributions p Y ( y ) with strong inter-dimensional correlations and high multimodality by transforming a simple base density p Z ( z ) through an invertible neural network under the change of variables formula.

Original: Such behavior is desirable in multivariate structured prediction tasks, where handcrafted per-pixel loss-based methods inadequately capture strong correlations between output dimensions.
Revised: Such behavior is desirable in multivariate structured prediction tasks, where handcrafted per-pixel loss-based methods inadequately capture strong correlations between output dimensions.

Original: We present a study of conditional normalizing ﬂows (CNFs), a class of NFs where the base density to output space mapping is conditioned on an input x, to model conditional densities p Y | X ( y | x ) .
Revised: We present a study of conditional normalizing ﬂows (CNFs), a class of NFs where the base density to output space mapping is conditioned on an input x, to model conditional densities p Y | X ( y | x ) .

Original: CNFs are efﬁcient in sampling and inference, they can be trained with a likelihood-based objective and CNFs, being generative ﬂows, do not suffer from mode collapse or training instabilities.
Revised: CNFs are efﬁcient in sampling and inference, they can be trained with a likelihood-based objective and CNFs, being generative ﬂows, do not suffer from mode collapse or training instabilities.

Original: We provide an effective method to train continuous CNFs for binary problems and in particular, we apply these CNFs to super-resolution and vessel segmentation tasks demonstrating competitive performance on standard benchmark datasets in terms of likelihood and conventional metrics.
Revised: We provide an effective method to train continuous CNFs for binary problems and in particular, we apply these CNFs to super-resolution and vessel segmentation tasks demonstrating competitive performance on standard benchmark datasets in terms of likelihood and conventional metrics.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Learning conditional distributions p Y | X ( y | x ) is one of the oldest problems in machine learning.
Revised: Learning conditional distributions p Y | X ( y | x ) is one of the oldest problems in machine learning.

Original: When the output y is high-dimensional this is a particularly challenging task, and the practitioner is left with many design choices.
Revised: When the output y is high-dimensional this is a particularly challenging task, and the practitioner is left with many design choices.

Original: Do we factorize the conditional?
Revised: Do we factorize the conditional?

Original: If not, do we model correlations with, say, a conditional random ﬁeld (Prince, 2012)?
Revised: If not, do we model correlations with, say, a conditional random ﬁeld (Prince, 2012)?

Original: Do we use a unimodal distribution?
Revised: Do we use a unimodal distribution?

Original: How fat should the tails be?
Revised: How fat should the tails be?

Original: Do we use an explicit likelihood at all, or use implicit methods (Mohamed &
Revised: Do we use an explicit likelihood at all, or use implicit methods (Mohamed &

Original: Rezende, 2015) such as a GAN (Goodfellow et al., 2014)?
Revised: Rezende, 2015) such as a GAN (Goodfellow et al., 2014)?

Original: Do we quantize the output?
Revised: Do we quantize the output?

Original: Ideally, the practitioner should not have to make design choices at all, and the distribution should be learned from the data.
Revised: Ideally, the practitioner should not have to make design choices at all, and the distribution should be learned from the data.

Original: In the ﬁeld of density estimation normalizing ﬂows (NFs) are a relatively new family of models (Rezende & Mohamed, 2015).
Revised: In the ﬁeld of density estimation normalizing ﬂows (NFs) are a relatively new family of models (Rezende & Mohamed, 2015).

Original: NFs model complicated high dimensional marginal distributions p Y
Revised: NFs model complicated high dimensional marginal distributions p Y

Original: ( y ) by transforming a simple base distribution or prior p Z ( z ) through a learnable, invertible mapping f φ
Revised: ( y ) by transforming a simple base distribution or prior p Z ( z ) through a learnable, invertible mapping f φ

Original: and then applying the change of variables formula.
Revised: and then applying the change of variables formula.

Original: NFs are efﬁcient in inference and sampling, are able to learn inter-dimensional correlations and multi-modality, and they are exact likelihood models, amenable to gradient-based optimization.
Revised: NFs are efﬁcient in inference and sampling, are able to learn inter-dimensional correlations and multi-modality, and they are exact likelihood models, amenable to gradient-based optimization.

Original: Flow-based generative models (Dinh et al., 2016) are generally trained on the image space, and are in some cases computationally efﬁcient in both the forward and inverse direction.
Revised: Flow-based generative models (Dinh et al., 2016) are generally trained on the image space, and are in some cases computationally efﬁcient in both the forward and inverse direction.

Original: These are advantageous over other likelihood based methods because i) sampling is efﬁcient opposed to autoregressive models (Van Oord et al., 2016), and ii) ﬂows admit exact likelihood optimization in contrast with variational autoencoders (Kingma & Welling, 2014).
Revised: These are advantageous over other likelihood based methods because i) sampling is efﬁcient opposed to autoregressive models (Van Oord et al., 2016), and ii) ﬂows admit exact likelihood optimization in contrast with variational autoencoders (Kingma & Welling, 2014).

Original: Conditional random ﬁelds directly model correlations between pixels, and have been fused with deep learning (Chen et al., 2016).
Revised: Conditional random ﬁelds directly model correlations between pixels, and have been fused with deep learning (Chen et al., 2016).

Original: However, they require the practitioner to choose which pixels have pairwise interactions.
Revised: However, they require the practitioner to choose which pixels have pairwise interactions.

Original: Another approach uses adversarial training (Goodfellow et al., 2014).
Revised: Another approach uses adversarial training (Goodfellow et al., 2014).

Original: A downside is that the training procedure can be unstable and they are difﬁcult to evaluate quantitatively.
Revised: A downside is that the training procedure can be unstable and they are difﬁcult to evaluate quantitatively.

Original: We propose to learn the likelihood of conditional distributions with few modeling choices using Conditional Normalizing Flows (CNFs).
Revised: We propose to learn the likelihood of conditional distributions with few modeling choices using Conditional Normalizing Flows (CNFs).

Original: CNFs can be harnessed for conditional distributions p Y | X ( y | x ) by conditioning the prior and the invertible mapping on the input x .
Revised: CNFs can be harnessed for conditional distributions p Y | X ( y | x ) by conditioning the prior and the invertible mapping on the input x .

Original: In particular, we apply conditional ﬂows to super-resolution (Wang et al., 2018) and vessel segmentation (Staal et al., 2004).
Revised: In particular, we apply conditional ﬂows to super-resolution (Wang et al., 2018) and vessel segmentation (Staal et al., 2004).

Original: We evaluate their performance gains on multivariate prediction tasks along side architecturally-matched factored baselines by comparing likelihood and application speciﬁc evaluation metrics.
Revised: We evaluate their performance gains on multivariate prediction tasks along side architecturally-matched factored baselines by comparing likelihood and application speciﬁc evaluation metrics.

Original:  B ACKGROUND
Revised:  B ACKGROUND

Original: In the following, we present the relevant background material on normalizing ﬂows and structured prediction.
Revised: In the following, we present the relevant background material on normalizing ﬂows and structured prediction.

Original: This section covers the change of variables formula, invertible modules, variational dequantization and conventional likelihood optimization.
Revised: This section covers the change of variables formula, invertible modules, variational dequantization and conventional likelihood optimization.

Original: 2. N ORMALIZING F LOWS
Revised: 2. N ORMALIZING F LOWS

Original: A standard NF in continuous space is based on a simple change of variables formula.
Revised: A standard NF in continuous space is based on a simple change of variables formula.

Original: Given two spaces of equal dimension Z and Y ; a once-differentiable, parametric, bijective 1 mapping f φ : Y →
Revised: Given two spaces of equal dimension Z and Y ; a once-differentiable, parametric, bijective 1 mapping f φ : Y →

Original: Z , where φ are the parameters of f ; and a prior distribution p Z ( z ) , we can model a complicated distribution p Y ( y ) as
Revised: Z , where φ are the parameters of f ; and a prior distribution p Z ( z ) , we can model a complicated distribution p Y ( y ) as

Original: [Equation]
Revised: [Equation]

Original: The term | ∂f φ ( y ) /∂
Revised: The term | ∂f φ ( y ) /∂

Original: y | is the Jacobian determinant of f φ , evaluated at y
Revised: y | is the Jacobian determinant of f φ , evaluated at y

Original: and it accounts for volume changes induced by f φ .
Revised: and it accounts for volume changes induced by f φ .

Original: The transformation f φ introduces correlations and multi-modality in p Y .
Revised: The transformation f φ introduces correlations and multi-modality in p Y .

Original: The main challenge in the ﬁeld of normalizing ﬂows is designing the transformation f φ .
Revised: The main challenge in the ﬁeld of normalizing ﬂows is designing the transformation f φ .

Original: It has to be i) bijective, ii) have an efﬁcient and tractable Jacobian determinant, iii) be from a ‘ﬂexible’ model class.
Revised: It has to be i) bijective, ii) have an efﬁcient and tractable Jacobian determinant, iii) be from a ‘ﬂexible’ model class.

Original: In addition, iv) for fast sampling the inverse needs to be efﬁciently computable.
Revised: In addition, iv) for fast sampling the inverse needs to be efﬁciently computable.

Original: Below we brieﬂy state which invertible modules are used in our architectures, obeying the aforementioned points.
Revised: Below we brieﬂy state which invertible modules are used in our architectures, obeying the aforementioned points.

Original: Coupling layers Afﬁne coupling layers (Dinh et al., 2016) are invertible, nonlinear layers.
Revised: Coupling layers Afﬁne coupling layers (Dinh et al., 2016) are invertible, nonlinear layers.

Original: They work by splitting the input z into two components z 0 and z 1 and nonlinearly transforming z 0 as a function of z 1 , before reconcatenating the result.
Revised: They work by splitting the input z into two components z 0 and z 1 and nonlinearly transforming z 0 as a function of z 1 , before reconcatenating the result.

Original: If z = [ z 0 , z 1 ] and y =
Revised: If z = [ z 0 , z 1 ] and y =

Original: [ y 0 , y 1 ] this is
Revised: [ y 0 , y 1 ] this is

Original: [Equation]
Revised: [Equation]

Original: Where the scale s ( · ) and translation t ( · ) functions can be any function, typically implemented with a CNN.
Revised: Where the scale s ( · ) and translation t ( · ) functions can be any function, typically implemented with a CNN.

Original: Similar conditioning with normalizing ﬂows has been done in previous works by Mohamed & Rezende (2015) and Kingma et al.
Revised: Similar conditioning with normalizing ﬂows has been done in previous works by Mohamed & Rezende (2015) and Kingma et al.

Original: (2016b).
Revised: (2016b).

Original: Invertible 1 x 1 Convolutions Proposed in Kingma & Dhariwal (2018), invertible 1 × 1 convolutions help mix information across channel dimensions.
Revised: Invertible 1 x 1 Convolutions Proposed in Kingma & Dhariwal (2018), invertible 1 × 1 convolutions help mix information across channel dimensions.

Original: We implement them as regular 1 ×convolutions and for the inverse, we convolve with the inverse of the kernel.
Revised: We implement them as regular 1 ×convolutions and for the inverse, we convolve with the inverse of the kernel.

Original: Squeeze layers Squeeze layers (Dinh et al., 2016) are used to compress the spatial resolution of activations.
Revised: Squeeze layers Squeeze layers (Dinh et al., 2016) are used to compress the spatial resolution of activations.

Original: These also help with increasing spatial receptive ﬁeld of pixels in the deeper activations.
Revised: These also help with increasing spatial receptive ﬁeld of pixels in the deeper activations.

Original: Split Prior Split priors (Dinh et al., 2016) work by spliting a set of activations z into two components z 0 and z 1 .
Revised: Split Prior Split priors (Dinh et al., 2016) work by spliting a set of activations z into two components z 0 and z 1 .

Original: We then condition z 1 on z 0 using a simple base density e.g. p ( z 1 | z 0 ) =
Revised: We then condition z 1 on z 0 using a simple base density e.g. p ( z 1 | z 0 ) =

Original: N ( z 1 ; µ ( z 0 ) , σ 2 ( z 0 )) , where µ ( · ) and σ 2 ( · ) are neural networks.
Revised: N ( z 1 ; µ ( z 0 ) , σ 2 ( z 0 )) , where µ ( · ) and σ 2 ( · ) are neural networks.

Original: The component z 0 can be modeled by further ﬂow layers.
Revised: The component z 0 can be modeled by further ﬂow layers.

Original: This prior, is useful for modeling hierarchical correlations between dimensions, and also helps reduce computation, since z 0 is reduced in size.
Revised: This prior, is useful for modeling hierarchical correlations between dimensions, and also helps reduce computation, since z 0 is reduced in size.

Original: Variational dequantization When modeling discrete data, Theis et al.
Revised: Variational dequantization When modeling discrete data, Theis et al.

Original: (2016) introduced the concept of dequantization .
Revised: (2016) introduced the concept of dequantization .

Original: For this, they modeled the probability mass function over y as a latent variable model
Revised: For this, they modeled the probability mass function over y as a latent variable model

Original: [Equation]
Revised: [Equation]

Original: where the latent variables v ∈ V are continuous-valued.
Revised: where the latent variables v ∈ V are continuous-valued.

Original: This is a convenient model to use, since the marginal p ( v ) , living on a continuous sample space, can be modelled with a continuous NF.
Revised: This is a convenient model to use, since the marginal p ( v ) , living on a continuous sample space, can be modelled with a continuous NF.

Original: The distribution P ( y | v ) is known as the quantizer and is typically an indicator function P ( y | v ) =
Revised: The distribution P ( y | v ) is known as the quantizer and is typically an indicator function P ( y | v ) =

Original: I [ v ∈ y + [0 , 1) D ] .
Revised: I [ v ∈ y + [0 , 1) D ] .

Original: Other works (Hoogeboom et al., 2019a; Tran et al., 2019) directly model P model ( y ) with a discrete-valued ﬂow, but these are known to be difﬁcult to optimize.
Revised: Other works (Hoogeboom et al., 2019a; Tran et al., 2019) directly model P model ( y ) with a discrete-valued ﬂow, but these are known to be difﬁcult to optimize.

Original: As an extension of dequantization, Ho et al.
Revised: As an extension of dequantization, Ho et al.

Original: (2019) introduced a variational distribution q ( v | y ) , called a dequantizer , and write a lower bound on the data log-likelihood using Jensen’s inequality as follows
Revised: (2019) introduced a variational distribution q ( v | y ) , called a dequantizer , and write a lower bound on the data log-likelihood using Jensen’s inequality as follows

Original: [Equation]
Revised: [Equation]

Original: Noting that the joint p ( y , v ) =
Revised: Noting that the joint p ( y , v ) =

Original: I [ v ∈ y + [0 , 1) D ] p ( v ) , we see that the dequantizer distribution q ( v | y ) must be deﬁned such that v ∈ y + [0 , 1) D , otherwise p ( y , v ) = 0 and the lower-bound is undeﬁned.
Revised: I [ v ∈ y + [0 , 1) D ] p ( v ) , we see that the dequantizer distribution q ( v | y ) must be deﬁned such that v ∈ y + [0 , 1) D , otherwise p ( y , v ) = 0 and the lower-bound is undeﬁned.

Original: Restricting q ( v | y ) to satisfy this condition, results in the following variational dequantization bound
Revised: Restricting q ( v | y ) to satisfy this condition, results in the following variational dequantization bound

Original: [Equation]
Revised: [Equation]

Original: 2. S TRUCTURED PREDICTION
Revised: 2. S TRUCTURED PREDICTION

Original: Structured prediction tasks such as image segmentation or super-resolution, can be probabilistically framed as learning an unknown target distribution p ∗ ( y | x ) , with an input x ∈ X and a target y ∈ Y .
Revised: Structured prediction tasks such as image segmentation or super-resolution, can be probabilistically framed as learning an unknown target distribution p ∗ ( y | x ) , with an input x ∈ X and a target y ∈ Y .

Original: In practice with deep learning models, the unknown distribution is often learned by a factored model:
Revised: In practice with deep learning models, the unknown distribution is often learned by a factored model:

Original: [Equation]
Revised: [Equation]

Original: where y d represents the d th dimension of y .
Revised: where y d represents the d th dimension of y .

Original: Several loss-based optimization methods are a special case of this factored model.
Revised: Several loss-based optimization methods are a special case of this factored model.

Original: The mean squared error is equivalent to a product of normal distributions with equal and ﬁxed standard deviation.
Revised: The mean squared error is equivalent to a product of normal distributions with equal and ﬁxed standard deviation.

Original: Other examples are: cross entropy, equivalent to a product of log categorical distributions, and binary cross entropy, equivalent to a product of log Bernoulli distributions.
Revised: Other examples are: cross entropy, equivalent to a product of log categorical distributions, and binary cross entropy, equivalent to a product of log Bernoulli distributions.

Original: With factorized independent likelihoods, individual dimensions of y are assumed to be conditionally independent.
Revised: With factorized independent likelihoods, individual dimensions of y are assumed to be conditionally independent.

Original: As a result, sampling leads to results with uncorrelated noise over the output dimensions.
Revised: As a result, sampling leads to results with uncorrelated noise over the output dimensions.

Original: In the literature, a ﬁx for this problem is to visualize the mode of the distribution and interpret that as a prediction.
Revised: In the literature, a ﬁx for this problem is to visualize the mode of the distribution and interpret that as a prediction.

Original: However, because the likelihood was optimized assuming a conditionally independent noise distribution, these modes tend to be blurry and lack crisp details.
Revised: However, because the likelihood was optimized assuming a conditionally independent noise distribution, these modes tend to be blurry and lack crisp details.

Original:  M ETHOD
Revised:  M ETHOD

Original: In this section we present our main innovations.
Revised: In this section we present our main innovations.

Original: i) learning conditional likelihoods using CNFs and ii) a variational dequantization framework for binary random variables.
Revised: i) learning conditional likelihoods using CNFs and ii) a variational dequantization framework for binary random variables.

Original: 3. C ONDITIONAL N ORMALIZING F LOWS
Revised: 3. C ONDITIONAL N ORMALIZING F LOWS

Original: We propose to learn conditional likelihoods using conditional normalizing ﬂows for complicated target distributions in multivariate prediction tasks.
Revised: We propose to learn conditional likelihoods using conditional normalizing ﬂows for complicated target distributions in multivariate prediction tasks.

Original: Take an input x ∈ X and a regression target y ∈ Y .
Revised: Take an input x ∈ X and a regression target y ∈ Y .

Original: We learn a complicated distribution p Y | X
Revised: We learn a complicated distribution p Y | X

Original: ( y | x ) using a conditional prior p Z | X ( z | x ) and a mapping f φ : Y × X
Revised: ( y | x ) using a conditional prior p Z | X ( z | x ) and a mapping f φ : Y × X

Original: → Z , which is bijective in Y and Z .
Revised: → Z , which is bijective in Y and Z .

Original: The likelihood of this model is:
Revised: The likelihood of this model is:

Original: [Equation]
Revised: [Equation]

Original: (cid:12)(cid:12)(cid:12)(cid:12)
Revised: (cid:12)(cid:12)(cid:12)(cid:12)

Original: [Equation]
Revised: [Equation]

Original: Notice that the difference between Equations 1 and 7 is that all distributions are conditional and the ﬂow has a conditioning argument of x .
Revised: Notice that the difference between Equations 1 and 7 is that all distributions are conditional and the ﬂow has a conditioning argument of x .

Original: The generative process from x to y (shown in Figure 1) can be described by ﬁrst sampling z ∼ p
Revised: The generative process from x to y (shown in Figure 1) can be described by ﬁrst sampling z ∼ p

Original: Z | X ( z | x ) from a simple base density with its parameters conditioned on x (for us this is a diagonal
Revised: Z | X ( z | x ) from a simple base density with its parameters conditioned on x (for us this is a diagonal

Original: Gaussian) and then passing it through a sequence of bijective mappings f − 1 φ ( z ; x ) .
Revised: Gaussian) and then passing it through a sequence of bijective mappings f − 1 φ ( z ; x ) .

Original: This allows for modelling multimodal conditional distributions in y , which is typically uncommon. 
Revised: This allows for modelling multimodal conditional distributions in y , which is typically uncommon. [Figure]

Original: [Equation]
Revised: [Equation]

Original: Training
Revised: [Figure] [Equation] [Figure] [Equation]

Original: Sampling
Revised: [Figure]

Original: For the training procedure, the process runs in reverse.
Revised: For the training procedure, the process runs in reverse.

Original: We begin with label y and conditioning input x .
Revised: We begin with label y and conditioning input x .

Original: We ‘ﬂow’ the label back through f φ to yield z = f φ ( y ; x ) , and then we evaluate the log-likelihood of the parameters of the prior, given this transformed label z .
Revised: We ‘ﬂow’ the label back through f φ to yield z = f φ ( y ; x ) , and then we evaluate the log-likelihood of the parameters of the prior, given this transformed label z .

Original: The ﬂow and prior parameters can be optimized using stochastic gradient descent and training in minibatches in the usual fashion.
Revised: The ﬂow and prior parameters can be optimized using stochastic gradient descent and training in minibatches in the usual fashion.

Original: Note that this style of training a conditional density model p Y | X ( y | x ) differs fundamentally from traditional models, because we compute the log-likelihood in z -space and not y -space.
Revised: Note that this style of training a conditional density model p Y | X ( y | x ) differs fundamentally from traditional models, because we compute the log-likelihood in z -space and not y -space.

Original: As a result, we are not biasing our results with an arbitrary choice of output-space likelihood or in the case of this paper, handcrafted image loss.
Revised: As a result, we are not biasing our results with an arbitrary choice of output-space likelihood or in the case of this paper, handcrafted image loss.

Original: Instead, one could interpret this method as learning the correlational and multimodal structure of the likelihood or simply put loss-learning.
Revised: Instead, one could interpret this method as learning the correlational and multimodal structure of the likelihood or simply put loss-learning.

Original: Conditional modules In our work, the conditioning is introduced in the prior, the split priors, and the afﬁne coupling modules.
Revised: Conditional modules In our work, the conditioning is introduced in the prior, the split priors, and the afﬁne coupling modules.

Original: For the prior, we set the mean and variance as functions of x .
Revised: For the prior, we set the mean and variance as functions of x .

Original: For the split prior, we add x as a conditoning argument to the conditional p ( z 1 | z 0 , x ) .
Revised: For the split prior, we add x as a conditoning argument to the conditional p ( z 1 | z 0 , x ) .

Original: And for the afﬁne coupling layers, we pass x to the scale and translation networks so that
Revised: And for the afﬁne coupling layers, we pass x to the scale and translation networks so that

Original: [Equation]
Revised: [Equation]

Original: In practice these functions are implemented using deep neural networks.
Revised: In practice these functions are implemented using deep neural networks.

Original: First the conditioning term x is transformed into a rich representation
Revised: First the conditioning term x is transformed into a rich representation

Original: h
Revised: h

Original: = g ( x ) using a large network g .
Revised: = g ( x ) using a large network g .

Original: Subsequently, each function in the ﬂow is applied to a concatenation [ · , · ] of h and the relevant part of z .
Revised: Subsequently, each function in the ﬂow is applied to a concatenation [ · , · ] of h and the relevant part of z .

Original: For example, the translation of a conditional coupling is computed as t ( z 1 , x ) =
Revised: For example, the translation of a conditional coupling is computed as t ( z 1 , x ) =

Original: NN ([ z 1 , h ]) .
Revised: NN ([ z 1 , h ]) .

Original: 3. V ARIATIONAL D EQUANTIZATION FOR B INARY R ANDOM V ARIABLES
Revised: 3. V ARIATIONAL D EQUANTIZATION FOR B INARY R ANDOM V ARIABLES

Original: We generalize the variational dequantization scheme for the binary setting.
Revised: We generalize the variational dequantization scheme for the binary setting.

Original: Let y ∈ { 0 , 1 } D be a multivariate binary random variable and v ∈ R D its dequantized representation.
Revised: Let y ∈ { 0 , 1 } D be a multivariate binary random variable and v ∈ R D its dequantized representation.

Original: In Ho et al.
Revised: In Ho et al.

Original: the bound is not guaranteed to be tight, since there is a domain mismatch in the support of p ( v ) and the variational dequantizer q ( v | y ) .
Revised: the bound is not guaranteed to be tight, since there is a domain mismatch in the support of p ( v ) and the variational dequantizer q ( v | y ) .

Original: Technically, if p ( v ) is modeled as a bijective mapping from a
Revised: Technically, if p ( v ) is modeled as a bijective mapping from a

Original: Gaussian distribution where the mapping only has ﬁnite volume changes, then the support of p ( v ) is unbounded.
Revised: Gaussian distribution where the mapping only has ﬁnite volume changes, then the support of p ( v ) is unbounded.

Original: On the otherhand, the support of the dequantizer is bounded and so we have to redeﬁne either the dequantizer to map to all of R D or restrict the support of the ﬂow to a bounded volume inside R D .
Revised: On the otherhand, the support of the dequantizer is bounded and so we have to redeﬁne either the dequantizer to map to all of R D or restrict the support of the ﬂow to a bounded volume inside R D .

Original: We resolve this by dequantizing with half-inﬁnite noise, where
Revised: We resolve this by dequantizing with half-inﬁnite noise, where

Original: [Equation]
Revised: [Equation]

Original: The softplus guarantees that samples from the neural network NN are only positive.
Revised: The softplus guarantees that samples from the neural network NN are only positive.

Original: If y is 1, the term sign ( y − 0 .
Revised: If y is 1, the term sign ( y − 0 .

Original: 5) outputs positive-valued noise and if y is 0 the noise is negative-valued.
Revised: 5) outputs positive-valued noise and if y is 0 the noise is negative-valued.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Normalizing ﬂows were originally introduced to machine learning to learn a ﬂexible variational posterior, a conditional distribution, in VAEs (Rezende & Mohamed, 2015; Kingma et al., 2016a; van den Berg et al., 2018).
Revised: Normalizing ﬂows were originally introduced to machine learning to learn a ﬂexible variational posterior, a conditional distribution, in VAEs (Rezende & Mohamed, 2015; Kingma et al., 2016a; van den Berg et al., 2018).

Original: Flow-based generative models (Dinh et al., 2016; Papamakarios et al., 2017; Huang et al., 2018; Kingma & Dhariwal, 2018;
Revised: Flow-based generative models (Dinh et al., 2016; Papamakarios et al., 2017; Huang et al., 2018; Kingma & Dhariwal, 2018;

Original: Hoogeboom et al., 2019b; Grathwohl et al., 2019; Cao et al., 2019; Chen et al., 2019) are typically trained directly in the data space.
Revised: Hoogeboom et al., 2019b; Grathwohl et al., 2019; Cao et al., 2019; Chen et al., 2019) are typically trained directly in the data space.

Original: Several of these are designed to be fast to invert, which makes them suitable for drawing samples after training.
Revised: Several of these are designed to be fast to invert, which makes them suitable for drawing samples after training.

Revised: Different versions and applications of conditional normalizing ﬂows include Agrawal & Dukkipati (2016) who utilize ﬂows in the decoder of Variational AutoEncoders Kingma & Welling (2014), which are conditioned on the latent variable.

Original: Trippe & Turner (2018)  utilize conditional ﬂows for prediction problems in a Bayesian framework for density estimation.
Revised: Trippe & Turner (2018) who utilize conditional ﬂows for prediction problems in a Bayesian framework for density estimation.

Original: Atanov et al.
Revised: Atanov et al.

Original: (2019) introduce a semi-conditional ﬂow that provides an efﬁcient way to learn from unlabeled data for semi-supervised classiﬁcation problems.
Revised: (2019) introduce a semi-conditional ﬂow that provides an efﬁcient way to learn from unlabeled data for semisupervised classiﬁcation problems.

Original: Other likelihood based methods include autoregressive models (Van Oord et al., 2016), which can be computationally expensive to draw samples from, and variational autoencoders (Kingma & Welling, 2014), which optimize a lowerbound of the likelihood.
Revised: Very recently, Ardizzone et al. (2019) have proposed conditional ﬂow-based generative models for image colorization, which differs from our work in training objective, architecture and applicability to binary segmentation. Autoregressive models (Van Oord et al., 2016) have also been studied for conditional image generation van den Oord et al. but are generally slow to sample from.

Original: Adversarial methods (Goodfellow et al., 2014) have widely been applied to (conditional) image density modeling tasks (Vu et al., 2019; Sajjadi et al., 2017b; Yuan et al., 2018; Mechrez et al., 2018), because they tend to generate high-ﬁdelity images.
Revised: Adversarial methods (Goodfellow et al., 2014) have widely been applied to (conditional) image density modeling tasks (Vu et al., 2019; Sajjadi et al., 2017b; Yuan et al., 2018; Mechrez et al., 2018), because they tend to generate high-ﬁdelity images.

Original: Disadvantages of adversarial methods are that they can be complicated to train, and it is difﬁcult to obtain likelihoods.
Revised: Disadvantages of adversarial methods are that they can be complicated to train, and it is difﬁcult to obtain likelihoods.

Original: For this reason, it can be hard to assess whether they are overﬁtting or generalizing.
Revised: For this reason, it can be hard to assess whether they are overﬁtting or generalizing.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Here we explain our experiments into super-resolution and vessel segmentation.
Revised: Here we explain our experiments into super-resolution and vessel segmentation.

Original: All models were implemented using the PyTorch framework.
Revised: All models were implemented using the PyTorch framework.

Original: 5. S INGLE I MAGE S UPER R ESOLUTION
Revised: 5. S INGLE I MAGE S UPER R ESOLUTION

Original: Single Image Super Resolution (SISR) methods aims to ﬁnd a high resolution image x hr given a single (downsampled) low resolution image x lr .
Revised: Single Image Super Resolution (SISR) methods aims to ﬁnd a high resolution image x hr given a single (downsampled) low resolution image x lr .

Original: Framing this problem as learning a likelihood, we utilize a CNF to learn the distribution p ( x hr | x lr ) .
Revised: Framing this problem as learning a likelihood, we utilize a CNF to learn the distribution p ( x hr | x lr ) .

Original: To compare our method we also train a factorized baseline likelihood model with comparable architectures and parameter budget.
Revised: To compare our method we also train a factorized baseline likelihood model with comparable architectures and parameter budget.

Original: The factorized baseline uses a product of discretized logistic distributions (Kingma et al., 2016a; Salimans et al., 2017).
Revised: The factorized baseline uses a product of discretized logistic distributions (Kingma et al., 2016a; Salimans et al., 2017).

Original: All methods are compared on negative log 2 -likelihood if available, which has the information theoretic interpretation bits per dimensions .
Revised: All methods are compared on negative log 2 -likelihood if available, which has the information theoretic interpretation bits per dimensions .

Original: In addition, we evaluate on SSIM (Wang et al., 2004) and PSNR.
Revised: In addition, we evaluate using SSIM (Wang et al., 2004) and PSNR metrics.

Original: Implementation Details
Revised: Implementation Details

Original: The ﬂow is based on the (Dinh et al., 2016; Kingma & Dhariwal, 2018) multi-scale architectures.
Revised: The ﬂow is based on the (Dinh et al., 2016; Kingma & Dhariwal, 2018) multi-scale architectures.

Original: Each step of ﬂow consists of K subﬂows and L levels.
Revised: Each step of ﬂow consists of K subﬂows and L levels.

Original: One subﬂow consists of an activation normalization, 1 × 1 convolution, and our conditional coupling layer.
Revised: One subﬂow consists of an activation normalization, 1 × 1 convolution, and our conditional coupling layer.

Original: After completing a level, half of the representation is factored-out and modeled using our conditional split prior.
Revised: After completing a level, half of the representation is factored-out and modeled using our conditional split prior.

Original: After all levels have been completed, our conditional prior is used to model the ﬁnal part of the latent variable.
Revised: After all levels have been completed, our conditional prior is used to model the ﬁnal part of the latent variable.

Original: The conditioning variable x lr is transformed into the feature representation h using Residual-inResidual Dense Block (RRDB) architecture (Wang et al., 2018), consisting of 16 residual-in-residual blocks.
Revised: The conditioning variable x lr is transformed into the feature representation h using Residual-inResidual Dense Block (RRDB) architecture (Wang et al., 2018), consisting of 16 residual-in-residual blocks.

Original: To match the parameter budget, the channel growth is 55 for the baseline and the growth is 32 for the CNF.
Revised: To match the parameter budget, the channel growth is 55 for the baseline and the growth is 32 for the CNF.

Original: Data
Revised: Data

Original: The models are trained on natural image datasets, Imagenet32 and ImageNet64 (Chrabaszcz et al., 2017).
Revised: The models are trained on natural image datasets, Imagenet32 and Imagenet64 (Chrabaszcz et al., 2017).

Original: Since the dataset has no test set, we use its validation images as a test set.
Revised: Since the dataset has no test set, we use its validation images as a test set.

Original: For validation we take 10000 images from the train images.
Revised: For validation we take 10000 images from the train images.

Original: The performance is always reported on the test set unless speciﬁed otherwise.
Revised: The performance is always reported on the test set unless speciﬁed otherwise.

Original: We evaluate our models on widely used benchmark datasets Set (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2012) and BSD100 (Huang et al., 2015).
Revised: We evaluate our models on widely used benchmark datasets Set (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2012) and BSD100 (Huang et al., 2015).

Original: At test time, we pad the test images with zeros at right and bottom to them square and suitable for our model.
Revised: At test time, we pad the test images with zeros at right and bottom so that they are square and compatible with squeeze layers.

Original: When evaluating on SSIM and PSNR, we crop out the patch of the original image shape.
Revised: When evaluating on SSIM and PSNR, we can extract the patch with the exact image shape.

Original: For all datasets the LR images are obtained using MATLABs bicubic kernel function with reducing aliasing artifacts following Wang et al.
Revised: For all datasets the LR images are obtained using MATLABs bicubic kernel function with reducing aliasing artifacts following Wang et al.

Original: For these experiments, the pixel values are dequantized by adding uniform noise in [0, 1].
Revised: For these experiments, the pixel values are dequantized by adding uniform noise (Theis et al., 2016).

Original: Training Settings We train on ImageNet32 and ImageNet64 for 200 , 000 iterations with mini batches of size 64, and a learning rate of 0.0001 using the Adam optimizer (Kingma & Ba, 2014).
Revised: Training Settings We train on ImageNet32 and ImageNet64 for 200 , 000 iterations with mini batches of size 64, and a learning rate of 0.0001 using the Adam optimizer (Kingma & Ba, 2015). [Figure]

Original: The high-resolution image x hr either the original 32 × 32 or 64 × 64 original input images.
Revised: The high-resolution image x hr either the original 32 × 32 or 64 × 64 original input images.

Original: The ﬂow architecture is build with L = 2 levels and K = 8 .[Figure]
Revised: The ﬂow architecture is build with L = 2 levels and K = 8 .

Original: 5.1. E VALUATION
Revised: 5.1. E VALUATION

Original: In this section the performance of CNFs for SISR is compared against a baseline likelihood model on ImageNet32 and ImageNet64.
Revised: In this section the performance of CNFs for SISR is compared against a baseline likelihood model on ImageNet32 and ImageNet64.

Original: Their performance measured in log 2 -likelihood (bits per dimension) is shown in Table 1.
Revised: Their performance measured in log 2 -likelihood (bits per dimension) is shown in Table 1, which show that the CNF outperforms the factorized baseline in likelihood.

Original: Recall that the baseline model is factorized and conditionally independent.
Revised: Recall that the baseline model is factorized and conditionally independent.

Original: These results indicate thatthe CNF outperforms the factorized baseline in likelihood, as it can capture the correlations and multi-modality present in the data.
Revised: These results indicate that it is advantageous to capture the correlations and multi-modality present in the data.

Original: [Table]
Revised: [Table]

Original: Super resolution samples x hr ∼ p ( x hr | x lr ) from the Imagenet64 test data are shown in Figure 2.
Revised: Super resolution samples x hr ∼ p ( x hr | x lr ) from the Imagenet64 test data are shown in Figure 2.

Original: The distribution mode of the factorized baseline is displayed in Figure 2 panel c).
Revised: The distribution mode of the factorized baseline is displayed in Figure 2 panel c).

Original: We show that the baseline is able to learn a relationship between conditioning variable x and y , but lacks crisp details.
Revised: We show that the baseline is able to learn a relationship between conditioning variable x and y , but lacks crisp details.

Original: The superresolution images from the CNF are shown in panel b).
Revised: The super-resolution images from the CNF are shown in panel b).

Original: Notice there are more high-frequency components modelled, for instance in grass and in hairs.
Revised: Notice there are more high-frequency components modelled, for instance in grass and in hairs.

Original: Following Kingma &
Revised: Following Kingma &

Original: Dhariwal (2018), we sample from the base distributions with a temperature τ of 0.8 to achieve the best perceptual quality for the distribution learned by the CNF.
Revised: Dhariwal (2018), we sample from the base distributions with a temperature τ of 0.8 to achieve the best perceptual quality for the distribution learned by the CNF.

Original: As there are no standard metric for measuring perceptual quality, we measure performance results on
Revised: As there are no standard metric for measuring perceptual quality, we measure performance results on

Original: PSNR and SSIM between our predicted image and the ground truth image in Table 2.
Revised: PSNR and SSIM between our predicted image and the ground truth image in Table 2.

Original: We compare
Revised: We compare

Original: CNFs to other state-of-the-art per-pixel loss based methods and the factorized baseline for a 2x upsampling task on standard super-resolution benchmarks.
Revised: CNFs to other state-of-the-art per-pixel loss based methods and the factorized baseline for a 2x upsampling task on standard super-resolution benchmarks.

Original: If available, we report negative log 2likelihood (bpd) (computed as an average over 1000 randomly cropped 128 x 128 patches).
Revised: If available, we report negative log 2likelihood (bpd) (computed as an average over 1000 randomly cropped 128 x 128 patches).

Original: Note that without any hyperparameter tuning or compositional loss weighting, as is typical in SISR, the
Revised: Note that without any hyperparameter tuning or compositional loss weighting, as is typical in SISR, the

Original: CNF performs competitively with state-of-the-art super-resolution methods by simply optimizing the likelihood.
Revised: CNF performs competitively with state-of-the-art super-resolution methods by simply optimizing the likelihood.

Original: The SSIM scores for the baseline perform on par or better than the adversarial methods and the CNF on all benchmarks.
Revised: The SSIM scores for the baseline perform on par or better than the adversarial methods and the CNF on all benchmarks.

Original: On PSNR scores however, the CNF beats the factorized discrete baseline.
Revised: On PSNR scores however, the CNF beats the factorized discrete baseline.

Original: Samples shown in Figure 3 show that the CNF predictions have more ﬁne grained texture details.
Revised: Samples shown in Figure 3 show that the CNF predictions have more ﬁne grained texture details.

Original: Comparing this ﬁnding with the performance results, show that metrics can be misleading.
Revised: Comparing this ﬁnding with the baseline that outperforms every method on SSIM, show that metrics can be misleading.

Original: Notice how samples from a independent factorized likelihood model has a lot of color noise on the contrary to the CNF.
Revised: Notice how samples from a independent factorized likelihood model have a lot of color noise, whereas samples from the CNF do not have such problems.

Original: Increasing temperature increases high-level detail, with τ = 0 .
Revised: Increasing temperature increases highlevel detail, where we ﬁnd that τ = 0 .

Original: 5 striking a balance between noise smoothing and detail.
Revised: 5 strikes a balance between noise smoothing and detail.

Original: This can be attributed to the property of ﬂows to model pixel correlations among output dimensions for high-dimensional data such as images.
Revised: This can be attributed to the property of ﬂows to model pixel correlations among output dimensions for high-dimensional data such as images.

Original: [Table]
Revised: [Table]

Original: V ESSEL SEGMENTATION
Revised: V ESSEL SEGMENTATION

Original: Vessel segmentation is an important, long-standing, medical imaging problem, where we seek to segment blood vessels from pictures of the retina (the back of the eye).
Revised: Vessel segmentation is an important, long-standing, medical imaging problem, where we seek to segment blood vessels from pictures of the retina (the back of the eye).

Original: This is a difﬁcult task, because the vessels are thin and of varying thickness.
Revised: This is a difﬁcult task, because the vessels are thin and of varying thickness.

Original: A likelihood function used in segmentation is
Revised: A likelihood function used in segmentation is

Original: [Table]
Revised: [Table]

Revised: distribution

Original: [Equation]
Revised: [Equation]

Original: 9) where p j = p ( y j = 1 | x ) is the prediction probability that pixel y j is positive (vessel class) and β is a class balancing constant set to ∼ 10% for us.
Revised: [Table] [Equation] (9) where p j = p ( y j = 1 | x ) is the prediction probability that pixel y j is positive (vessel class) and β is a class balancing constant set to ∼ 10% for us.

Original: This loss function is preferred, because it accounts for the apparent class imbalance in the ratio of vessels to background.
Revised: This loss function is preferred, because it accounts for the apparent class imbalance in the ratio of vessels to background.

Original: In practice, the numerator of this likelihood is used as a loss function and the normalizer is ignored.
Revised: In practice, the numerator of this likelihood is used as a loss function and the normalizer is ignored.

Original: The resulting loss is called a weighted cross-entropy.
Revised: The resulting loss is called a weighted cross-entropy.

Original: In our experiments we train using the weighted cross-entropy (as in the literature) [Table] but we report likelihood values including the normalizer, for a meaningful comparison.
Revised: In our experiments we train using the weighted cross-entropy (as in the literature), but we report likelihood values including the normalizer, for a meaningful comparison.

Original: Dataset and comparisons We test on the DRIVE database Staal et al.
Revised: Dataset and comparisons We test on the DRIVE database Staal et al.

Original: (2004) consisting of 584 × 565 , 8-bit RGB images, split into 20 train, and 20 test images.
Revised: (2004) consisting of 584 × 565 , 8-bit RGB images, split into 20 train, and 20 test images.

Original: To compare against other methods, we plot precision-recall curves, report the maximum F-score along each curve (shown as a dot in the graph), report the bits per dimension, and plot distributions in PR-space.
Revised: To compare against other methods, we plot precision-recall curves, report the maximum F-score along each curve (shown as a dot in the graph), report the bits per dimension, and plot distributions in PR-space.

Original: The main CNN-based contenders are Deep Retinal Image Understanding (DRIU) (Maninis et al., 2016) and HolisticallyNested Edge Detection (HED) (Xie & Tu, 2015), both of which are instances of Deeply-Supervised
Revised: The main CNN-based contenders are Deep Retinal Image Understanding (DRIU) (Maninis et al., 2016) and HolisticallyNested Edge Detection (HED) (Xie & Tu, 2015), both of which are instances of Deeply-Supervised

Original: Nets (Lee et al., 2015).
Revised: Nets (Lee et al., 2015).

Original: The main difference between DRIU and HED is that DRIU is pretrained on
Revised: The main difference between DRIU and HED is that DRIU is pretrained on

Original: ImageNet Krizhevsky et al.
Revised: ImageNet Krizhevsky et al.

Original: (2012); whereas, HED is not.
Revised: (2012); whereas, HED is not.

Original: Other competing methods are reported with results cited from (Maninis et al., 2016).
Revised: Other competing methods are reported with results cited from (Maninis et al., 2016).

Original: For fairness, we also train a model which we call the factored baseline , which uses the exact same architecture as the ﬂow but run a feedforward model and trained with the weighted Bernoulli loss. 
Revised: For fairness, we also train a model which we call the likelihood baseline , which uses the exact same architecture as the ﬂow but run a feedforward model and trained with the weighted Bernoulli loss. [Table]

Original: Implementation The ﬂow is identical to the ﬂow used in the previous section, with some key differences.
Revised: Implementation The ﬂow is identical to the ﬂow used in the previous section, with some key differences.

Original: i) instead of activation normalization, we use instance normalization (Ulyanov et al., 2016), ii) the conditional afﬁne coupling layers do not contain a scaling component s ( · ) but just the translation t ( · ) , hence it is volume preserving, and iii) we train using variational dequantization.
Revised: i) instead of activation normalization, we use instance normalization (Ulyanov et al., 2016), ii) the conditional afﬁne coupling layers do not contain a scaling component s ( · ) but just the translation t ( · ) , hence it is volume preserving, and iii) we train using variational dequantization.

Original: Since the data is binary-valued, we dequantize according to the Flow++ scheme of Ho et al., modiﬁed to binary variables (see Section 3.2), using a CNF at just a single scale for the dequantizer.
Revised: Since the data is binary-valued, we dequantize according to the Flow++ scheme of Ho et al., modiﬁed to binary variables (see Section 3.2), using a CNF at just a single scale for the dequantizer.

Original: The CNF is conditioned on resolution-matched features extracted from a VGGlike network Simonyan & Zisserman (2015).
Revised: The CNF is conditioned on resolution-matched features extracted from a VGG-like network Simonyan & Zisserman (2015).

Original: This model is composed of blocks of the form block =
Revised: This model is composed of blocks of the form block =

Original: [InstanceNorm, ReLU, conv], and 2x2 max-pooling layer, shown in Table 5.
Revised: [InstanceNorm, ReLU, conv], and 2x2 max-pooling layer, shown in Table 4.

Original: All ﬁlter sizes are 3x3.
Revised: All ﬁlter sizes are 3x3.

Original: The outputs are at layers 4 and 7.
Revised: The outputs are at layers 4 and 7.

Original: These are used to condition the resolution 512 and 256 levels of the CNF, respectively.
Revised: These are used to condition the resolution 512 and 256 levels of the CNF, respectively.

Original: Training/test settings We train using the Adam optimizer at learning rate 0.001, and a minibatch size of 2 for 2000 epochs.
Revised: Training/test settings We train using the Adam optimizer at learning rate 0.001, and a minibatch size of 2 for 2000 epochs.

Original: All images are padded to 1024x1024 pixels, so we can use squeeze layers.
Revised: All images are padded to 1024x1024 pixels, so that they are compatible with squeeze layers.

Original: We use 360 ◦ rotation augmentation, isotropic scalings in the range [0 . 8 , 1 . 2] , and shears drawn from a normal distribution with standard deviation 10 ◦ .
Revised: We use 360 ◦ rotation augmentation, isotropic scalings in the range [0 . 8 , 1 . 2] , and shears drawn from a normal distribution with standard deviation 10 ◦ .

Original: At test time we draw samples from our model and compare those against the groundtruth labels.
Revised: At test time we draw samples from our model and compare those against the groundtruth labels.

Original: This contrasts with other methods, that measure labels against thresholded versions of a factorized predictive distribution.
Revised: This contrasts with other methods, that measure labels against thresholded versions of a factorized predictive distribution.

Original: To create the PR curve in Figure 6 we take the average of 100 samples and threshold the resulting map (example shown in Figure 4).
Revised: To create the PR curve in Figure 5 we take the average of 100 samples and threshold the resulting map (example shown in Figure 4).

Original: While crude, this mean image is useful in deﬁning a PR-curve, since there is not great topology change between samples.
Revised: While crude, this mean image is useful in deﬁning a PR-curve, since there is not great topology change between samples.

Original: Evaluation The results of our experiments are shown in Table 3 and Figure 6, with a visualization in Figure 4.
Revised: Evaluation The results of our experiments are shown in Table 3 and Figure 5, with a visualization in Figure 4.

Original: We see in the table that the CNF acheives the best bits per dimension, with comparable F-score to the state of the art model (DRIU), but our model does not require pretraining on ImageNet.
Revised: We see in the table that the CNF trained with our binary dequantization achieves the best bits per dimension, with comparable F-score to the state of the art model (DRIU), but our model does not require pretraining on ImageNet.

Revised: Interestingly, we found training a ﬂow with uniform dequantization slightly unstable and the results were far from satisfactory.

Original: This result is corroborated by the PR-curve Figure 6, which shows a comparable curve  to the DRIU model.
Revised: In the PR-curve Figure 5, we show a comparable curve for our binary dequantized CNF to the DRIU model.

Original: These results, however, say nothing about the calibration of the probability outputs, but just that the various probability predictions are well ranked.
Revised: These results, however, say nothing about the calibration of the probability outputs, but just that the various probability predictions are well ranked.

Original: To gain an insight into the calibration of the probabilities, we measure the distribution of precision and recall values for point samples drawn from all models, including a second human grader, present in the original DRIVE dataset.
Revised: To gain an insight into the calibration of the probabilities, we measure the distribution of precision and recall values for point samples drawn from all models, including a second human grader, present in the original DRIVE dataset.

Original: We synthesized samples from factored distributions (all except ours and ‘human’), by sampling images from a factored Bernoulli with mean as the soft image.
Revised: We synthesized samples from factored distributions (all except ours and ‘human’), by sampling images from a factored Bernoulli with mean as the soft image.

Original: We see the results in the right hand plot of Figure 6, which shows that while the other CNN-based methods such as DRIU or HED have good precision, they suffer in term of recall.
Revised: We see the results in the right hand plot of Figure 5, which shows that while the other CNN-based methods such as DRIU or HED have good precision, they suffer in term of recall.

Original: On the other hand, the CNF drops in precision a little bit, but makes up for this in terms of high recall, with a PR distribution overlapping the human grader.
Revised: On the other hand, the CNF drops in precision a little bit, but makes up for this in terms of high recall, with a PR distribution overlapping the human grader.

Original: This indicates that the CNF has learned a well-callibrated distribution, compared to the baseline methods.
Revised: This indicates that the CNF has learned a well-calibrated distribution, compared to the baseline methods.

Original: Further evidence of this is seen in the visualization in Figure 4, which shows details from the predicted means (soft images).
Revised: Further evidence of this is seen in the visualization in Figure 4, which shows details from the predicted means (soft images).

Original: This shows that the DRIU and factor model baseline overdilate segmentations and the CNF does not.
Revised: This shows that the DRIU and likelihood baseline overdilate segmentations and the CNF does not.

Original: This can be explained from the fact that in the weighted Bernoulli it is cheaper to overdilate than to underdilate.
Revised: This can be explained from the fact that in the weighted Bernoulli it is cheaper to overdilate than to underdilate.

Original: Since the CNF contains no handcrafted loss function, we circumvent this pathology.[Figure] [Figure] ) PR
Revised: Since the CNF contains no handcrafted loss function, we circumvent this pathology.

Original: [Figure]
Revised: [Figure]

Original:  C ONCLUSION& D ISCUSSION
Revised:  C ONCLUSION

Original: In this paper we propose to learn likelihoods of conditional distributions using conditional normalizing ﬂows.
Revised: In this paper we propose to learn likelihoods of conditional distributions using conditional normalizing ﬂows.

Original: In this setting, supervised prediction tasks can be framed probabilistically.
Revised: In this setting, supervised prediction tasks can be framed probabilistically.

Original: In addition, we propose a generalization of variational dequantization for binary random variables, which is useful for binary segmentation problems.
Revised: In addition, we propose a generalization of variational dequantization for binary random variables, which is useful for binary segmentation problems.

Original: Experimentally we show competitive performance with competing methods in the domain of super-resolution and binary image segmentation
Revised: Experimentally we show competitive performance with competing methods in the domain of super-resolution and binary image segmentation

Original: L
Revised: This section describes architecture and optimization details of the conditional normalizing ﬂow network, low-resolution image feature extractor, and shallow convolutional neural network in the conditional coupling layers. The conditional coupling layer is shown schematically in Figure 6. This shows that conditioned on an input x , we are able to build a relatively straight-forward invertible mapping between latent representations z and y , which have been partitioned into vectors of equal dimension.

Original: T YPE
Revised: [Equation] neural network transformation is Details for the CNFs are given in Table 5 and the details of the individual coupling layers in Table 6 and 7. The architecture of the feature extractor is given in Table 8.

Original: R ES .
Revised: The architecture has levels and subﬂows, following (Dinh et al., 2016; Kingma & Dhariwal, 2018). All networks are optimized using Adam (Kingma & Ba, 2015) for 200000 iterations. [Table] ImageNet

Revised: [Table]

Revised: ImageNet

Revised: [Table]

Revised: DRIVE

Original: input  block  max-pool  block  block  max-pool  block  block
Revised: [Table] [Table] [Table] [Table]

Original: A C ONDITIONAL I MAGE G ENERATION
Revised: B C ONDITIONAL I MAGE G ENERATION

Revised: In this section, larger versions of the ImageNet64 samples are provided, sampled at different temperatures τ .

Revised: [Figure]