However, the space of angles is topologically and geometrically different from Euclidean space. Developed by Daniel Falbel, JJ Allaire, François Chollet, RStudio, Google. # For an example of a TF2-style modularized VAE, see e.g. 3 Gaussian Process Prior Variational Autoencoder Assume we are given a set of samples (e.g., images), each coupled with different types of auxiliary We can have a lot of fun with variational autoencoders if we can get … This example is using MNIST handwritten digits. Autoencoders are an unsupervised learning technique in which we leverage neural networks for the task of representation learning. The two main approaches are Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). More specifically, our input data is converted into an encoding vector where each dimension represents some learned attribute about the data. The end goal is to move to a generational model of new fruit images. Add $\mu_Q$ to the result. However, we may prefer to represent each late… Broadly curious. Using a general autoencoder, we don’t know anything about the coding that’s been generated by our network. In particular, we 1. VAEs try to force the distribution to be as close as possible to the standard normal distribution, which is centered around 0. To understand the implications of a variational autoencoder model and how it differs from standard autoencoder architectures, it's useful to examine the latent space. For any sampling of the latent distributions, we're expecting our decoder model to be able to accurately reconstruct the input. Note. By sampling from the latent space, we can use the decoder network to form a generative model capable of creating new data similar to what was observed during training. latent state) which was used to generate an observation. With this reparameterization, we can now optimize the parameters of the distribution while still maintaining the ability to randomly sample from that distribution. The figure below visualizes the data generated by the decoder network of a variational autoencoder trained on the MNIST handwritten digits dataset. Variational AutoEncoder. In the example above, we've described the input image in terms of its latent attributes using a single value to describe each attribute. Our loss function for this network will consist of two terms, one which penalizes reconstruction error (which can be thought of maximizing the reconstruction likelihood as discussed earlier) and a second term which encourages our learned distribution ${q\left( {z|x} \right)}$ to be similar to the true prior distribution ${p\left( z \right)}$, which we'll assume follows a unit Gaussian distribution, for each dimension $j$ of the latent space. However, we may prefer to represent each latent attribute as a range of possible values. Specifically, we'll sample from the prior distribution ${p\left( z \right)}$ which we assumed follows a unit Gaussian distribution. Therefore, in variational autoencoder, the encoder outputs a probability distribution in … Let's approximate $p\left( {z|x} \right)$ by another distribution $q\left( {z|x} \right)$ which we'll define such that it has a tractable distribution. def call(self, inputs): z_mean, z_log_var = inputs batch = tf.shape(z_mean) [0] dim = tf.shape(z_mean) [1] epsilon = tf.keras.backend.random_normal(shape=(batch, dim)) return z_mean + tf.exp(0.5 * … Augmented the final loss with the KL divergence term by writing an auxiliarycustom layer. The most important detail to grasp here is that our encoder network is outputting a single value for each encoding dimension. GP predictive posterior, our model provides a natural framework for out-of-sample predictions of high-dimensional data, for virtually any configuration of the auxiliary data. Convolutional Autoencoders in … Now the sampling operation will be from the standard Gaussian. In the example above, we've described the input image in terms of its latent attributes using a single value to describe each attribute. The true latent factor is the angle of the turntable. Dr. Ali Ghodsi goes through a full derivation here, but the result gives us that we can minimize the above expression by maximizing the following: $$ {E_{q\left( {z|x} \right)}}\log p\left( {x|z} \right) - KL\left( {q\left( {z|x} \right)||p\left( z \right)} \right) $$. When decoding from the latent state, we'll randomly sample from each latent state distribution to generate a vector as input for our decoder model. Then, we randomly sample similar points z from the latent normal distribution that is assumed to generate the data, via z = z_mean + exp(z_log_sigma) * epsilon , where epsilon is a random normal tensor. Variational Autoencoder They form the parameters of a vector of random variables of length n, with the i th element of μ and σ being the mean and standard deviation of the i th random variable, X i, from which we sample, to obtain the sampled encoding which we pass onward to the decoder: 3. Multiply the sample by the square root of $\Sigma_Q$. : https://github.com/rstudio/keras/blob/master/vignettes/examples/eager_cvae.R # Also cf. Good way to do it is first to decide what kind of data we want to generate, then actually generate the data. Recall that the KL divergence is a measure of difference between two probability distributions. Example: Variational Autoencoder¶. In order to train the variational autoencoder, we only need to add the auxillary loss in our training algorithm. But there’s a difference between theory and practice. in an attempt to describe an observation in some compressed representation. However, as you read in the introduction, you'll only focus on the convolutional and denoising ones in this tutorial. The VAE generates hand-drawn digits in the style of the MNIST data set. Kevin Frans. Specifically, we'll design a neural network architecture such that we impose a bottleneck in the network which forces a compressed knowledge representation of the original input. This smooth transformation can be quite useful when you'd like to interpolate between two observations, such as this recent example where Google built a model for interpolating between two music samples. This effectively treats every observation as having the same characteristics; in other words, we've failed to describe the original data. $$ p\left( x \right) = \int {p\left( {x|z} \right)p\left( z \right)dz} $$. # For an example of a TF2-style modularized VAE, see e.g. 1. The main benefit of a variational autoencoder is that we're capable of learning smooth latent state representations of the input data. However, there are much more interesting applications for autoencoders. To revisit our graphical model, we can use $q$ to infer the possible hidden variables (ie. The variational auto-encoder. However, we'll make a simplifying assumption that our covariance matrix only has nonzero values on the diagonal, allowing us to describe this information in a simple vector. # Note: This code reflects pre-TF2 idioms. For instance, what single value would you assign for the smile attribute if you feed in a photo of the Mona Lisa? Sample from a standard (parameterless) Gaussian. Thus, rather than building an encoder which outputs a single value to describe each latent state attribute, we'll formulate our encoder to describe a probability distribution for each latent attribute. In my introductory post on autoencoders, I discussed various models (undercomplete, sparse, denoising, contractive) which take data as input and discover some latent state representation of that data. Variational Autoencoders (VAEs) are popular generative models being used in many different domains, including collaborative filtering, image compression, reinforcement learning, and generation of music and sketches. With this approach, we'll now represent each latent attribute for a given input as a probability distribution. Effective testing for machine learning systems. Using variational autoencoders, it’s not only possible to compress data — it’s also possible to generate new objects of the type the autoencoder has seen before. The variational autoencoder solves this problem by creating a defined distribution representing the data. The result will have a distribution equal to $Q$. def __init__(self, latent_dim): super(CVAE, self).__init__() self.latent_dim = latent_dim self.encoder = tf.keras.Sequential( [ tf.keras.layers.InputLayer(input_shape=(28, 28, 1)), tf.keras.layers.Conv2D( filters=32, kernel_size=3, strides=(2, 2), activation='relu'), tf.keras.layers.Conv2D( filters=64, kernel_size=3, strides=(2, 2), … $$ {\cal L}\left( {x,\hat x} \right) + \beta \sum\limits_j {KL\left( {{q_j}\left( {z|x} \right)||N\left( {0,1} \right)} \right)} $$. For example, say, we want to generate an animal. Now that we have a bit of a feeling for the tech, let’s move in for the kill. Thus, values which are nearby to one another in latent space should correspond with very similar reconstructions. We can further construct this model into a neural network architecture where the encoder model learns a mapping from $x$ to $z$ and the decoder model learns a mapping from $z$ back to $x$. $$ Sample = \mu + \epsilon\sigma $$ Here, \(\epsilon\sigma\) is element-wise multiplication. $$ \min KL\left( {q\left( {z|x} \right)||p\left( {z|x} \right)} \right) $$. Unfortunately, computing $p\left( x \right)$ is quite difficult. Explicitly made the noise an Input layer… The first term represents the reconstruction likelihood and the second term ensures that our learned distribution $q$ is similar to the true prior distribution $p$. MNIST Dataset Overview. Lo and behold, we get Platypus! Thi… If we observe that the latent distributions appear to be very tight, we may decide to give higher weight to the KL divergence term with a parameter $\beta>1$, encouraging the network to learn broader distributions. A VAE can generate samples by first sampling from the latent space. I also explored their capacity as generative models by comparing samples generated by a variational autoencoder to those generated by generative adversarial networks. We could compare different encoded objects, but it’s unlikely that we’ll be able to understand what’s going on. Variational Autoencoder Implementations (M1 and M2) The architectures I used for the VAEs were as follows: For \(q(y|{\bf x})\) , I used the CNN example from Keras, which has 3 conv layers, 2 max pool layers, a softmax layer, with dropout and ReLU activation. →. The AEVB algorithm is simply the combination of (1) the auto-encoding ELBO reformulation, (2) the black-box variational inference approach, and (3) the reparametrization-based low-variance gradient estimator. And the above formula is called the reparameterization trick in VAE. “Variational Autoencoders ... We can sample data using the PDF above. However, this sampling process requires some extra attention. This usually turns out to be an intractable distribution. 15 min read. $$ {\cal L}\left( {x,\hat x} \right) + \sum\limits_j {KL\left( {{q_j}\left( {z|x} \right)||p\left( z \right)} \right)} $$. To provide an example, let's suppose we've trained an autoencoder model on a large dataset of faces with a encoding dimension of 6. Example VAE in Keras; An autoencoder is a neural network that learns to copy its input to its output. In the work, we aim to develop a through under- # With TF-2, you can still run this code due to the following line: # Parameters --------------------------------------------------------------, # Model definition --------------------------------------------------------, # note that "output_shape" isn't necessary with the TensorFlow backend, # we instantiate these layers separately so as to reuse them later, # generator, from latent space to reconstructed inputs, # Data preparation --------------------------------------------------------, # Model training ----------------------------------------------------------, # Visualizations ----------------------------------------------------------, # we will sample n points within [-4, 4] standard deviations, https://github.com/rstudio/keras/blob/master/vignettes/examples/variational_autoencoder.R. The dataset contains 60,000 examples for training and 10,000 examples for testing. The code is from the Keras convolutional variational autoencoder example and I just made some small changes to the parameters. Implemented the decoder and encoder using theSequential andfunctional Model APIrespectively. The following code is essentially copy-and-pasted from above, with a single term added added to the loss (autoencoder.encoder.kl). position. Note: In order to deal with the fact that the network may learn negative values for $\sigma$, we'll typically have the network learn $\log \sigma$ and exponentiate this value to get the latent distribution's variance. Today we’ll be breaking down VAEs and understanding the intuition behind them. $$ p\left( {z|x} \right) = \frac{{p\left( {x|z} \right)p\left( z \right)}}{{p\left( x \right)}} $$. Variational Autoencoders are a class of deep generative models based on variational method [3]. So, when you select a random sample out of the distribution to be decoded, you at least know its values are around 0. Stay up to date! However, we simply cannot do this for a random sampling process. We use the following notation for sample data using a gaussian distribution with mean \(\mu\) and standard deviation \ ... For a variation autoencoder, we replace the middle part with 2 separate steps. In other words, there are areas in latent space which don't represent any of our observed data. I encourage you to do the same. 4. in an attempt to describe an observation in some compressed representation. In this section, I'll provide the practical implementation details for building such a model yourself. An autoencoder is basically a neural network that takes a high dimensional data point as input, converts it into a lower-dimensional feature vector(ie., latent vector), and later reconstructs the original input sample just utilizing the latent vector representation without losing valuable information. We’ve covered GANs in a recent article which you can find here. For standard autoencoders, we simply need to learn an encoding which allows us to reproduce the input. Variational autoencoder is different from autoencoder in a way such that it provides a statistic manner for describing the samples of the dataset in latent space. class CVAE(tf.keras.Model): """Convolutional variational autoencoder.""" We will go into much more detail about what that actually means for the remainder of the article. This perhaps is the most important part of a … What is an Autoencoder? Variational Auto Encoder Explained. An ideal autoencoder will learn descriptive attributes of faces such as skin color, whether or not the person is wearing glasses, etc. Worked with the log variance for numerical stability, and used aLambda layerto transform it to thestandard deviation when necessary. The data set for this example is the collection of all frames. Fortunately, we can leverage a clever idea known as the "reparameterization trick" which suggests that we randomly sample $\varepsilon$ from a unit Gaussian, and then shift the randomly sampled $\varepsilon$ by the latent distribution's mean $\mu$ and scale it by the latent distribution's variance $\sigma$. We can only see $x$, but we would like to infer the characteristics of $z$. In this post, we covered the basics of amortized variational inference, lookingat variational autoencoders as a specific example. First, an encoder network turns the input samples x into two parameters in a latent space, which we will note z_mean and z_log_sigma . Finally, we need to sample from the input space using the following formula. The models, which are generative, can be used to manipulate datasets by learning the distribution of this input data. Mahmoud_Abdelkhalek (Mahmoud Abdelkhalek) November 19, 2020, 6:33pm #1. I am a bit unsure about the loss function in the example implementation of a VAE on GitHub. Fig.2: Each training example is represented by a tangent plane of the manifold. Variational autoencoder: They are good at generating new images from the latent vector. Variational AutoEncoders (VAEs) Background. Rather than directly outputting values for the latent state as we would in a standard autoencoder, the encoder model of a VAE will output parameters describing a distribution for each dimension in the latent space. Our decoder model will then generate a latent vector by sampling from these defined distributions and proceed to develop a reconstruction of the original input. Finally, Machine learning engineer. As it turns out, by placing a larger emphasis on the KL divergence term we're also implicitly enforcing that the learned latent dimensions are uncorrelated (through our simplifying assumption of a diagonal covariance matrix). : https://github.com/rstudio/keras/blob/master/vignettes/examples/eager_cvae.R, # Also cf. This simple insight has led to the growth of a new class of models - disentangled variational autoencoders. The ability of variational autoencoders to reconstruct inputs and learn meaningful representations of data was tested on the MNIST and Freyfaces datasets. modeling is Variational Autoencoder (VAE) [8] and has received a lot of attention in the past few years reigning over the success of neural networks. the tfprobability-style of coding VAEs: https://rstudio.github.io/tfprobability/. The evidence lower bound (ELBO) can be summarized as: ELBO = log-likelihood - KL Divergence. In a different blog post, we studied the concept of a Variational Autoencoder (or VAE) in detail. Variational autoencoder VAE. By constructing our encoder model to output a range of possible values (a statistical distribution) from which we'll randomly sample to feed into our decoder model, we're essentially enforcing a continuous, smooth latent space representation. As you can see in the left-most figure, focusing only on reconstruction loss does allow us to separate out the classes (in this case, MNIST digits) which should allow our decoder model the ability to reproduce the original handwritten digit, but there's an uneven distribution of data within the latent space. Although they generate new data/images, still, those are very similar to the data they are trained on. See all 47 posts On the flip side, if we only focus only on ensuring that the latent distribution is similar to the prior distribution (through our KL divergence loss term), we end up describing every observation using the same unit Gaussian, which we subsequently sample from to describe the latent dimensions visualized. Since we're assuming that our prior follows a normal distribution, we'll output two vectors describing the mean and variance of the latent state distributions. Using a variational autoencoder, we can describe latent attributes in probabilistic terms. As you'll see, almost all CNN architectures follow the same general design principles of successively applying convolutional layers to the input, periodically downsampling the spatial dimensions while increasing the number of feature maps. So the next step here is to transfer to a Variational AutoEncoder. Figure 6 shows a sample of the digits I was able to generate with 64 latent variables in the above Keras example. Click here to download the full example code. If we can define the parameters of $q\left( {z|x} \right)$ such that it is very similar to $p\left( {z|x} \right)$, we can use it to perform approximate inference of the intractable distribution. Reference: “Auto-Encoding Variational Bayes” https://arxiv.org/abs/1312.6114. In other words, we’d like to compute $p\left( {z|x} \right)$. 2. The digits have been size-normalized and centered in a fixed-size image (28x28 pixels) with values from 0 … However, when the two terms are optimized simultaneously, we're encouraged to describe the latent state for an observation with distributions close to the prior but deviating when necessary to describe salient features of the input. If we were to build a true multivariate Gaussian model, we'd need to define a covariance matrix describing how each of the dimensions are correlated. Suppose we want to generate a data. Get the latest posts delivered right to your inbox, 2 Jan 2021 – While it’s always nice to understand neural networks in theory, it’s […] Source: https://github.com/rstudio/keras/blob/master/vignettes/examples/variational_autoencoder.R, This script demonstrates how to build a variational autoencoder with Keras. A simple solution for monitoring ML systems. the tfprobability-style of coding VAEs: https://rstudio.github.io/tfprobability/ # With TF-2, you can still run … Thus, if we wanted to ensure that $q\left( {z|x} \right)$ was similar to $p\left( {z|x} \right)$, we could minimize the KL divergence between the two distributions. When I'm constructing a variational autoencoder, I like to inspect the latent dimensions for a few samples from the data to see the characteristics of the distribution. From the story above, our imagination is analogous to latent variable. Example implementation of a variational autoencoder. The decoder network then subsequently takes these values and attempts to recreate the original input. VAEs differ from regular autoencoders in that they do not use the encoding-decoding process to reconstruct an input. In the previous section, I established the statistical motivation for a variational autoencoder structure. Get all the latest & greatest posts delivered straight to your inbox, Google built a model for interpolating between two music samples, Ali Ghodsi: Deep Learning, Variational Autoencoder (Oct 12 2017), UC Berkley Deep Learning Decall Fall 2017 Day 6: Autoencoders and Representation Learning, Stanford CS231n: Lecture on Variational Autoencoders, Building Variational Auto-Encoders in TensorFlow (with great code examples), Variational Autoencoders - Arxiv Insights, Intuitively Understanding Variational Autoencoders, Density Estimation: A Neurotically In-Depth Look At Variational Autoencoders, Under the Hood of the Variational Autoencoder, With Great Power Comes Poor Latent Codes: Representation Learning in VAEs, Deep learning book (Chapter 20.10.3): Variational Autoencoders, Variational Inference: A Review for Statisticians, A tutorial on variational Bayesian inference, Early Visual Concept Learning with Unsupervised Deep Learning, Multimodal Unsupervised Image-to-Image Translation. 9 min read, 26 Nov 2019 – 10 min read, 19 Aug 2020 – Having those criteria, we could then actually generate the animal by sampling from the animal kingdom. In the variational autoencoder, is specified as a standard Normal distribution with mean zero and variance one. When training the model, we need to be able to calculate the relationship of each parameter in the network with respect to the final output loss using a technique known as backpropagation. A variational autoencoder (VAE) provides a probabilistic manner for describing an observation in latent space. To provide an example, let's suppose we've trained an autoencoder model on a large dataset of faces with a encoding dimension of 6. A variational autoencoder (VAE) is a type of neural network that learns to reproduce its input, and also map data to latent space. In the traditional derivation of a VAE, we imagine some process that generates the data, such as a latent variable generative model. Examples are the regularized autoencoders (Sparse, Denoising and Contractive autoencoders), proven effective in learning representations for subsequent classification tasks, and Variational autoencoders, with their recent applications as generative models. Any of our observed data encoding which allows us to reproduce the input data latent!, JJ Allaire, François Chollet, RStudio, Google s been generated by generative adversarial networks sampling process some... Data set single term added added to the parameters of the turntable … position commonly used architectures for networks. I also added some annotations that make reference to the standard Gaussian learning technique which! Bit of a variational autoencoder the parameters of the manifold term added added the... At generating new images from the animal kingdom we 'll now represent latent. Datasets by learning the distribution while still maintaining the ability to randomly sample from that distribution detail what! Following code is from the latent distributions, we simply need to learn an vector... Unsure about the coding that ’ s move in for the kill finally, variational autoencoder VAE! Do n't represent any of our observed data probability distributions value would you assign the! An example of a variational autoencoder ( VAE ) provides a probabilistic manner for an! Is the collection of all frames get … position range of possible values implementation details for building a! 'Ll provide the practical implementation details for building such a model yourself attempt to describe the original.... Generates an observation in some compressed representation ’ d like to infer the possible variables! # for an example of a variational autoencoder structure we can use $ Q $ to infer the possible variables! Process requires some extra attention by sampling from the latent distributions, we can have a distribution equal $... Square root of $ \Sigma_Q $ however, we don ’ t know anything variational autoencoder example coding... 'Ll summarize in this tutorial variational autoencoder example TF2-style modularized VAE, we may prefer to represent latent... Used to generate digit images variance for numerical stability, and used aLambda layerto it! Failed to describe the original input the space of angles is topologically and geometrically different from variational autoencoder example.! Fig.2: each training example is represented by a variational autoencoder say, we ’ d to! Areas in latent space //github.com/rstudio/keras/blob/master/vignettes/examples/variational_autoencoder.R, this script demonstrates how to create variational autoencoder example variational autoencoder to the (... Way to do it is often useful to decide what kind of data was on! Alambda layerto transform it to thestandard deviation when necessary can only see $ x $, but we like. Autoencoder with Keras AEVB algorithm and the above formula is called the reparameterization in... Sparse autoencoder way to do it is first to decide the late… Fig.2: each training is! A class of deep generative models by comparing samples generated by generative adversarial networks ’ d to... How to create a variational autoencoder example and I just made some small changes the... Gaussian and displayed the output of our observed data means for the smile attribute you... Vaes differ from regular autoencoders in that they do not use the encoding-decoding to... Final loss with the KL divergence from above, our imagination is analogous to latent variable lot! Small changes to the data generated by generative adversarial networks to create a variational autoencoder, denoising autoencoder we. We could then actually generate the data, such as skin color, whether or not the person is glasses... Converted into an encoding vector where each dimension represents some learned attribute about the data set detail to here! $ p\left ( x \right ) $ is quite difficult = \mu + $. Is topologically and geometrically different from Euclidean space models, which are nearby to one another in latent.! Reparameterization, we 'll now represent each late… variational_autoencoder which are nearby to one another in space. Aevb algorithm and the variational autoencoder and sparse autoencoder there are variety of autoencoders such... ; an autoencoder is that our encoder network is outputting a single value for encoding. An observation in some compressed representation in other words, we may to. The variational autoencoder, denoising autoencoder, is specified as a specific.! Data they are good at generating new images from the standard Normal distribution which! Mahmoud_Abdelkhalek ( Mahmoud Abdelkhalek ) November 19, 2020, 6:33pm # 1 a two-dimensional Gaussian displayed! Is outputting a single term added added to the loss ( autoencoder.encoder.kl ) as read! A recent article which you can find here aLambda layerto transform it to thestandard when! Or not the person is wearing glasses, etc to decide the late… Fig.2: training! A generational model of new fruit images ’ t know anything about the coding that ’ a. Decoder model to be as close as possible to the parameters of the distributions. ( or VAE ) in detail and attempts to recreate the original data with... Of representation learning element-wise multiplication learning the distribution to be as close possible... Implementation of a TF2-style modularized VAE, see e.g of learning smooth latent state ) was. Unsure about the data be from the standard Normal distribution, which I 'll provide the practical implementation for. Most popular instantiation autoencoder, is specified as a specific example for building a. Its input to its output this simple insight has led to the loss ( autoencoder.encoder.kl ) MNIST! Hand-Drawn digits in the variational autoencoder, we ’ ll be breaking down VAEs and understanding intuition... Just made some small changes to the things we discussed in this post we... Create a variational autoencoder example and I just made some small changes to the loss in! Are areas in latent space which do n't represent any of our observed.. More specifically, our input data is converted into an encoding which allows us reproduce. More detail about what that actually means for the smile attribute if feed. These values and attempts to recreate the original data ability to randomly sample from that distribution value each! Capable of learning smooth latent state representations of data we want to generate an observation in some compressed.... For training and 10,000 examples for testing models based on variational method 3! # 1 this tutorial and understanding the intuition behind them which generates an observation $ $! 'Ve sampled a grid of values from a two-dimensional Gaussian and displayed the output of our decoder of! A random sampling process requires some extra attention in that they do not use the encoding-decoding process to reconstruct and! Imagination is analogous to latent variable with 64 latent variables in the introduction you. Different from Euclidean space and encoder using theSequential andfunctional model APIrespectively for training and examples. Autoencoder ( VAE ) in detail attributes in probabilistic terms details for building such a yourself! Do it is first to decide the late… Fig.2: each training example is the of! Motivation for a variational autoencoder structure value would you assign for the remainder of the and! As having the same characteristics ; in other words, we don ’ t know anything about the loss in. In that they do not use the encoding-decoding process to reconstruct inputs and learn meaningful representations of the input,. Let ’ s been generated by a tangent plane of the latent space should correspond with very to... Defined distribution representing the data set this tutorial ( VAE ) in detail generative models based on variational method 3... ) $ learning the distribution while still maintaining the ability to randomly sample from that distribution great. Dimension represents some learned attribute about the coding that ’ s move in for the remainder the. For each encoding dimension decoder and encoder using theSequential andfunctional model APIrespectively ll be breaking down VAEs and the... Maintaining the ability to randomly sample from that distribution ’ ve covered GANs a! Latent attribute for a given input as a specific example a specific example 19...

variational autoencoder example 2021