O N THE L ATENT S PACE OF F LOW - BASED M ODELS
Anonymous authors
Paper under double-blind review
A BSTRACT
Flow-based generative models typically deﬁne a latent space with dimensionality identical to the observational space.
In many problems, however, the data does not populate the full ambient data-space that they natively reside in, but rather inhabit a lower-dimensional manifold.
In such scenarios, ﬂow-based models are unable to represent data structures exactly as their density will always have support off the data manifold, potentially resulting in degradation of model performance.
In addition, the requirement for equal latent and data space dimensionality can unnecessarily increase model complexity for contemporary ﬂow models.
Towards addressing these problems, we propose to learn a manifold prior that affords beneﬁts to both the tasks of sample generation and representation quality.
An auxiliary product of our approach is that we are able to identify the intrinsic dimension of the data distribution.
I NTRODUCTION
Normalizing ﬂows (Rezende and Mohamed, 2015; Kobyzev et al., 2020) have shown considerable potential for the tasks of modelling and inferring expressive distributions through the learning of well-speciﬁed probabilistic models.
Contemporary ﬂow-based approaches deﬁne a latent space with dimensionality identical to the data space, typically by parameterizing a complex model p X ( x | θ ) using an invertible neural network f θ .
Samples drawn from an initial, simple distribution p Z ( z ) ( e.g .
Gaussian) can be mapped to a complex distribution as x = f θ ( z ) .
The process results in a tractable density that inhabits the full data space.
However, contemporary ﬂow models may make for an inappropriate choice to represent data that resides in a lower-dimensional manifold and thus does not populate the full ambient space.
In such cases, the estimated model will necessarily have mass lying off the data manifold, which may result in under-ﬁtting and poor generation qualities.
Furthermore, principal objectives such as Maximum Likelihood Estimation (MLE) and Kullback-Leibler (KL) divergence minimization are ill-deﬁned, bringing additional challenges for model training.
In this work, we propose a principled strategy to model a data distribution that lies on a continuous manifold and we additionally identify the intrinsic dimension of the data manifold.
Speciﬁcally, by using the connection between MLE and KL divergence minimization in Z space, we can address the important problem of ill-deﬁned KL divergence under typical ﬂow based assumptions.
Flow models are based on the idea of “change of variable”.
Assume a random variable Z with distribution P Z and probability density p
Z ( z ) .
We can transform Z to get a random variable X :
X = f ( Z ) , where f : R D → R D is an invertible function with inverse f − 1 = g .
Suppose X has distribution P X and density function p X ( x ) , then log p X ( x ) will have the following form
[Equation]
where log (cid:12)(cid:12)(cid:12) det (cid:16) ∂g∂x
(cid:17)(cid:12)(cid:12)(cid:12) is the log determinant of the Jacobian matrix.
We call f (or g ) a volumepreserving function if the log determinant is equal to 0.
Training of ﬂow models typically makes use of MLE.
We denote X d as the random variable of the data with distribution P d and density p
d ( x ) .
In addition to the well-known connection between MLE and minimization of the KL divergence KL( p d ( x ) ||
p
X ( x )) in X space (see Appendix A for detail),
MLE is also (approximately) equivalent to minimizing the KL divergence in Z space, this is due to the KL divergence is invariant under invertible transformation (Yeung, 2008;
Papamakarios et al.,
Speciﬁcally, we deﬁne Z Q : Z Q = g ( X d ) with distribution Q Z and density function q ( z ) , the KL divergence in Z space can be written as
[Equation]
The full derivation can be found in Appendix A. Since we can only access samples x 1 , x 2 , . . .
, x N from p d ( x ) , we approximate the integral by Monte Carlo sampling
[Equation]
We thus highlight the connection between MLE and KL divergence minimization, in Z space, for ﬂow based models.
The prior distribution p ( z ) is usually chosen to be a D -dimensional Gaussian distribution.
However, if the data distribution P d is singular, for example a measure on a low dimensional manifold, the induced latent distribution Q Z is also singular.
In this case, the KL divergence in equation 2 is typically not well-deﬁned under the considered ﬂow based model assumptions.
This issue brings both theoretical and practical challenges that we will discuss in the following section.
F LOW MODELS WITH MANIFOLD DATA
We assume a data sample x ∼ P d to be a D dimensional vector x ∈ R D and deﬁne the ambient dimensionality of P d , denoted by Amdim ( P d ) , to be D .
However for many datasets of interest, e.g . natural images, the data distribution P d is commonly believed to be supported on a lower dimensional manifold (Beymer and Poggio, 1996).
We assume the dimensionality of the manifold to be K where K < D , and deﬁne the intrinsic dimension of P d , denoted by Indim ( P d ) , to be the dimension of this manifold.
Figure 1a provides an example of this setting where P d is a 1D distribution in 2D space.
Speciﬁcally, each data sample x ∼ P d is a 2D vector x = { x 1 , x 2 } where x 1 ∼ N (0 , 1) and x 2 = sin(2 x 1 ) .
Therefore, this example results in Amdim ( P d ) = 2 and Indim ( P d ) = 1 .
In ﬂow-based models, function f is constructed such that it is both bijective and differentiable.
When the prior P Z is a distribution whose support is R D ( e.g . Multivariate Gaussian distribution), the marginal distribution P X will also have support R D and Amdim ( P X ) =
Indim ( P X ) =
D .
When the support of the data distribution lies on a K -dimensional manifold and K < D , P d and P X are constrained to have different support.
That is, the intrinsic dimensions of P X and P d are always different; Indim ( P X ) (cid:54) =
Indim ( P d ) .
In this case it is impossible to learn a model distribution P X identical to the data distribution P d .
Nevertheless, ﬂow-based models have shown strong empirical success in real-world problem domains such as the ability to generate high quality and realistic images (Kingma and Dhariwal, 2018).
Towards investigating the cause and explaining this disparity between theory and practice, we employ a toy example to provide intuition for the effects and consequences resulting from model and data distributions that possess differing intrinsic dimension.
Consider the toy dataset introduced previously; a 1D distribution lying in a 2D space (Figure 1a).
The prior density p ( z ) is a standard 2D Gaussian p ( z ) =
N (0 , I Z ) and the function f is a nonvolume preserving ﬂow with two coupling layers (see Appendix C.1).
In Figure 1b we plot samples from the ﬂow model; the sample x is generated by ﬁrst sampling a 2D datapoint z ∼ N (0 , I Z ) and then letting x = f ( z ) .
Figure 1c shows samples from the prior distributions P Z and Q Z .
Q Z is deﬁned as the transformation of P d using the bijective function g , such that Q Z is constrained to support a 1D manifold in 2D space, and Indim ( Q Z ) =
Indim ( P d ) = 1 .
Training of Q Z to match P
Z (which has intrinsic dimension 2), can be seen in Figure 1c to result in curling up of the manifold in the latent space, contorting it towards satisfying a distribution that has intrinsic dimension 2.
This ill-behaved phenomenon causes several potential problems for contemporary ﬂow models:
Poor sample quality.
Figure 1b shows examples where incorrect assumptions in turn result in the model generating bad samples.
Low quality data representations.
The discussed characteristic that results in “curling up” of the latent space may cause degradations of the representation quality.
Inefﬁcient use of network capacity .
Neural network capacity is spent on contorting the distribution Q Z to satisfy imposed dimensionality constraints.
A natural solution to the problem of intrinsic dimension mismatch is to select a prior distribution P Z with the same dimensionality as the intrinsic dimension of the data distribution such that:
Indim ( P Z ) =
Indim ( P d ) .
However, since we do not know Indim ( P d ) explicitly, one option involves to instead learn it from the data distribution.
In the following section, we will introduce a parameterization approach that enables us to learn Indim ( P d ) .
[Figure]
L EARNING A M ANIFOLD P RIOR
Consider a data vector x ∈ R D , then a ﬂow based model prior P Z is usually given by a Ddimensional Gaussian distribution or alternative simple distribution that is also absolutely continuous (a.c.) in R D .
Therefore, the intrinsic dimension Indim ( P Z ) =
D .
To allow a prior to have intrinsic dimension strictly less than D , we let P Z have the ‘ generalized density ’ 1 p ( z ) with the form
[Equation]
where z ∈ R D and A is a D × D lower triangular matrix with D ( D +1) parameters, such that AA T is constrained to be a positive semi-deﬁnite matrix.
When AA T has full rank D , then P Z is a (nondegenerate) multivariate Gaussian on R D .
When Rank ( AA T ) = K and K < D , then P Z will degenerate to a Gaussian supported on a K -dimensional manifold, such that the intrinsic dimension
Indim ( P Z ) =
K .
Figure 2 illustrates a sketch of this scenario.
In practice, we initialize A to be an identity matrix, so AA T will also be an identity matrix and P Z is initialized as a standard Gaussian.
When Rank ( AA T ) < D , the degenerate covariance AA T is no longer invertible and we are unable to evaluate the density value of p ( z ) for a given random vector z .
Furthermore, when the data distribution P d is supported on a K -dimensional manifold, Q Z will also be supported on a Kdimensional manifold and no longer has a valid density function.
Using equation 2 to train the ﬂow model then becomes impossible as the KL divergence between P Z and Q Z is not well deﬁned 2 .
Recent work by Zhang et al.
(2020) proposed a new family of divergence to address this problem.
In the following section we brieﬂy review the key concepts pertaining to this family of divergences.
F IX THE I LL - DEFINED KL DIVERGENCE
random Let Z Q and Z P be two random variables with distribution Q Z and P Z .
The KL divergence between Q Z and P Z is not deﬁned if Q Z or P Z does not have valid density function.
Let K be an a.c. random variable that is independent of Z Q and Z P and has density p K , We deﬁne Z ˜ P =
Z P + K ; Z ˜ Q = Z Q + K with distributions ˜ P Z and ˜ Q Z
respectively.
Then ˜ P Z and ˜ Q Z are a.c. with density functions
[Equation]
We can thus deﬁne the spread KL divergence between Q Z and P Z as the KL divergence between ˜ Q
Z and ˜ P Z as:
[Equation]
In this work we let K be a Gaussian with diagonal covariance σ 2 Z I to satisfy the sufﬁcient conditions
such that (cid:102) KL is a valid divergence (see Zhang et al.
(2020) for details)
and has the properties:
[Equation]
Since Q Z and P Z are transformed from P d and P X using an invertible function g , we have
[Equation]
Therefore, the spread KL divergence can be used to train ﬂow based models with a manifold prior in order to ﬁt a dataset that lies on a lower-dimensional manifold.
I DENTIFIABILITY OF INTRINSIC DIMENSION
A byproduct of our model is that the intrinsic dimension of the data manifold can be identiﬁed.
Section 4 shows that when (cid:102) KL( Q Z || P Z ) = 0 ⇔ P X
= P d , the supports of P X and P d will also have the same intrinsic dimension: Indim ( P X ) =
Indim ( P d ) .
The ﬂow function g , and its inverse g = f − 1 , are bijective and continuous so f is a diffeomorphism (Kobyzev et al., 2020).
Due to the invariance of dimension property of diffeomorphisms (Lee, 2013, Theorem 2.17), the manifold that supports P Z will have the same dimension as the manifold that supports P X .
Therefore, we have
[Equation]
Since the intrinsic dimension of P Z is equal to the rank of the matrix AA T , we can calculate
Rank ( AA T ) by counting the number of non-zero eigenvalues of the matrix AA T .
This allows for identiﬁcation of the intrinsic dimension of the data distribution as
[Equation]
We have shown that we can identify the intrinsic dimension of P d using the spread KL divergence.
In the next section, we will discuss how to estimate the spread KL divergence in practice.
E STIMATION OF THE SPREAD KL DIVERGENCE
Our goal is to minimize the spread KL divergence between Q Z and P Z .
Using our deﬁnition of spread divergence (equation 7), this is equivalent to minimizing
[Equation]
where q (˜ z ) and p (˜ z ) are deﬁned in equation 6.
We separate the objective into two terms and now discuss the estimation for each of them.
Term 1: We use H( · ) to denote the differential entropy.
Term 1 is the negative entropy − H( Z ˜ Q ) .
For a volume preserving g and X d is a.c., the entropy H( Z Q ) =
H( X d ) and is independent of the model parameters and can be ignored during training.
However, the entropy H( Z ˜ Q )
=
H( Z Q + K ) will still depend on g , see Appendix B.1 for an example.
We claim that when the variance of K is small, the dependency between the H( Z ˜ Q ) and volume preserving function g is weak, thus we can approximate equation 12 by leaving out term 1 and will not affect the training.
To build intuitions, we ﬁrst assume
X d is a.c., so Z Q = g ( X d ) is also a.c.. Using standard entropic properties (Kontoyiannis and Madiman, 2014), we can pose the following relationship
[Equation]
where I( · , · ) denotes the mutual information.
Since Z Q is independent of function g and if σ 2 Z → 0 , then I( Z Q + K, K ) → 0 (see Appendix B.2 for a proof), the contribution of the I( Z Q + K, K ) term, with respect to training g , becomes negligible in the case of small σ 2 Z .
Unfortunately, equation 13 is no longer valid when P d lies on a manifold since Z Q will be a singular random variable and the differential entropy H( Z Q ) is not deﬁned.
In Appendix B.3, we show that leaving out the entropy H( Z ˜ Q ) corresponds to minimizing an upper bound of the spread KL divergence.
To further ﬁnd out the contribution of the negative entropy term, we compare between leaving out − H( Z ˜ Q ) and approximating − H( Z ˜ Q ) during training.
In Appendix B.4, we discuss the approximation technique and give the empirical evidence which shows that the ignoring − H( Z ˜ Q ) will not affect the training of our model.
Therefore, we make use of volume preserving g and small variance σ 2 Z = 1 × 10 − 4 in our experiments.
In contrast to other volume preserving ﬂows, that utilize a ﬁxed prior P Z , our method affords additional ﬂexibility by way of allowing for changes to the ‘volume’ of the prior towards matching the distribution of the target data.
In this way, our decision to employ volume-preserving ﬂow functions does not limit the expressive power or abilities of the model, in principle.
Popular non-volume preserving ﬂow structures, e.g. afﬁne coupling ﬂow, may also easily be normalized to become volume preserving, thus further extending the applicability of our approach (see Appendix C.1 for an example).
Term 2: The noisy prior p (˜ z ) is deﬁned to be a degenerate Gaussian N (0 , AA T ) , convolved with a
Gaussian noise N (0 , σ 2 Z I ) , and has a closed form density
[Equation]
Therefore, the log density log p (˜ z ) is well deﬁned, we can approximate term 2 by Monte Carlo
[Equation]
where q (˜ z )
= (cid:82) p (˜ z | z ) d Q Z .
To sample from q (˜ z ) , we ﬁrst get a data sample x ∼ P d , use function g to get z
= g ( x ) (so z is a sample of Q Z ) and ﬁnally sample ˜
z ∼ p (˜ z | z ) .
E XPERIMENTS
To demonstrate the effectiveness of our approach, Sections 7.1–7.4 report experiments on four datasets; toy 2D and 3D data, the fading square dataset and a synthesized adaptation of MNIST.
We use the Adam optimizer (Kingma and Ba, 2014) in all our experiments.
Our ﬂow networks are consisted of incompressible afﬁne coupling layer introduced by Sorrenson et al.
Dinh et al.
We compare a volume preserving ﬂow with a learnable prior (our method) to a non-volume preserving ﬂow with ﬁxed prior, so both models have the ability to adapt their ‘volume’ to ﬁt the target distribution and retains fair comparisons.
See Appendix C.1 for a detailed discussion of the incompressible afﬁne coupling layer and the network structures we use for all the experiments.
7. 2D TOY DATA
We ﬁrstly verify our method using the toy dataset described in Section 2 and Figure 1a.
The ﬂow function has two coupling layers.
We train our model using learning rate 3 × 10 − 4 and batch sizefor 10k iterations.
Figure 3 shows the samples from the model, the learned prior and the eigenvalues of AA T .
We observe the sample quality is better than that in Figure 1b and the prior has learned a degenerate Gaussian with Indim ( P Z ) = 1 , which matches Indim ( P d ).
In Appendix C.2, we show that our model can not only learn the manifold support of target distribution but also capture the ‘density’ allocation on the manifold.
[Figure]
S- CURVE DATASET
We ﬁt our model to the S-curve dataset shown in Figure 4a.
The data distribution lies on a 2D manifold in a 3D space, therefore Indim ( P d ) = 2 .
Speciﬁc network structure and training details can be found in Appendix C.1.
After training, our model learns a nonlinear function g to transform P d to Q Z , where the latter lies on a 2D linear subspace in 3D space (see Figure 4b).
Following this, a linear dimensionality reduction can be conducted to generate 2D data representations, we now brieﬂy outline a general procedure for this.
For Q Z with Amdim ( Q Z )
=
D and Indim ( Q Z )
=
K , we ﬁrst ﬁnd the eigenvectors e 1 , . .
, e D of
AA T , sorted by their eigenvalues.
When Rank ( AA T ) =
K ≤ D , there exist K eigenvectors with positive eigenvalues.
We select the ﬁrst K eigenvectors and form the matrix E =
[ e 1 , . .
, e K ] with dimension D × K .
We then transform each data sample x ∈ R D into Z space:
z = g ( x ) , such that z ∈ R D .
Afterwards, a linear projection is carried out z proj =
zE to obtain the lower dimensional representation z proj ∈ R K .
This procedure can be seen as a nonlinear dimensionality reduction, where the nonlinear component is solved using the learned function g .
We plot the resulting representations in Figure 4b.
The colormap indicates correspondence between the data in 3D space and the representation in 2D space.
We observe that our method can successfully (1) identify that the intrinsic dimension of the data is two and (2) project the data into a 2D space that faithfully preserves the structure of the original data distribution.
We also compare the sample generation quality with a ﬂow that has a ﬁxed Gaussian prior, see Appendix C.3 for details.
[Figure]
7. F ADING S QUARE DATASET
The fading square dataset (Rubenstein et al., 2018) was proposed in order to diagnose model behavior when data distribution and model possess differing intrinsic dimension and therefore constitutes a further relevant test bed for our current work.
The dataset consists of 32 × 32 pixel images with 6 × 6 grey squares on a black background.
The grey scale values are sampled from a uniform distribution with range [0 , 1] , so Indim ( P d ) = 1 .
Figure 5a shows the data samples.
We ﬁt our model to the dataset, the network structure and the training details can be found in Appendix C.
Figure 5b shows samples from our trained model.
Figure 5d shows the ﬁrst 20 eigenvalues of the
AA T (ranked from high to low), we can see only one eigenvalue is larger than zero and the others have converged to zero.
This illustrates that we have successfully identiﬁed the intrinsic dimension of P d .
We further carry out the dimensionality reduction process that was introduced in Section 7.2; the latent representation z is projected onto a 1D line and we visualize the correspondence between the projected representation and the data in Figure 5e.
Pixel grey-scale values can be observed to decay as the 1D representation is traversed from left to right, indicating that our representations are consistent with the properties of the original data distribution.
In contrast, we ﬁnd that the traditional ﬂow model, with a ﬁxed 1024D Gaussian p ( z ) , fails to learn such a data distribution, see Figure 5c.
[Figure]
7. MNIST DATA
We further investigate training of our model using images of digits.
However, for datasets like
MNIST (LeCun, 1998), the true intrinsic dimension is unknown.
In order to verify the correctness of our model’s ability to identify the intrinsic data dimension, we construct synthetic datasets by ﬁrst ﬁtting an implicit model p θ ( x ) = (cid:82) δ ( x − g
( z ) p ( z ))
d z to the MNIST dataset, and then use samples from the trained model x ∼ p θ ( x ) as training data.
The intrinsic dimension of the training dataset is the same as the dimension of the latent variable z in the implicit model Indim ( P d )
=
dim ( z ) .
We construct two datasets with dim ( z ) = 5 and dim ( z ) = 10 such that Indim ( P d ) = 5 and
Indim ( P d ) = 10 , respectively.
Further details on the implicit model, ﬂow network structure, training method and samples from the learned models are found in Appendix C.
In contrast to the fading square dataset, we ﬁnd that in order to successfully train the model ( i.e . such that valid image samples are generated), it is necessary to add small Gaussian noise to the training data.
This trick is commonly used in the training of ﬂow based models for image data (Sorrenson et al., 2020).
We note that adding Gaussian noise breaks the assumption that the data lies on a manifold and, alternatively, the intrinsic dimension of the training data distribution will be equal to its ambient dimension.
Towards alleviating this undesired effect, we ﬁrstly add Gaussian noise with standard deviation σ x = 0 .
05 and anneal σ x after 2000 k iterations with a factor of 0 .
9 every 10 k iterations.
However, to help retain successful model training, we disallow the annealing procedure to reach a state where zero noise is added due to the outlined model behavior observed when considering this image dataset.
Experimentally, we cap a lower-bound Gaussian noise level of 0 . 01 and leave further investigation of this phenomenon to future work.
In Figure 6a and 6b, we plot the ﬁrst 20 eigenvalues of the AA T (ranked from high to low) after training on two synthetic MNIST datasets with intrinsic dimension 5 and 10.
It can be observed that 5 and 10 eigenvalues are signiﬁcantly larger than the remaining values, respectively.
It can thus be concluded that the intrinsic dimension of the two datasets are 5 and 10.
Remaining non-zero eigenvalues can be attributed to the small Gaussian noise added to the training data.
For the original MNIST dataset, it was shown that digits have different intrinsic dimension (Costa and Hero, 2006).
This suggests the distribution of MNIST may lie on several, disconnected manifolds with differing intrinsic dimension.
Although our model assumes that P d lies on one continuous manifold, it is interesting to investigate the case when this model assumption is not fulﬁlled.
We thus ﬁt our model to the original MNIST data and plot the eigenvalues in Figure 6c.
In contrast to Figures 6a and 6b, the gap between eigenvalues can be seen to be less pronounced, with no obvious step change.
However the values suggest that the intrinsic dimension of MNIST is between 11 and 14 .
This result is consistent with previous estimations stating that the intrinsic dimension of MNIST is between 12 and 14 (Facco et al., 2017; Hein and Audibert, 2005; Costa and Hero, 2006).
Recent work Cornish et al.
(2019) discusses ﬁtting ﬂow models to a P d that lies on disconnected components, by introducing a mixing prior.
Such techniques may be easily combined with our method towards constructing more powerful ﬂow models; a promising direction for future work.
[Figure]
R ELATED W ORK
Classic latent variable generative models assume that data distributions lie around a low-dimensional manifold, for example the Variational Auto-Encoder (Kingma and Welling, 2013) or, recently introduced, Noisy Injective Flows (Cunningham et al., 2020).
Such methods typically assume that observational noise is not degenerated (e.g. a ﬁxed Gaussian distribution), therefore the model distribution is absolutely continuous and maximum likelihood learning is thus well deﬁned.
However, common distributions such as natural images usually don’t have Gaussian observational noise (Zhao et al., 2017).
Therefore, in this work, we focus on modeling distributions that lie on a low-dimensional manifold.
The study of manifold learning for nonlinear dimensionality reduction Cayton (2005) and intrinsic dimension estimation Camastra and Staiano (2016) is a rich ﬁeld with an extensive set of tools.
However, most methods commonly do not model data density on the manifold and are thus not used for the same purpose as the models introduced here.
There are however a number of recent works that introduced normalizing ﬂows on manifolds that we now highlight and relate to our approach.
Several works deﬁne ﬂows on manifolds with prescribed charts.
Gemici et al.
(2016) generalized ﬂows from Euclidean spaces to Riemannian manifolds by proposing to map points from the manifold
M to R K , apply a normalizing ﬂow in this space and then map back to M .
The technique has been further extended to Tori and Spheres (Rezende et al., 2020).
These methods require knowledge of the intrinsic dimension K and a parameterization of the coordinate chart of the data manifold.
Without providing a chart mapping a priori, M-ﬂow (Brehmer and Cranmer, 2020) recently proposed an algorithm that learns the chart mapping and distribution density simultaneously.
However, their method still requires that the dimensionality of the manifold is known.
They propose that the dimensionality can be learned either by a brute-force solution or through a trainable variance in the density function.
The brute-force solution is clearly infeasible for data embedded in extremely high dimensional space, as is often encountered in deep learning tasks.
Use of a trainable variance is natural and similar to our approach.
However, as discussed at the beginning of this paper, without carefully handling the KL or MLE term in the objective, a vanishing variance parameter will result in wild behavior of the optimization process since these terms are not well deﬁned.
While the GIN model considered in Sorrenson et al.
(2020) could recover the low dimensional generating latent variables following their identiﬁability theorem, the assumptions therein require knowledge of an auxiliary variable, e.g . the label, which is not required in our model.
Behind this superﬁcial difference is the essential discrepancy between the concept of informative dimensions and intrinsic dimensions.
The GIN model discovers the latent variables that are informative in a given context, deﬁned by the auxiliary variable u instead of the true intrinsic dimensions.
In their synthetic example, the ten dimensional data is a nonlinear transformation of ten dimensional latent variables where two out of ten are correlated with the labels of the data and the other eight are not.
In this example, there are two informative dimensions, but there are ten intrinsic dimensions.
Nevertheless, our method for intrinsic dimension discovery can be used together with informative dimension discovery methods to discover ﬁner structures of data.
C ONCLUSION
We presented a principled strategy to learn the data distribution that lies on a manifold and identify its intrinsic dimension.
We ﬁx the ill-deﬁned KL divergence and show, across multiple datasets, the resulting beneﬁts for ﬂow based models under both sample generation and representation quality.
There remain a number of open questions and interesting directions for future work.
Namely; further exploration of the effects of the entropy term in the case of non-volume preserving networks and, additionally, investigation of the phenomenon concerning the apparent necessity of noise addition in cases pertaining to complex real-world distributions e g .
image data
A M AXIMUM L IKELIHOOD ESTIMATION AND KL DIVERGENCE
Given data x 1 , x 2 , . . .
, x N sampled independently from the true data distribution P d , with density function p d ( x ) , we want to ﬁt the model density p ( x ) 3 to the data.
A popular choice to achieve this involves minimization of the KL divergence where:
[Equation]
Since we can only access samples from p d ( x ) , we approximate the integral by Monte Carlo sampling
[Equation]
Therefore, minimizing the KL divergence between the data distribution and the model is (approximately) equivalent to Maximum Likelihood Estimation (MLE).
When p ( x ) is a ﬂow-based model with invertible ﬂow function f : Z → X, g = f − 1 , minimizing the KL divergence in X space is equivalent to minimizing the KL divergence in the Z space.
We let
X d be the random variable of data distribution and deﬁne Z Q : Z Q = g
( X d ) with density q ( z ) , so q ( z ) can be represented as
[Equation]
[Equation]
Let p ( z ) be the density of the prior distribution P Z , the KL divergence in Z space can be written as
[Equation] [Equation]
Term 1: using the properties of transformation of random variable (Papoulis and Pillai, 2002, pp.
660), the negative entropy can be written as
[Equation]
Term2: the cross entropy can be written as
[Equation]
Therefore, the KL divergence in Z space is equivalent to the KL divergence in X space
[Equation]
[Equation]
We thus build the connection between MLE and minimizing the KL divergence in Z space.
B E NTROPY B. A N EXAMPLE
Assume a 2D Gaussian random variable X with covariance (cid:20)
[Equation]
, g is a volume preserving ﬂow with parameter θ .
Let Z 1 =
g θ 1 ( Z ) be a Gaussian with covariance (cid:20)   (cid:21) and Z 2 = g θ 2 ( X ) be a Gaussian with covariance (cid:20)   (cid:21) .
Therefore the entropy H( Z 1 ) =
H( Z 2 ) =
H( X ) and doesn’t not depend on θ .
Let K be an Gaussian with zero mean and covariance (cid:20) 10(cid:21) , so Z 1 + K is a
Gaussian with covariance (cid:20)   (cid:21) and Z 2 + K is a Gaussian with covariance (cid:20)   (cid:21) .
Therefore
H( Z 1 + K ) (cid:54) =
H( Z 2 + K ) and H( g θ ( X ) + K ) depends on θ .
A similar example can be constructed when X is not absolutely continuous.
B. Z IS AN ABSOLUTELY CONTINUOUS RANDOM VARIABLE
For two mutually independent absolutely continuous random variable Z and K , the mutual information between Z + K and K is
[Equation]
[Equation] [Equation]
The last equality holds because Z and K are independent.
Since mutual information I( Z + K, K ) ≥ 0 , we have
[Equation]
Assume K has a Gaussian distribution with 0 mean and variance σ 2 Z 4 .
When σ 2 Z = 0 , K degenerates to a delta function, so
Z + K
=
Z
and
[Equation]
(29) this is because the mutual information between an a.c. random variable and a singular random variable is still well deﬁned, see (Yeung, 2008, Theorem 10.33).
Assume K 1 , K 2 are Gaussian random variables with 0 mean and variances σ 21 and σ 22 respectively.
Without loss of generality, we assume σ 21 > σ 22 and σ 21 =
σ 22 + σ 2 δ , and let K δ be the random variable of a Gaussian that hasmean and variance σ 2 δ such that K 1 = K 2 + K δ .
By the data-processing inequality, we have
[Equation]
Therefore, I( Z + K, K ) is a monotonically decreasing function when σ 2 Z decreases and when σ 2 Z → 0 , I ( Z + K, K ) → 0 .
B. U PPER BOUND OF THE SPREAD KL DIVERGENCE
In this section, we show that leaving out the entropy term H( Z ˜ Q ) in equation 12 is equivalent to minimizing an upper bound of the spread KL divergence.
For singular random variable Z Q = g ( X d ) and absolutely continuous random variable K that are independent, we have
[Equation]
The second equation is from the deﬁnition of Mutual Information (MI); the MI between an a.c. random variable and a singular random variable is well deﬁned and always positive, see (Yeung, 2008, Theorem 10.33) for a proof.
Therefore, we can construct an upper bound of the spread KL objective in equation
[Equation]
Therefore, ignoring the negative entropy term during training is equivalent to minimizing an upper bound of the spread KL objective.
B. E MPIRICAL EVIDENCE FOR IGNORING THE NEGATIVE ENTROPY
In this section, we ﬁrst introduce the approximation technique to compute the negative entropy term, and then discuss the contribution of this term for the training.
The negative entropy of random variable Z ˜ Q is
[Equation]
where
[Equation]
) and p K is the density of a Gaussian with diagonal covariance σ 2 Z I .
We ﬁrst approximate ˜ q
(˜ z ) by a mixture of Gaussians
[Equation]
where z n is the n th sample from distribution Q z by ﬁrst sampling x n ∼ P d and letting z n = g ( x n ) .
We denote the random variable of this Gaussian mixture as ˆ Z N ˜ Q and approximate
[Equation]
However, the entropy of a Gaussian mixture distribution does not have a closed form, so we further conduct a ﬁrst order Taylor expansion approximation (Huber et al., 2008)
[Equation]
this approximation is accurate for small σ 2 Z .
Finally we have our approximation
[Equation]
To evaluate the contribution of the negative entropy, we train our ﬂow model on both low dimensional data (Toy datasets: 2D) and high dimensional data (Fading square dataset: 1024D) by optimization that uses two objectives: (1) ignoring the negative entropy term in equation 7 and (2) approximating the negative entropy term in equation 7 using the approximation discussed above.
During training, we keep tracking of the value of the entropy H( Z ˜ Q ) (using the approximation value) in both objectives.
We let N equal the batch size when approximating the entropy.
Additional training details remain consistent with those described in Appendix C.
To evaluate the contribution of the negative entropy, we train our ﬂow model on both low dimensional data (Toy datasets: 2D) and high dimensional data (Fading square dataset: 1024D) by optimization that uses two objectives: (1) ignoring the negative entropy term in equation 7 and (2) approximating the negative entropy term in equation 7 using the approximation discussed above.
During training, we keep tracking of the value of the entropy H( Z ˜ Q ) (using the approximation value) in both objectives.
We let N equal the batch size when approximating the entropy.
Additional training details remain consistent with those described in Appendix C.
In Figure 7, we plot the (approximated) entropy value H( Z ˜ Q ) during training for both experiments.
We ﬁnd the difference between having the (approximated) negative entropy term and ignoring the negative entropy to be negligible.
We leave theoretical investigation on the effects of leaving out the entropy term during training to future work.
[Figure]
C E XPERIMENTS C. N ETWORK A RCHITECTURE
The ﬂow network we use consists of incompressible afﬁne coupling layers (Sorrenson et al., 2020;
Dinh et al., 2016).
Each coupling layer splits a D -dimensional input x into two parts x 1: d and x d +1: D .
The output of the coupling layer is
[Equation]
where s : R d → R D − d and t : R d → R D − d are scale and translation functions parameterized by neural networks, (cid:12) is the element-wise product.
The log-determinant of the Jacobian of a coupling layer is the sum of the scaling function (cid:80)
j s ( x 1: d ) j .
To make the coupling transform volume preserving, we normalize the output of the scale function, so the i -th dimension of the output is
[Equation]
[Equation]
and the log-determinant of the Jacobian is (cid:80) i ˜ s ( x 1: d ) i = 0 .
We compare a volume preserving ﬂow with a learnable prior (normalized s ( · ) ) to a non-volume preserving ﬂow with ﬁxed prior (unnormalized s ( · ) ).
In this way both models have the ability to adapt their ‘volume’ to ﬁt the target distribution, retaining comparison fairness.
In our afﬁne coupling layer, the scale function s and the translation function t have two types of structure: fully connected net and convolution net.
Each fully connected network is a 4-layer neural network with hidden-size 24 and Leaky ReLU with slope 0.2.
Each convolution net is a 3-layer convolutional neural network with hidden channel size 16, kernel size 3 × 3 and padding size 1.
The activation is Leaky ReLU with slope 0.2.
The downsampling decreases the image width and height by a factor of 2 and increases the number of channels by 4 in a checkerboard-like fashion (Sorrenson et al., 2020; Jacobsen et al., 2018).
When multiple convolutional nets are connected together, we randomly permute the channels of the output of each network except the ﬁnal one.
In Table 1, 2, 3, 4, we show the network structures for our four main paper experiments.
[Table] [Table]
[Table] [Table]
C. T OY DATASET
We also construct a second dataset and train a ﬂow model using the same training procedure discussed in Section 7.1.
Figure 8a shows the samples from the data distribution P d , each data point is a 2D vector x
= [ x 1 , x 2 ] where x 1 ∼ N (0 , 1) and x 2 = x 1 , so Indim ( P d ) = 1 .
Figure 8f shows that the prior P Z has learned the true intrinsic dimension Indim ( P Z ) =
Indim ( P d ) = 1 .
We compare to samples drawn from a ﬂow model that uses a ﬁxed 2D Gaussian prior, with results shown in Figure 8.
We can observe, for this simple dataset, ﬂow with a ﬁx Gaussian prior can generate reasonable samples, but the ‘curling up’ behavior, discussed in the main paper, remains highly evident in the Z space, see Figure 8c.
We also plot the ‘density allocation’ on the manifold for the two toy datasets.
For example, for the data generation process x = [ x 1 , x 2 ] where x 1 ∼ p = N (0 , 1) and x 2 = x 1 , we use the density value p ( x = x 1 ) to indicate the ‘density allocation’ on the 1D manifold.
To plot the ‘density allocation’ of our learned model, we ﬁrst sample x s uniformly from the support of the data distribution, the subscript ‘s’ here means that they only contain the information of the support .
Speciﬁcally, since x s = [ x s 1 , x s 2 ] , we sample x s 1 ∼ p
=
U ( − 3 σ, 3 σ ) ( σ is the standard deviation of N (0 , 1) , U is the uniform distribution) and let x s
= x s 1 or x s 2 = sin( x s 1 ) , depending on which dataset is used.
We use the projection procedure that was described in Section 7.2 to obtain the projected samples z projs , so z proj s ∈ R Indim ( P d ) .
We also project the learned prior P Z to R Indim ( P d ) by constructing P proj Z as a Gaussian with zero mean and a diagonal covariance contains the non-zeros eigenvalues of AA T .
Therefore P projZ is a.s. in R Indim ( P d ) and we denote its density function as p proj ( z ) .
We then use the density value p proj ( z = z projs ) to indicate the ‘density allocation’ at the location of x s on the manifold support.
In Figure 9, we compare our model with the ground truth and ﬁnd that we can successfully capture the ‘density allocation’ on the manifold.
[Figure]
C. S-C URVE DATASET
To ﬁt our model to the data, we use the Adam optimizer with learning rate 5 × 10 − 4 and batch size 500 and train the model for 200 k iterations.
We anneal the learning rate with a factor of 0 .
9 every 10 k iterations.
We compare our method with a traditional normalizing ﬂow with a ﬁxed 3D Gaussian prior.
Both models have the same network architecture and training procedure.
Figure 10a and 10b show the samples form our model and the traditional ﬂow with a ﬁxed 3D Gaussian prior.
We can observe more samples lying outwith the true data distribution in Figure 10b than in Figure 10a.
We can conclude that our model has better sample quality considering this S-Curve dataset.
We also compare the latent representation for both models, see Figure 10c and 10d.
We can see the representation distribution Q Z captures the structure of the data distribution well whereas the distribution Q Z in
Figure 10d is unable to do so.
C. F ADING S QUARE DATASET
To ﬁt the data, we train our model for 20 k iterations with batch-size 100 using the Adam optimizer.
The learning rate is initialized to 5 × 10 − 4 and decays with a factor of 0 .
9 at every 1 k iterations.
We additionally use an L 2 weight decay with factor 0 .
C. MNIST D ATASET C.5. I MPLICIT DATA GENERATION MODEL
To ﬁt an implicit model to the MNIST dataset, we ﬁrst train a Variational Auto-Encoder
[Figure]
feed-forward neural network with a ReLU activation and the size of the two hidden outputs areand 200 .
We use a Gaussian decoder p ( x | z ) =
N ( g θ ( z ) , σ 2 x I ) with ﬁxed variance σ x = 0 . 3 .
The g θ is parameterized by a 3-layer feed-forward neural network with hidden layer sizes 200 and 400 .
The activation of the hidden output uses a ReLU and we utilize a Sigmoid function in the ﬁnal layer output to constrain the output between 0 and 1.
The training objective is to maximize the lower bound of the log likelihood
[Equation]
see Kingma and Welling (2013) for further details.
We use an Adam optimizer with learning rate 1 × 10 − 4 and batch size 100 to train the model for 100 epochs.
After training, we sample from the model by ﬁrst taking a sample z ∼ p ( z ) and letting x = g θ
( z ) .
This is equivalent to taking a sample from an implicit model p θ ( x ) = (cid:82) δ
( x − g θ ( z ))
d ( z ) d z , see Tolstikhin et al.
(2017) for further discussion regarding this implicit model construction.
In Figure 11, we plot samples from the trained implicit model with dim ( z ) = 5 , dim ( z ) = 10 and the original MNIST data.
C.5. F LOW MODEL TRAINING
We train our ﬂow models to ﬁt the synthetic MNIST dataset with intrinsic dimensions 5 andand the original MNIST dataset.
In all models, we use the Adam optimizer with learning rate 5 × 10 − 4 and batch size 100 .
We train the model for 3000 k iterations and, following the initial 1000 k iterations, the learning rate decays every 10 k iterations by a factor 0 .
In Figure 12, we plot the samples from our models trained on three different training datasets.
In Figure 13, we also plot the samples from traditional ﬂow models that trained on these three datasets using the same experiment settings , we found the samples from our models are sharper than that from traditional ﬂow models.
[Figure] [Figure]
[Figure] [Figure]