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Variational Autoencoders

After Deep Autoregressive Models and Deep Generative Modelling, we will continue our discussion with Variational AutoEncoders (VAEs) after covering up DGM basics and AGMs. Variational autoencoders (VAEs) are a deep learning method to produce synthetic data (images, texts) by learning the latent representations of the training data. AGMs are sequential models and generate data based on previous data points by defining tractable conditionals. On the other hand, VAEs are using latent variable models to infer hidden structure in the underlying data by using the following intractable distribution function: 

(1)   \begin{equation*} p_\theta(x) = \int p_\theta(x|z)p_\theta(z) dz. \end{equation*}

The generative process using the above equation can be expressed in the form of a directed graph as shown in Figure ?? (the decoder part), where latent variable z\sim p_\theta(z) produces meaningful information of x \sim p_\theta(x|z).

Architectures AE and VAE based on the bottleneck architecture. The decoder part work as a generative model during inference.

Figure 1: Architectures AE and VAE based on the bottleneck architecture. The decoder part work as
a generative model during inference.

Autoencoders

Autoencoders (AEs) are the key part of VAEs and are an unsupervised representation learning technique and consist of two main parts, the encoder and the decoder (see Figure ??). The encoders are deep neural networks (mostly convolutional neural networks with imaging data) to learn a lower-dimensional feature representation from training data. The learned latent feature representation z usually has a much lower dimension than input x and has the most dominant features of x. The encoders are learning features by performing the convolution at different levels and compression is happening via max-pooling.

On the other hand, the decoders, which are also a deep convolutional neural network are reversing the encoder’s operation. They try to reconstruct the original data x from the latent representation z using the up-sampling convolutions. The decoders are pretty similar to VAEs generative models as shown in Figure 1, where synthetic images will be generated using the latent variable z.

During the training of autoencoders, we would like to utilize the unlabeled data and try to minimize the following quadratic loss function:

(2)   \begin{equation*} \mathcal{L}(\theta, \phi) = ||x-\hat{x}||^2, \end{equation*}


The above equation tries to minimize the distance between the original input and reconstructed image as shown in Figure 1.

Variational autoencoders

VAEs are motivated by the decoder part of AEs which can generate the data from latent representation and they are a probabilistic version of AEs which allows us to generate synthetic data with different attributes. VAE can be seen as the decoder part of AE, which learns the set parameters \theta to approximate the conditional p_\theta(x|z) to generate images based on a sample from a true prior, z\sim p_\theta(z). The true prior p_\theta(z) are generally of Gaussian distribution.

Network Architecture

VAE has a quite similar architecture to AE except for the bottleneck part as shown in Figure 2. in AES, the encoder converts high dimensional input data to low dimensional latent representation in a vector form. On the other hand, VAE’s encoder learns the mean vector and standard deviation diagonal matrix such that z\sim \matcal{N}(\mu_z, \Sigma_x) as it will be performing probabilistic generation of data. Therefore the encoder and decoder should be probabilistic.

Training

Similar to AGMs training, we would like to maximize the likelihood of the training data. The likelihood of the data for VAEs are mentioned in Equation 1 and the first term p_\theta(x|z) will be approximated by neural network and the second term p(x) prior distribution, which is a Gaussian function, therefore, both of them are tractable. However, the integration won’t be tractable because of the high dimensionality of data.

To solve this problem of intractability, the encoder part of AE was utilized to learn the set of parameters \phi to approximate the conditional q_\phi (z|x). Furthermore, the conditional q_\phi (z|x) will approximate the posterior p_\theta (z|x), which is intractable. This additional encoder part will help to derive a lower bound on the data likelihood that will make the likelihood function tractable. In the following we will derive the lower bound of the likelihood function:

(3)   \begin{equation*} \begin{flalign} \begin{aligned} log \: p_\theta (x) = & \mathbf{E}_{z\sim q_\phi(z|x)} \Bigg[log \: \frac{p_\theta (x|z) p_\theta (z)}{p_\theta (z|x)} \: \frac{q_\phi(z|x)}{q_\phi(z|x)}\Bigg] \\ = & \mathbf{E}_{z\sim q_\phi(z|x)} \Bigg[log \: p_\theta (x|z)\Bigg] - \mathbf{E}_{z\sim q_\phi(z|x)} \Bigg[log \: \frac{q_\phi (z|x)} {p_\theta (z)}\Bigg] + \mathbf{E}_{z\sim q_\phi(z|x)} \Bigg[log \: \frac{q_\phi (z|x)}{p_\theta (z|x)}\Bigg] \\ = & \mathbf{E}_{z\sim q_\phi(z|x)} \Big[log \: p_\theta (x|z)\Big] - \mathbf{D}_{KL}(q_\phi (z|x), p_\theta (z)) + \mathbf{D}_{KL}(q_\phi (z|x), p_\theta (z|x)). \end{aligned} \end{flalign} \end{equation*}


In the above equation, the first line computes the likelihood using the logarithmic of p_\theta (x) and then it is expanded using Bayes theorem with additional constant q_\phi(z|x) multiplication. In the next line, it is expanded using the logarithmic rule and then rearranged. Furthermore, the last two terms in the second line are the definition of KL divergence and the third line is expressed in the same.

In the last line, the first term is representing the reconstruction loss and it will be approximated by the decoder network. This term can be estimated by the reparametrization trick \cite{}. The second term is KL divergence between prior distribution p_\theta(z) and the encoder function q_\phi (z|x), both of these functions are following the Gaussian distribution and has the closed-form solution and are tractable. The last term is intractable due to p_\theta (z|x). However, KL divergence computes the distance between two probability densities and it is always positive. By using this property, the above equation can be approximated as:

(4)   \begin{equation*} log \: p_\theta (x)\geq \mathcal{L}(x, \phi, \theta) , \: \text{where} \: \mathcal{L}(x, \phi, \theta) = \mathbf{E}_{z\sim q_\phi(z|x)} \Big[log \: p_\theta (x|z)\Big] - \mathbf{D}_{KL}(q_\phi (z|x), p_\theta (z)). \end{equation*}

In the above equation, the term \mathcal{L}(x, \phi, \theta) is presenting the tractable lower bound for the optimization and is also termed as ELBO (Evidence Lower Bound Optimization). During the training process, we maximize ELBO using the following equation:

(5)   \begin{equation*} \operatorname*{argmax}_{\phi, \theta} \sum_{x\in X} \mathcal{L}(x, \phi, \theta). \end{equation*}

.

Furthermore, the reconstruction loss term can be written using Equation 2 as the decoder output is assumed to be following Gaussian distribution. Therefore, this term can be easily transformed to mean squared error (MSE).

During the implementation, the architecture part is straightforward and can be found here. The user has to define the size of latent space, which will be vital in the reconstruction process. Furthermore, the loss function can be minimized using ADAM optimizer with a fixed batch size and a fixed number of epochs.

Figure 2: The results obtained from vanilla VAE (left) and a recent VAE-based generative model NVAE (right)

Figure 2: The results obtained from vanilla VAE (left) and a recent VAE-based generative
model NVAE (right)

In the above, we are showing the quality improvement since VAE was introduced by Kingma and
Welling [KW14]. NVAE is a relatively new method using a deep hierarchical VAE [VK21].

Summary

In this blog, we discussed variational autoencoders along with the basics of autoencoders. We covered
the main difference between AEs and VAEs along with the derivation of lower bound in VAEs. We
have shown using two different VAE based methods that VAE is still active research because in general,
it produces a blurry outcome.

Further readings

Here are the couple of links to learn further about VAE-related concepts:
1. To learn basics of probability concepts, which were used in this blog, you can check this article.
2. To learn more recent and effective VAE-based methods, check out NVAE.
3. To understand and utilize a more advance loss function, please refer to this article.

References

[KW14] Diederik P Kingma and Max Welling. Auto-encoding variational bayes, 2014.
[VK21] Arash Vahdat and Jan Kautz. Nvae: A deep hierarchical variational autoencoder, 2021.

Deep Generative Modelling

Nowadays, we see several real-world applications of synthetically generated data (see Figure 1), for example solving the data imbalance problem in classification tasks, performing style transfer for artistic images, generating protein structure for scientific analysis, etc. In this blog, we are going to explore synthetic data generation using deep neural networks with the mathematical background.

 Synthetic images generated by deep generative models - deep learning generates images

Figure 1 – Synthetic images generated by deep generative models

What is Deep Generative modelling?

Deep generative modelling (DGM) falls in the category of unsupervised learning and addresses a challenging task of the distribution estimation of the given data. To approximate the underlying distribution of a complicated and high dimensional data, Deep generative models (DGM) utilize various deep neural networks architectures e.g., CNN and RNN. Furthermore, the trained DGMs generate samples which have the same distribution as the training data distribution. In other words, if the given training data has the distribution function 𝑝𝑑 (𝑥), then DGMs learn to
generate the samples from a distribution 𝑝𝜃 (𝑥) such that 𝑝𝑑 (𝑥) ≈ 𝑝𝜃 (𝑥).

Deep Learning as unsupervised learner - DGMs pipeline

Figure 2 – DGMs pipeline

Figure 2 represents the general idea about the deep generative modeling, where DGMs are generating data samples with distribution of 𝑝𝜃 (𝑥), which is quite similar to the data distribution of training samples 𝑝𝑑 (𝑥).

Why Deep Generative modelling is important?

DGMs are mainly used to generate synthetic data, which can be used in different applications. The followings are a few examples:

  1. To avoid the data imbalance problems in several real-life classification problems
  2. Text-to-image, image-to-image conversion, image inpainting, super-resolution
  3. Speech and music synthesis.
  4. Computer graphics: rendering, texture generation, character movement, fluid dynamics
    simulation.

How DGMs work?

The above figure is representing a complete workflow of DGMs and it is not very precise because it is combining both training and inference process. During the inference/generation, there will be a slight modification, which is shown in the following figure:

Data generation with random input and a trained DGM

Figure 3 – Data generation with random input and a trained DGM

As it is clear from the above figure, the user gives a random sample as the input to the trained generator to generate a sample which has the similar distribution to the training data. Let us consider that the random input z is sampled from a tractable distribution 𝑝(𝑧) and supported in 𝑅𝑚 and the training data distribution (intractable) is high dimensional and supported in 𝑅𝑛. Therefore, the main goal of trained generator can be written as:

    \begin{equation*} g_\theta:\mathbb{R}^m \to \mathbb{R}^n, \quad \textit{such that}, \quad \min_{\theta} d(p_d (x),p_\theta (x)) \end{equation*}

where d denotes the distance between the two probability distributions and every random vector z will mapped in an unknown vector x, which has an intractable distribution. The vector z is commonly referred as latent variable which is sample from a latent space and in general, follows a tractable Gaussian distribution. The distance minimization problem can be addressed using maximum likelihood. Let us assume that the generator function 𝑔𝜃 is known then we can compute the likelihood of the generated sample x from the latent variable z:

(1)   \begin{equation*} p_\theta (x)= \int p_\theta (x|z) p(z)dz \end{equation*}

The term 𝑝𝜃(𝑥|𝑧) measures the closeness between the generated sample 𝑔𝜃(𝑧) to the original sample x. Based on the data, the likelihood function can be Gaussian for real valued data or Bernoulli for the binary data. From the above discussion, it is clear that the approximating the generator function is most challenging task and that is performed suing deep neural network with high dimensional data. A deep neural network approximates the generator function by computing the generator parameters 𝜃.

Types of DGMs

There are several different types of DGMs to approximate the generator functions, which can generate the new data points with the similar distribution of the training data. In this series of the blogs, we will discuss these methods which are mentioned in the following figure.

In general, DGMs can be separated into implicit and explicit methods, where explicit method are basically likelihood-based methods and learn the data distribution based on an explicitly defined 𝑝𝜃(𝑥). On the other hand, implicit methods learn data distribution directly without any prior model structure. Furthermore, explicit methods are split into tractable and approximation-based methods, where tractable methods are utilizing the model structures which have exact likelihood evaluation and approximation-based methods are applying different forms of approximation in the likelihood estimation.

Summary

In this blog article, we covered the mathematical foundation of DGMs including the different types. In further blog articles, we will cover the above mentioned different DGMs with theoretical background and applications.