FFJORD: F REE - FORM C ONTINUOUS D YNAMICS FOR S CALABLE R EVERSIBLE G ENERATIVE M ODELS
A BSTRACT
Reversible generative models map points from a simple distribution to a complex distribution through an easily invertible neural network.
Likelihood-based training of these models requires restricting their architectures to allow cheap computation of Jacobian determinants.
Alternatively, the Jacobian trace can be used if the transformation is speciﬁed by an ordinary differential equation.
In this paper, we use Hutchinson’s trace estimator to give a scalable unbiased estimate of the logdensity.
The result is a continuous-time invertible generative model with unbiased density estimation and one-pass sampling, while allowing unrestricted neural network architectures.
We demonstrate our approach on high-dimensional density estimation, image generation, and variational inference, improving the state-ofthe-art among exact likelihood methods with efﬁcient sampling
I NTRODUCTION
[Equation]
[Equation]
z
[Equation]
Reversible generative models use cheaply invertible neural networks to transform samples from a ﬁxed base distribution.
Examples include NICE (Dinh et al., 2014), Real
NVP (Dinh et al., 2017), and Glow (Kingma & Dhariwal, 2018).
These models are easy to sample from, and can be trained by maximum likelihood using the change of variables formula.
However, this requires placing awkward restrictions on their architectures, such as partitioning dimensions or using rank one weight matrices, in order to avoid an O ( D 3 ) cost determinant computation.
Recently, Chen et al.
(2018) introduced continuous normalizing ﬂows (CNF), deﬁning the mapping from latent variables to data using ordinary differential equations (ODE).
In their model, the likelihood can be computed using trace operations costing only O ( D 2 ) .
This allows a more ﬂexible, but still restricted, family of network architectures to be used.
Extending this work, we introduce an unbiased stochastic estimator of the likelihood that has O ( D ) time cost, allowing completely unrestricted architectures.
Furthermore, we have implemented GPU-based adaptive ODE solvers to train and evaluate these models on modern hardware.
We call our approach Free-form Jacobian of
Reversible Dynamics (FFJORD).
Figure 1 shows FFJORD smoothly transforming a Gaussian distribution into a multi-modal distribution.
B ACKGROUND : G ENERATIVE M ODELS AND C HANGE OF V ARIABLES
In contrast to directly parameterizing a normalized distribution (e.g. Oord et al. (2016); Germain et al.
, the change of variables formula allows one to specify a complex normalized distribution implicitly by warping a normalized base distribution p z ( z ) through an invertible function f : R D
→ R D .
Given a random variable z ∼ p
z ( z ) the log density of x = f ( z ) follows
[Equation]
where ∂f ( z ) / ∂ z is the Jacobian of f .
In general, computing the log determinant has a time cost of
O ( D 3 ) .
Much work has gone into developing restricted neural network architectures which make computing the Jacobian’s determinant more tractable.
These approaches broadly fall into three categories:
Normalizing ﬂows.
By restricting the functional form of f , various determinant identities can be exploited (Rezende & Mohamed, 2015; Berg et al., 2018).
These models cannot be trained as generative models from data because they do not have a tractable inverse f − 1 .
However, they are useful for specifying approximate posteriors for variational inference (Kingma & Welling, 2014).
Autoregressive transformations.
By using an autoregressive model and specifying an ordering of the dimensions, the Jacobian of f is enforced to be lower triangular (Kingma et al., 2016; Oliva et al., 2018).
These models excel at density estimation for tabular datasets (Papamakarios et al., 2017), but require D sequential evaluations of f to invert, which is prohibitive when D is large.
Partitioned transformations.
Partitioning the dimensions and using afﬁne transformations makes the determinant of the Jacobian cheap to compute, and the inverse f − 1 computable with the same cost as f (Dinh et al., 2014; 2017).
This method allows the use of convolutional architectures, excelling at density estimation for image data (Dinh et al., 2017; Kingma & Dhariwal, 2018).
Throughout this work, we refer to reversible generative models those that use the change of variables to transform a base distribution to the model distribution while maintaining both efﬁcient density estimation and efﬁcient sampling capabilities using a single pass of the model.
2. O THER G ENERATIVE M ODELS
There exist several approaches to generative modeling approaches which don’t use the change of variables equation for training.
Generative adversarial networks (GANs) (Goodfellow et al., 2014) use large, unrestricted neural networks to transform samples from a ﬁxed base distribution.
Lacking a closed-form likelihood, an auxiliary discriminator model must be trained to estimate divergences or density ratios in order to provide a training signal.
Autoregressive models (Germain et al., 2015;
Oord et al., 2016) directly specify the joint distribution p ( x ) as a sequence of explicit conditional distributions using the product rule.
These models require at least O ( D ) evaluations to sample from.
Variational autoencoders (VAEs) (Kingma & Welling, 2014) use an unrestricted architecture to explicitly specify the conditional likelihood p ( x | z ) , but can only efﬁciently provide a stochastic lower bound on the marginal likelihood p ( x ) .
2. C ONTINUOUS N ORMALIZING F LOWS
Chen et al.
(2018) deﬁne a generative model for data x ∈ R D similar to those based on (1), but which replaces the warping function with an integral of continuous-time dynamics.
The generative process works by ﬁrst sampling from a base distribution z 0 ∼ p z 0 ( z 0 ) .
Then, given an ODE deﬁned by the parametric function f ( z ( t ) , t ; θ ) , we solve the initial value problem z ( t 0 ) =
z 0 , ∂ z ( t ) / ∂t = f
( z ( t ) , t ; θ ) to obtain a data sample x
= z ( t 1 ) .
These models are called Continous Normalizing
Flows (CNF).
The change in log-density under this model follows a second differential equation, called the instantaneous change of variables formula (Chen et al., 2018):
[Equation]
We can compute total change in log-density by integrating across time:
[Equation]
[Table]
Sampling
[Table]
likelihood
[Table]
-form
Jacobian
Variational Autoencoders (cid:51) (cid:51) (cid:55) (cid:51)
Generative Adversarial Nets (cid:51) (cid:51) (cid:55) (cid:51)
Likelihood-based Autoregressive (cid:51) (cid:55) (cid:51) (cid:55)
C h a ng e o f
V a r i a b l e s
Normalizing Flows (cid:55) (cid:51) (cid:51) (cid:55)
Reverse-NF, MAF, TAN (cid:51) (cid:55) (cid:51) (cid:55)
NICE, Real NVP, Glow, Planar CNF (cid:51) (cid:51) (cid:51) (cid:55)
FFJORD (cid:51) (cid:51) (cid:51) (cid:51)
Given a datapoint x , we can compute both the point z 0 which generates x , as well as log p ( x ) under the model by solving the combined initial value problem:
[Equation]
dt (cid:124) [Equation] (cid:125) dynamics , (cid:20) z ( t 1 ) log p ( x ) − log p (
[Equation]
( t
[Equation]
[Table] [Equation] [Equation] which integrates the combined dynamics of z ( t ) and the log-density of the sample backwards in time from t 1 to t 0 .
We can then compute log p ( x ) using the solution of (4) and adding log p z 0 ( z 0 ) .
The existence and uniqueness of (4) require that f and its ﬁrst derivatives be Lipschitz continuous (Khalil, 2002), which can be satisﬁed in practice using neural networks with smooth Lipschitz activations, such as softplus or tanh.
2.2. B ACKPROPAGATING THROUGH ODE S OLUTIONS WITH THE A DJOINT M ETHOD
CNFs are trained to maximize (3).
This objective involves the solution to an initial value problem with dynamics parameterized by θ .
For any scalar loss function which operates on the solution to an initial value problem
[Equation]
then Pontryagin (1962) shows that its derivative takes the form of another initial value problem dL
[Equation] [Equation]
The quantity − ∂L / ∂
z ( t ) is known as the adjoint state of the ODE.
Chen et al.
(2018) use a black-box
ODE solver to compute z ( t 1 ) , and then a separate call to a solver to compute (6) with the initial value ∂L / ∂
z ( t 1 ) .
This approach is a continuous-time analog to the backpropgation algorithm
(Rumelhart et al., 1986; Andersson, 2013) and can be combined with gradient-based optimization to ﬁt the parameters θ by maximum likelihood.
S CALABLE DENSITY EVALUATION WITH UNRESTRICTED ARCHITECTURES
Switching from discrete-time dynamics to continuous-time dynamics reduces the primary computational bottleneck of normalizing ﬂows from O ( D 3 ) to O
( D 2 ) , at the cost of introducing a numerical
ODE solver.
This allows the use of more expressive architectures.
For example, each layer of the original normalizing ﬂows model of Rezende & Mohamed (2015) is a one-layer neural network with only a single hidden unit.
In contrast, the instantaneous transformation used in planar continuous normalizing ﬂows (Chen et al., 2018) is a one-layer neural network with many hidden units.
In this section, we construct an unbiased estimate of the log-density with O ( D ) cost, allowing completely unrestricted neural network architectures to be used.
3. U NBIASED L INEAR - TIME L OG -D ENSITY E STIMATION
In general, computing Tr ( ∂f / ∂
z ( t ) ) exactly costs O
( D 2 ) , or approximately the same cost as D evaluations of f , since each entry of the diagonal of the Jacobian requires computing a separate derivative of f .
However, there are two tricks that can help.
First, vector-Jacobian products v T ∂f∂ z can be computed for approximately the same cost as evaluating f using reverse-mode automatic differentiation.
Second, we can get an unbiased estimate of the trace of a matrix by taking a double product of that matrix with a noise vector:
[Equation]
The above equation holds for any D -by- D matrix A and distribution p ( (cid:15) ) over D -dimensional vectors such that E [ (cid:15) ]
=
and Cov( (cid:15) )
=
I .
The Monte Carlo estimator derived from (7) is known as
Hutchinson’s trace estimator (Hutchinson, 1989; Adams et al., 2018).
To keep the dynamics deterministic within each call to the ODE solver, we can use a ﬁxed noise vector (cid:15) for the duration of each solve without introducing bias:
[Equation]
Typical choices of p ( (cid:15) ) are a standard Gaussian or Rademacher distribution (Hutchinson, 1989).
3.1. R EDUCING V ARIANCE WITH B OTTLENECK C APACITY
Often, there exist bottlenecks in the architecture of the dynamics network, i.e. hidden layers whose width H is smaller than the dimensions of the input D .
In such cases, we can reduce the variance of Hutchinson’s estimator by using the cyclic property of trace.
Since the variance of the estimator for Tr( A ) grows asymptotic to ||
A || 2 F (Hutchinson, 1989), we suspect that having fewer dimensions should help reduce variance.
If we view the dynamics as a composition of two functions f = g ◦ h
( z ) then we observe
[Equation]
When f has multiple hidden layers, we choose H to be the smallest dimension.
This bottleneck trick can reduce the norm of the matrix which may also help reduce the variance of the trace estimator.
As introducing a bottleneck limits our model capacity, we do not use this trick in our experiments.
However this trick can reduce variance when a bottleneck is used, as shown in our ablation studies.
3. FFJORD: A C ONTINUOUS - TIME R EVERSIBLE G ENERATIVE M ODEL
Our complete method uses the dynamics deﬁned in (2) and the efﬁcient log-likelihood estimator of (8) to produce the ﬁrst scalable and reversible generative model with an unconstrained Jacobian.
This call this method Free-Form Jacobian of Reversible Dyanamics (FFJORD).
Pseudo-code of our method is given in Algorithm 1, and Table 1 summarizes the capabilities of our model compared to other modeling approaches.
Assuming the cost of evaluating f is on the order of O ( DH ) where D is the dimensionality of the data and H is the size of the largest hidden layer in f , then the cost of computing the likelihood in models with repeated use of invertible transformations (1) is O
(( DH + D 3 ) L ) where L is the number of transformations used.
For CNF, this reduces to O (( DH + D 2 )ˆ L ) for CNFs, where ˆ L is the number of evaluations of f used by the ODE solver.
With FFJORD, this reduces further to
O (( DH + D )ˆ L ) .
Require: dynamics f θ , start time t 0 , stop time t 1 , data samples x , data dimension D .
(cid:15) ← sample unit variance( x .shape)
(cid:46) Sample (cid:15) outside of the integral function
f aug ( [ z t , log p t ] , t ):
(cid:46) Augment f with log-density dynamics.
f t ← f θ ( z ( t ) , t ) (cid:46) Evaluate neural network g ← (cid:15) T ∂f∂ z (cid:12)(cid:12)
z ( t ) (cid:46)
Compute vector-Jacobian product with automatic differentiation (cid:101)
Tr = g (cid:15) (cid:46) Unbiased estimate of Tr( ∂f∂ z ) with (cid:15) T ∂f∂ z (cid:15) return [ f t , − (cid:101)
Tr ] (cid:46) Concatenate dynamics of state and log-density end function [ z 0 , ∆ logp ] ← odeint( f aug , [ x ,(cid:126) 0] , t 0 , t 1 ) (cid:46) Solve the ODE (cid:82) t 1 t 0 f
aug ([ z ( t ) , log p ( z ( t ))] , t ) dt log ˆ p ( x ) ← log p z 0 ( z 0 ) - ∆ logp (cid:46)
Add change in log-density return log ˆ p ( x )
E XPERIMENTS
[Figure]
We demonstrate FFJORD on a variety of density estimation tasks, and for approximate inference in variational autoencoders (Kingma &
Welling, 2014).
Experiments were conducted using a suite of GPU-based ODE-solvers and an implementation of the adjoint method for backpropagation 1 .
In all experiments the RungeKutta 4(5) algorithm with the tableau from
Shampine (1986) was used to solve the ODEs.
We ensure tolerance is set low enough so numerical error is negligible; see Appendix C.
We used Hutchinson’s trace estimator (7) during training and the exact trace when reporting test results.
This was done in all experiments except for our density estimation models trained on MNIST and CIFAR10 where computing the exact Jacobian trace was too expensive.
The dynamics of FFJORD are deﬁned by a neural network f which takes as input
the current state z ( t ) ∈
R D and the current time t ∈ R .
We experimented with several ways to incorporate t as an input to f , such as hyper-networks, but found that simply concatenating t on to z ( t ) at the input to every layer worked well and was used in all of our experiments.
4. D ENSITY E STIMATION O N T OY 2D D ATA
We ﬁrst train on 2 dimensional data to visualize the model and the learned dynamics.
In Figure 2, we show that by warping a simple isotropic Gaussian, FFJORD can ﬁt both multi-modal and even discontinuous distributions.
The number of evaluations of the ODE solver is roughly 70-100 on all datasets, so we compare against a Glow model with 100 discrete layers.
The learned distributions of both FFJORD and Glow can be seen in Figure 2.
Interestingly, we ﬁnd that Glow learns to stretch the single mode base distribution into multiple modes but has trouble modeling the areas of low probability between disconnected regions.
In contrast, FFJORD is capable of modeling disconnected modes and can also learn convincing approximations of discontinuous density functions (middle row in Figure 2).
Since the main beneﬁt of FFJORD is the ability to train with deeper dynamics networks, we also compare against planar CNF
(Chen et al., 2018) which can
[Figure]
[Table]
be viewed as a single hidden layer network.
Without the of a ﬂexible network planar complex distributions
4. D ENSITY E STIMATION ON R EAL D ATA
We perform density estimation on ﬁve tabular datasets preprocessed as in Papamakarios et al.
(2017) and two image datasets; MNIST and CIFAR10.
When reproducing Glow, we use the same conﬁgurations for Real NVP as Papamakarios et al.
and add invertible fully connected layer between all coupling layers.
On the tabular datasets, FFJORD performs the best out of reversible models by a wide margin but is outperformed by recent autoregressive models.
Of those, FFJORD outperforms
MAF (Papamakarios et al., 2017) on all but one dataset and manages to outperform TAN Oliva et al.
(2018) on the MINIBOONE dataset.
These models require O ( D ) sequential computations to sample from while the best performing method, MAF-DDSF (Huang et al., 2018), cannot be sampled from without resorting to correlated or expensive sampling algorithms such as MCMC.
On MNIST we ﬁnd that FFJORD can model the data as effectively as Glow and Real NVP using only a single ﬂow deﬁned by a single neural network.
This is in contrast to Glow and Real NVP which must compose many ﬂows to achieve similar performance.
When we use multiple ﬂows in a multiscale architecture (like those used by Glow and Real NVP) we obtain better performance on
MNIST and comparable performance to Glow on CIFAR10.
Notably, FFJORD is able to achieve this performance while using less than 2% as many parameters as Glow.
We also note that Glow uses a learned base distribution whereas FFJORD and Real NVP use a ﬁxed Gaussian.
A summary of our results on density estimation can be found in Table 2 and samples can be seen in Figure 3.
Full details on architectures used, our experimental procedure, and additional samples can be found in Appendix B.1.
In general, our approach is slower than competing methods, but we ﬁnd the memory-efﬁciency of the adjoint method allows us to use much larger batch sizes than those methods.
On the tabular datasets we used a batch sizes up to 10,000 and on the image datasets we used a batch size of 900.
4. V ARIATIONAL A UTOENCODER
We compare FFJORD to other normalizing ﬂows for use in variational inference.
We train a
VAE (Kingma & Welling, 2014) on four datasets using a FFJORD ﬂow and compare to VAEs with no ﬂow, Planar Flows
[Table]
In VAEs it is common for the encoder network to also output the parameters of the ﬂow as a function of the input x .
With FFJORD, we found this led to differential equations which were too difﬁcult to integrate numerically.
Instead, the encoder network outputs a low-rank update to a global weight matrix and an input-dependent bias vector.
When used in recognition nets, neural network layers deﬁning the dynamics inside FFJORD take the form
[Equation]
D out ×+ ˆ b ( x ) (cid:124)(cid:123)(cid:122)(cid:125)
D out × 
(10) where h is the input to the layer, σ is an element-wise activation function, D in and D out are the input and output dimension of this layer, and ˆ U ( x ) , ˆ V ( x ) , ˆ b ( x ) are input-dependent parameters returned from an encoder network.
A full description of the model architectures used and our experimental setup can be found in Appendix B.2.
On every dataset tested, FFJORD outperforms all other competing normalizing ﬂows.
A summary of our variational inference results can be found in Table 3.
A NALYSIS AND D ISCUSSION
[Figure]
We performed a series of ablation experiments to gain a better understanding of the proposed model.
5. F ASTER T RAINING WITH B OTTLENECK T RICK
We plotted the training losses on MNIST using an encoder-decoder architecture (see Appendix B.1 for details).
Loss during training is plotted in Figure 4, where we use the trace estimator directly on the D × D Jacobian, or we use the bottleneck trick to reduce the dimension to
H × H .
Interestingly, we ﬁnd that while the bottleneck trick (9) can lead to faster convergence when the trace is estimated using a Gaussian-distributed (cid:15) , we did not observe faster convergence when using a Rademacherdistributed (cid:15) .
5. N UMBER OF F UNCTION E VALUATIONS VS . D ATA D IMENSION
The full computational cost of integrating the instantaneous change of variables (2) is O ( DH (cid:98) L ) where D is dimensionality of the data, H is the size of the hidden state, and (cid:98) L is the number of function evaluations (NFE) that the adaptive solver uses to integrate the ODE.
In general, each evaluation of the model is O ( DH ) and in practice, H is typically chosen to be close to D .
Since the general form of the discrete change of variables equation (1) requires O ( D 3 ) -cost, one may wonder whether the number of evaluations (cid:98) L depends on D .
[Figure]
We train VAEs using FFJORD ﬂows with increasing latent dimension D .
The NFE throughout training is shown in Figure 5.
In all models, we ﬁnd that the NFE increases throughout training, but converges to the same value, independent of D .
We conjecture that the number of evaluations is not dependent on the dimensionality of the data but the complexity of its distribution, or more speciﬁcally, how difﬁcult it is to transform its density into the base distribution.
5. S INGLE - SCALE VS . M ULTI - SCALE FFJORD
[Figure]
Crucial to the scalability of Real NVP and Glow is the multiscale architecture originally proposed in Dinh et al.
We compare a single-scale encoder-decoder style FFJORD with a multiscale FFJORD on the MNIST dataset where both models have a comparable number of parameters and plot the total NFE–in both forward and backward passes–against the loss achieved in Figure 6.
We ﬁnd that while the single-scale model uses approximately one half as many function evaluations as the multiscale model, it is not able to achieve the same performance as the multiscale model.
S COPE AND L IMITATIONS
Number of function evaluations can be prohibitive.
The number of function evaluations required to integrate the dynamics is not ﬁxed ahead of time, and is a function of the data, model architecture, and model parameters.
This number tends to grow as the models trains and can become prohibitively large, even when memory stays constant due to the adjoint method.
Various forms of regularization such as weight decay and spectral normalization (Miyato et al., 2018) can be used to reduce the this quantity, but their use tends to hurt performance slightly.
Limitations of general-purpose ODE solvers.
In theory, our model can approximate any differential equation (given mild assumptions based on existence and uniqueness of the solution), but in practice our reliance on general-purpose ODE solvers restricts us to non-stiff differential equations that can be efﬁciently solved.
ODE solvers for stiff dynamics exist, but they evaluate f many more times to achieve the same error.
We ﬁnd that a small amount of weight decay regularizes the ODE to be sufﬁciently non-stiff.
C ONCLUSION
We have presented FFJORD, a reversible generative model for high-dimensional data which can compute exact log-likelihoods and can be sampled from efﬁciently.
Our model uses continuoustime dynamics to produce a generative model which is parameterized by an unrestricted neural network.
All required quantities for training and sampling can be computed using automatic differentiation, Hutchinson’s trace estimator, and black-box ODE solvers.
Our model stands in contrast to other methods with similar properties which rely on restricted, hand-engineered neural network architectures.
We demonstrated that this additional ﬂexibility allows our approach to achieve on-par or improved performance on density estimation and variational inference.
We believe there is much room for further work exploring and improving this method.
FFJORD is empirically slower to evaluate than other reversible models like Real NVP or Glow, so we are interested speciﬁcally in ways to reduce the number of function evaluations used by the ODE-solver without hurting predictive performance.
Advancements like these will be crucial in scaling this method to even higher-dimensional datasets.
A CKNOWLEDGEMENTS
We thank Roger Grosse and Yulia Rubanova for helpful discussions.
A PPENDIX A Q UALITATIVE S AMPLES
Samples from our FFJORD models trained on MNIST and CIFAR10 can be found in Figure 7.
[Figure]
l e s
D a t a
A PPENDIX B E XPERIMENTAL D ETAILS AND A DDITIONAL R ESULTS B. D ENSITY E STIMATION
On the tabular datasets we performed a grid-search over network architectures.
We searched over models with 1, 2, 5, or 10 ﬂows with 1, 2, 3, or 4 hidden layers per ﬂow.
Since each dataset has a different number of dimensions, we searched over hidden dimensions equal to 5, 10, or 20 times the data dimension (hidden dimension multiplier in Table 4).
We tried both the tanh and softplus nonlinearities.
The best performing models can be found in the Table 4.
On the image datasets we experimented with two different model architectures; a single ﬂow with an encoder-decoder style architecture and a multiscale architecture composed of multiple ﬂows.
While they were able to ﬁt MNIST and obtain competitive performance, the encoder-decoder architectures were unable to ﬁt more complicated image datasets such as CIFAR10 and Street View
House Numbers.
The architecture for MNIST which obtained the results in Table 2 was composed of four convolutional layers with 64 → 64 → 128 → 128 ﬁlters and down-sampling with strided convolutions by two every other layer.
There are then four transpose-convolutional layers who’s
The multiscale architectures were inspired by those presented in Dinh et al.
We compose multiple ﬂows together interspersed with “squeeze” operations which down-sample the spatial resolution of the images and increase the number of channels.
These operations are stacked into a “scale block” which contains N ﬂows, a squeeze, then N ﬂows.
For MNIST we use 3 scale blocks and for CIFAR10 we use 4 scale blocks and let N = 2 for both datasets.
Each ﬂow is deﬁned byconvolutional layers with 64 ﬁlters and a kernel size of 3.
The softplus nonlinearity is used in all layers.
Both models were trained with the Adam optimizer (Kingma & Ba, 2015).
We trained forepochs with a learning rate of .001 which was decayed to .0001 after 250 epochs.
Training took place on six GPUs and completed after approximately ﬁve days.
B. V ARIATIONAL A UTOENCODER
Our experimental procedure exactly mirrors that of Berg et al.
We use the same 7-layer encoder and decoder, learning rate (.001), optimizer (Adam Kingma & Ba (2015)), batch size (100), and early stopping procedure (stop after 100 epochs of no validaiton improvment).
The only difference was in the nomralizing ﬂow used in the approximate posterior.
We performed a grid-search over neural network architectures for the dynamics of FFJORD.
We searched over networks with 1 and 2 hidden layers and hidden dimension 512, 1024, and 2048.
We used ﬂows with 1, 2, or 5 steps and wight matrix updates of rank 1, 20, and 64.
We use the softplus activation function for all datasets except for Caltech Silhouettes where we used tanh.
The best performing models can be found in the Table 5.
Models were trained on a single GPU and training took between four hours and three days depending on the dataset.
Dataset nonlinearity
[Table] [Table]
B. S TANDARD D EVIATIONS FOR T ABULAR D ENSITY E STIMATION
[Table]
A PPENDIX C N UMERICAL E RROR FROM THE ODE S OLVER
ODE solvers are numerical integration methods so there is error inherent in their outputs.
Adaptive solvers (like those used in all of our experiments) attempt to predict the errors that they accrue and modify their step-size to reduce their error below a user set tolerance.
It is important to be aware of this error when we use these solvers for density estimation as the solver outputs the density that we report and compare with other methods.
When tolerance is too low, we run into machine precision errors.
Similarly when tolerance is too high, errors are large, our training objective becomes biased and we can run into divergent training dynamics.
Since a valid probability density function integrates to one, we take a model trained on Figure 1 and numerically ﬁnd the area under the curve using Riemann sum and a very ﬁne grid.
We do this for a range of tolerance values and show the resulting error in Figure 8.
We set both atol and rtol to the same tolerance.
[Figure]
The numerical error follows the same order as the tolerance, as expected.
During training, we ﬁnd that the error becomes non-negligible when using tolerance values higher than 10 − 5 .
For most of our experiments, we set tolerance to 10 − 5 as that gives reasonable performance while requiring few number of evaluations.
For the tabular experiments, we use atol = 10 − 8 and rtol = 10 − 6 .