are a class of latent variable models for unsupervised learning. The learned generative model and the corresponding (approximate) posterior distribution of the latent variables provide a decoder/encoder pair that might capture semantically meaningful features of the data. In this paper, we address the issue of learning informative latent encodings/representations of the data.
The vanilla VAE uses a standard normal prior distribution for the latent variables. However, it has been shown that this often leads to over-regularising the posterior distribution, resulting in a poor latent representation (Alemi et al., 2018). There are several approaches to alleviate this problem: (i) defining and learning complex prior distributions that can model the encoded data manifold (Chen et al., 2016b; Tomczak & Welling, 2018); (ii) using specialised optimisation algorithms that try to find local minima of the training objective that correspond to informative latent representations (Bowman et al., 2016; Sønderby et al., 2016; Higgins et al., 2017; Rezende & Viola, 2018); and (iii) adding mutual-information based constrains or regularisers to incentivise a good correspondence between the data and the latent variables (Alemi et al., 2018; Zhao et al., 2017; Chen et al., 2016a). In this paper, we address the first two approaches.
As a starting point, we use the approach from (Tomczak & Welling, 2018), where the authors note that the optimal prior (empirical Bayes) is the aggregated posterior—a uniform mixture of approximate posteriors evaluated at the data points. Using this insight, they propose a prior that is a uniform mixture of approximate posterior distributions, evaluated at a few learned pseudo data points. However, a finite mixture does not always provide a good prior (e.g., Sec. 4.2). In this paper, we propose to approximate the aggregated posterior through a continuous mixture/hierarchical distribution. This enables a highly flexible prior, and hence avoids over-regularising the approximate posterior.
In order to learn such hierarchical priors, we extend the optimisation framework introduced in (Rezende & Viola, 2018), where the authors reformulate the VAE objective as the Lagrangian of a constrained-optimisation problem. They impose an inequality constraint on the reconstruction error and choose the KL divergence between the approximate posterior and the prior as the optimisation objective. Instead of a standard normal, we use the hierarchical distribution described above as prior and approximate it by applying an importance-weighted bound (Burda et al., 2015). Concurrently, we introduce the associated optimisation algorithm, inspired by GECO (Rezende & Viola, 2018), the latter does not always lead to good encodings (e.g., Sec. 4.1). Our approach prevents posterior collapse and results in more informative latent representations than previous methods.
2.1 VAEs as a Constrained-Optimisation Problem
VAEs model the distribution of i.i.d. data as the marginal
The model parameters are learned through amortised variational EM, which requires learning an approximate posterior distribution . It is hoped that the learned and result in an informative latent representation of the data. For example, cluster w.r.t. some discrete features or important factors of variation in the data. In Sec. 4.1, we show a toy example, where the model can learn the true underlying factors of variation in .
are typically assumed to be diagonal Gaussians with their parameters defined as neural network functions of the conditioning variables.stands for the empirical distribution of the data . The (EM) optimisation problem (e.g. Neal & Hinton, 1998) is formulated as
The corresponding optimisation algorithm has been often implemented as a double-loop algorithm, however, in the context of VAEs—or neural inference models in general—it is a common practice to optimise jointly.
It has been shown that local minima with high ELBO values do not necessarily result in informative latent representations (Alemi et al., 2018; Higgins et al., 2017). In order to address this problem, several approaches have been developed, which typically result in some weighting schedule for either the negative expected log-likelihood or the KL term of the ELBO (Bowman et al., 2016; Sønderby et al., 2016). This is due to the fact that a different ratio targets different regions in the rate-distortion plane, either favouring better compression or reconstruction (Alemi et al., 2018).
In (Rezende & Viola, 2018), the authors reformulate the VAE objective as the Lagrangian of a constrained-opti- misation problem. They choose the as the optimisation objective and impose the inequality constraint . Typically is defined as the reconstruction-error-related term in . Since is the average reconstruction error, this formulation allows for a better control of the quality of generated data. In the resulting Lagrangian objective
the Lagrange multiplier can be viewed as a weighting term for . This approach leads to a similar optimisation objective as in (Higgins et al., 2017) with . The authors propose a descent-ascent algorithm (GECO) for finding the saddle point of the Lagrangian. The parameters are optimised through gradient descent and is updated as
corresponding to a quasi-gradient ascent due to ; is the update’s learning rate. In the context of stochastic batch gradient training,
is estimated as the running-average, where is the batch average . To the best of our understanding111The optimisation problem is not explicitly stated in (Rezende & Viola, 2018)., the GECO algorithm solves the optimisation problem
Here, can be viewed to correspond to the E-step of the EM algorithm. However, in general this objective can only be guaranteed to be the ELBO if , or in case of , a scaled lower bound on the ELBO.
2.2 Hierarchical Priors for Learning Informative Latent Representations
In this section, we propose a hierarchical prior for VAEs within the constrained-optimisation setting. Our goal is to incentivise the learning of informative latent representations and to avoid over-regularising the posterior distribution (i) by increasing the complexity of the prior distribution , and (ii) by providing an optimisation method to learn such models.
It has been shown that the optimal empirical Bayes prior is the aggregated posterior distribution
We follow (Tomczak & Welling, 2018) to approximate this distribution in the form of a mixture distribution. However, we opt for a continuous mixture/hierarchical model
with a standard normal . This leads to a hierarchical model with two stochastic layers. As a result, intuitively, our approach inherently favours the learning of continuous latent features. We refer to this model by variational-hierarchical prior (VHP).
In order to learn the parameters in Eq. (8), we propose to adapt the constrained-optimisation problem in Sec. 2.1 to hierarchical models. For this purpose we use an importance-weighted (IW) bound (Burda et al., 2015) to define a sequence of upper bounds (and constrained-optimisation problems). That is, we use
with importance weights, defining an upper bound on Eq. (2.1):
As a result, we arrive to the optimisation problem
which we can optimise by the following double-loop algorithm: (i) in the outer loop we update the bound w.r.t. ; (ii) in the inner loop we solve the optimisation problem by applying an update scheme for and , respectively. In the following, we use the -parameterisation to be in line with (e.g. Higgins et al., 2017; Sønderby et al., 2016).
In the GECO update scheme (Eq. (5)), increases/decreases until . However, provided the constraint is fulfilled, we want to obtain a lower lower bound on the log-likelihood. As discussed in Sec. 2.1, that can be guaranteed when (ELBO). To achieve this, we propose to replace the corresponding -version of Eq. (5) by
where we define
Here, is the Heaviside function and we introduce a slope parameter . This update can be interpreted as follows. If the constraint is violated, i.e. , the update scheme is equal to Eq. (5) due to the Heaviside function. In case the constraint is fulfilled, the term guarantees that we finish at , to obtain/optimise the ELBO at the end of the training. Thus, we impose , which is reasonable since does not violate the constraint. A visualisation of the -update scheme is shown in Fig. 1. Note that there are alternative ways to modify Eq. (5), see App. B.1, however, Eq. (12) led to the best results.
However, the double-loop approach in Eq. (11) is often computationally inefficient. Thus, we decided to run the inner loop only until the constraints are satisfied and then updating the bound. That is, we optimise Eq. (11) and skip the outer loop/bound updates when the constraints are not satisfied. It turned out that the bound updates were often skipped in the initial phase, but rarely skipped later on. Hence, the algorithm behaves as layer-wise pretraining (Bengio et al., 2007). For these reasons, we propose Alg. 1 (REWO) that separates training into two phases: an initial phase, where we only optimise the reconstruction error—and a main phase, where all parameters are updated jointly.
In the initial phase, we initialise to enforce a reconstruction optimisation. Thus, to train the first stochastic layer for achieving a good encoding of the data through , measured by the reconstruction error. For preventing to become smaller than the initial value during the first iteration steps, we start to update when the condition is fulfilled. A good encoding is required to learn the conditionals and in the second stochastic layer. Otherwise, would be strongly regularised towards , resulting in , from which it typically does not recover (Sønderby et al., 2016). In the main phase, after is fulfilled, we additionally optimise the parameters of the second stochastic layer and start to update . This approach avoids posterior collapse in both stochastic layers (see Sec. 4.1 and App. D.2), and thereby supports the hierarchical prior to learn an informative latent representation for preventing the aforementioned over-regularisation.
The proposed method, which is a combination of an ELBO-like Lagrangian and an IW bound, can be interpreted as follows: the posterior of the first stochastic layer can learn an informative latent representation due to the flexible hierarchical prior. Since a diagonal Gaussian is not flexible enough to capture the (true) posterior of the second stochastic layer, we propose to enhance it by using an importance-weighted bound (Cremer et al., 2017) (alternatively, one could use, for example, normalising flows (Rezende & Mohamed, 2015)). This has the following advantages: (i) it facilitates learning a precise conditional from the standard normal distribution to the aggregated posterior ; (ii) it allows to fully exploit its representational capacity—otherwise, the model could compensate a less expressive by regularising (see App. B.4).
2.3 Graph-Based Interpolations for Verifying Latent Representations
A key reason for introducing hierarchical priors was to facilitate an informative latent representation due to less over-regularisation of the posterior. To verify the quality of the latent representations, we build on the manifold hypothesis, defined in (Cayton, 2005; Rifai et al., 2011). The idea can be summarised by the following assumption: real-world data presented in high-dimensional spaces is likely to concentrate in the vicinity of nonlinear sub-manifolds of much lower dimensionality. Following this hypothesis, the quality of latent representations can be evaluated by interpolating between different data points along the learned data manifold in the latent space—and reconstructing the resulting path to the observable space.
To implement the above idea, we propose a graph-based method (Chen et al., 2018) which summarises the continuous latent space by a graph consisting of a finite number of nodes. The nodes can be obtained by randomly sampling samples from the learned prior (Eq. (8)):
The graph is henceforth constructed by connecting each of them by undirected edges to its k-nearest neighbours. The edge weights are Euclidean distances in the latent space between the related node pairs. Once the graph is built, interpolation between data points and can be done as follows. First, we encode the data points as , where is the mean of . Next, the encoded data is added as new nodes to the graph along with edges to the existing (nearest neighbour) nodes.
To find the shortest path through the graph between nodes and , a classic search algorithm such as can be used. The result is a sequence , where , representing the shortest path in the latent space along the learned latent manifold. Finally, to obtain the interpolation, we reconstruct to the observable space.
3 Related Work
Several works improve the VAE by learning more complex priors such as the stick-breaking prior (Nalisnick & Smyth, 2017), a nested Chinese Restaurant Process prior (Goyal et al., 2017), Gaussian mixture priors (Dilokthanakul et al., 2016), or autoregressive priors (Chen et al., 2016b). A closely related line of research is based on the insight that the optimal prior is the aggregated posterior. The VampPrior (Tomczak & Welling, 2018) approximates the aggregated posterior by a uniform mixture of approximate posterior distributions, evaluated at a few learned pseudo data points. In our approach, the aggregated posterior is approximated by using an IW bound. Compared to the VampPrior, the VHP can be viewed as a continuous mixture distribution.
In the context of VAEs, hierarchical latent variable models were already introduced earlier (Rezende et al., 2014; Burda et al., 2015; Sønderby et al., 2016). Compared to our approach, these works have in common the structure of the generative model but differ in the factorisation of the inference models and the optimisation procedure. In our proposed method, the VAE objective is reformulated as the Lagrangian of a constrained-optimisation problem. The prior of this ELBO-like Lagrangian is approximated by an IW bound. It is motivated by the fact that a single stochastic layer with a flexible prior can be sufficient for modelling an informative latent representation. The second stochastic layer is required to learn a sufficiently flexible prior.
The common problem of failing to use the full capacity of the model in VAEs (Burda et al., 2015) has been addressed by applying annealing/warm-up (Bowman et al., 2016; Sønderby et al., 2016). Here, the KL divergence between the approximate posterior and the prior is multiplied by a weighting factor, which is linearly increased from 0 to 1 during training. However, such predefined schedules might be suboptimal. Therefore, (Rezende & Viola, 2018) introduce a constrained-optimisation algorithm called GECO. By reformulating the objective as a constrained-optimisation problem, the above weighting term can be represented by a Lagrange multiplier and updated based on the reconstruction error. Our proposed algorithm builds on GECO, providing several modifications discussed in Sec. 2.2.
In (Higgins et al., 2017), the authors propose a constrained-optimisation framework, where the optimisation objective is the expected negative log-likelihood and the constraint is imposed in the KL term—recall that in (Rezende & Viola, 2018) the roles are reversed. Instead of optimising the resulting Lagrangian, the authors choose Lagrange multipliers (
parameter) that maximise a heuristic cost for disentanglement. In contrast to our approach, the goal is not to learn a latent representation that reflects the topology of the data but a disentangled representation, where different dimensions of the latent space correspond to different features of the data.
To validate our approach, we consider the following experiments. In Sec. 4.1
, we demonstrate that our method learns to represent the degree of freedom in the data of a moving pendulum. In Sec.4.2, we generate human movements based on the learned latent representations of real-world data (CMU Graphics Lab Motion Capture Database). In Sec. 4.3, the marginal log-likelihood on standard datasets such as MNIST, Fashion-MNIST, and OMNIGLOT is evaluated. In Sec. 4.4, we compare our method on the high-dimensional image datasets 3D Faces and 3D Chairs. The model architectures used in our experiments can be found in App. F.
4.1 Artificial Pendulum Dataset
We created a dataset of 15,000 images of a moving pendulum (see Fig. 4). Each image has a size of pixels and the joint angles are distributed uniformly in the range .
Fig. 2 shows latent representations of the pendulum data learned by the VHP when applying REWO and GECO, respectively; the same is used in both cases. In case of REWO, the approximate posterior (Fig. 2(a), top row) is optimised to reach a low reconstruction error at the beginning of the training due to . The variance of the posterior’s standard deviation, expressed by the greyscale, measures whether the contribution to the ELBO is equally distributed over all data points. Once is fulfilled (Fig. 2(a), iter=350), begins to be updated and the parameters of the second stochastic layer start to be optimised, leading to informative hierarchical prior distributions (Fig. 2(a), bottom row). Between iteration 2000 and 5000, the increase in results in a regularisation of the latent representation, and hence in a higher reconstruction error. At iteration 5000, stops to increase due to (see Eq. (12)). From iteration 5000 to 27,500, is updated driven by an interplay between the reconstruction error and the KL divergence, . After , the regularisation impact of the KL divergence does not increase anymore, leading to an improvement of the latent representation (Fig. 2(a), iter=150,000).
To validate whether the obtained latent representation is informative, we apply a linear regression after transforming the latent space to polar coordinates. The goal is to predict the joint angle of the pendulum. We compare REWO with GECO, and additionally withwarm-up (WU) (Sønderby et al., 2016), a linear annealing schedule of . In the latter, we use a VAE objective—defined as an ELBO/IW bound combination, similar to Eq. (2.2); the related plots are in App. B.2. Tab. 1 shows the absolute errors (OLS regression) for the different optimisation procedures; details on the regression can be found in App. B.3. REWO leads to the most precise prediction of the ground truth.
|VHP + REWO||0.054|
|VHP + GECO||0.53|
+ WU (20 epochs)
|VHP + WU (200 epochs)||0.31|
Furthermore, we demonstrate in App. B.4 that a less expressive posterior in the second stochastic layer leads to poor latent representations, since the model compensates it by restricting —as discussed in Sec. 2.2.
Finally, we compare the latent representations, learned by the VHP and the IWAE, using our graph-based interpolation method. The graphs, shown in Fig. 3, are built (see Sec. 2.3) based on 1000 samples from the prior of the respective model. The red curves depict the interpolation along resulting data manifold, between pendulum images with joint angles of 0 and 180 degrees, respectively. The reconstructions of the interpolations are shown in (Fig. 4). The top row (VHP + REWO) shows a smooth change of the joint angles, whereas the middle (VHP + GECO) and bottom row (IWAE) contain discontinuities resulting in an unrealistic interpolation.
4.2 Human Motion Capture Database
This section presents the evaluation on the CMU Graphics Lab Motion Capture Database222http://mocap.cs.cmu.edu/, which consists of several human motion recordings. Our experiments base on data of five different motions. Since different motions have similar body positions in certain frames, the corresponding manifolds are connected, making it a suitable dataset for interpolation experiments. We preprocess the data as in (Chen et al., 2015)
, such that each frame is represented by a 50-dimensional feature vector.
We compare our method with the VampPrior and the IWAE. The prior and aggregated approximate posterior of the three methods is shown in Fig. 5. In case of the VHP and the VampPrior the latent representations of different movements are separated and the learned prior matches the aggregated posterior. By contrast, the IWAE is restricted by the Gaussian prior and cannot represent the different motions separately in the latent space. Next, we generate four interpolations (Fig. 6) using our graph-based approach: between two frames within one motion (black line) and of different motions (orange, red, and maroon); the reconstructions are shown in Fig. 7 and App. C. The VampPrior and the VHP lead to smooth interpolations, whereas the IWAE interpolations show abrupt changes in the movements.
Fig. 8 depicts the movement smoothness factor, which we define as the RMS of the second order finite difference along the interpolated path. Thus, smaller values correspond to smoother movements. For each of the three methods, it is averaged across 10 graphs, each with 100 interpolations. The starting and ending points are randomly selected. As a result, the latent representation learned by the VHP leads to smoother movement interpolations than in case of the VampPrior and IWAE.
4.3 Evaluation on MNIST, Fashion-MNIST, and OMNIGLOT
We compare our method quantitatively to the VampPrior and the IWAE on MNIST (Lecun et al., 1998; Larochelle & Murray, 2011), Fashion-MNIST (Xiao et al., 2017), and OMNIGLOT (Lake et al., 2015). For this purpose, we report the marginal log-likelihood (LL) on the respective test set. Following the test protocol of previous work (Tomczak & Welling, 2018), we evaluate the LL using importance sampling with 5,000 samples (Burda et al., 2015). The results are reported in Tab. 2.
VHP + REWO performs as good or better than state-of-the-art on these datasets. The same was used for training VHP with REWO and GECO. The two stochastic layer hierarchical IWAE does not perform better than VHP + REWO, supporting our claim that a flexible prior in the first stochastic layer and a flexible approximate posterior in the second stochastic layer are sufficient. Additionally, we show that REWO leads to a similar amount of active units as WU (see App. D.2).
|dynamic MNIST||static MNIST||Fashion- MNIST||OMNI- GLOT|
|VHP + REWO||78.88||82.74||225.37||101.78|
|VHP + GECO||95.01||96.32||234.73||108.97|
4.4 Qualitative Results: 3D Chairs and 3D Faces
We generated 3D Faces (Paysan et al., 2009) based on images of 2000 faces with 37 views each. 3D Chairs (Aubry et al., 2014) consists of 1393 chair images with 62 views each. The images have a size of pixels.
Here, our approach is compared with the IWAE using a 32-dimensional latent space. The learned encodings are evaluated qualitatively by using the graph-based interpolation method. Fig. 9(a) and 9(c) show interpolations along the latent manifold learned by the VHP + REWO. Compared to the IWAE (Fig. 9(b) and 9(d)), they are less blurry and smoother. Further results can be found in App. E.
In this paper, we have proposed a hierarchical prior in the context of variational autoencoders and extended the constrained-optimisation framework in Taming VAEs to hierarchical models by using an importance-weighted bound on the marginal of the hierarchical prior. Concurrently, we have introduced the associated optimisation algorithm to facilitate good encodings.
We have shown that the learned hierarchical prior is indeed non-trivial, moreover, it is well-adapted to the latent representation, reflecting the topology of the encoded data manifold. Our method provides informative latent representations and performs particularly well on data where the relevant features change continuously. In case of the pendulum (Sec. 4.1), the prior has learned to represent the degrees of freedom in the data—allowing us to predict the pendulum’s angle by a simple OLS regression. The experiments on the human motion data (Sec. 4.2) and on the high-dimensional Faces and Chairs datasets (Sec. 4.4) have demonstrated that the learned hierarchical prior leads to smoother and more realistic interpolations than a standard normal prior (or the VampPrior). Moreover, we have obtained test log-likelihoods (Sec. 4.3) comparable to state-of-art on standard datasets.
We would like to thank Maximilian Soelch for valuable suggestions and discussions.
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Appendix A Comparison: -Update Scheme in REWO and in GECO
Appendix B Pendulum
b.1 Training Process with Alternative -Update Scheme
An alternative way to define the -update scheme such that is to use Eq. (5) with
As in Sec. 2, is defined as the update’s learning rate. This leads to the following -update scheme:
b.2 Training Process with/without WU
b.3 OLS Regression on Learned Latent Representations
Fig. 15 shows the joint angle versus , where is the second component of the latent space and the radius is estimated from the learned latent representation.
b.4 VHP with ELBO instead of IW Bound
b.5 Latent Representations Learned by VHP and IWAE
Appendix C CMU Human Motion
Appendix D Quantitative Results
d.1 Training Efficiency
d.2 Active Units
Furthermore, we evaluate whether REWO prevents over-pruning of the latent variables (Yeung et al., 2017). Following (Sønderby et al., 2016), we evaluate for different optimisation strategies, where . We show the results for the inner latent variable on several datasets in Fig. 21.
Appendix E Faces and Chairs
Appendix F Model Architectures
FC 256, 256, 256, 256. ReLU activation.
|FC 256, 256, 256, 256. ReLU activation. Gaussian.|
|FC 256, 256, 256, 256, ReLU activation.|
|FC 256, 256, 256, 256, ReLU activation.|
|Others||= 0.02, = 5, = 16.|
|Graph||1,000 nodes, 18 neighbours.|
|FC 256, 256, 256, 256. ReLU activation.|
|FC 256, 256, 256, 256. ReLU activation. Gaussian.|
|FC 256, 256, 256, 256, ReLU activation.|
|FC 256, 256, 256, 256, ReLU activation.|
|Others||= 0.02, = 1, = 32.|
|Graph||2,530 nodes, 15 neighbours.|
5(stride 2) , 3233(stride 1), 4855(stride 2).
|4833(stride 1), 6455(stride 2), 6433(stride 1).|
|9655(stride 2), 9633(stride 1), FC 256. ReLU activation|
|Deconv reverse of encoder. ReLU activation. Bernoulli.|
|FC 256, 256, ReLU activation.|
|FC 256, 256, ReLU activation.|
|others||= 0.2, = 1, = 16.|
|Graph||10,000 nodes (faces), 8,637 nodes (chairs), 18 neighbours.|
|Fashion-MNIST,||GatedConv 3277(stride 1) , 3233(stride 2),|
|OMNIGLOT||6455(stride 1), 6433(stride 2), 633(stride 1)|
|GatedFC 784, GatedConv 6433(stride 1),|
|6433(stride 1), 6433(stride 1), 6433(stride 1).|
|linear activation. Bernoulli.|
|GatedFC 256, 256, linear activation.|
|GatedFC 256, 256, linear activation.|
|others||= 0.18 (dynamicMNIST, staticMNIST, OMNIGLOT),|
|= 0.31 (Fashion-MNIST),|
|= 1, = 16.|