# variational autoencoder example

The evidence lower bound (ELBO) can be summarized as: ELBO = log-likelihood - KL Divergence. The first term represents the reconstruction likelihood and the second term ensures that our learned distribution $q$ is similar to the true prior distribution $p$. Thi… We are now ready to define the AEVB algorithm and the variational autoencoder, its most popular instantiation. However, as you read in the introduction, you'll only focus on the convolutional and denoising ones in this tutorial. variational_autoencoder. Implemented the decoder and encoder using theSequential andfunctional Model APIrespectively. As you can see, the distinct digits each exist in different regions of the latent space and smoothly transform from one digit to another. First, we imagine the animal: it must have four legs, and it must be able to swim. Recall that the KL divergence is a measure of difference between two probability distributions. Although they generate new data/images, still, those are very similar to the data they are trained on. This example shows how to create a variational autoencoder (VAE) in MATLAB to generate digit images. 9 min read, 26 Nov 2019 – See all 47 posts There are variety of autoencoders, such as the convolutional autoencoder, denoising autoencoder, variational autoencoder and sparse autoencoder. Example: Variational Autoencoder¶. The digits have been size-normalized and centered in a fixed-size image (28x28 pixels) with values from 0 … 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. The VAE generates hand-drawn digits in the style of the MNIST data set. 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. 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. Sample from a standard (parameterless) Gaussian. Machine learning engineer. VAEs try to force the distribution to be as close as possible to the standard normal distribution, which is centered around 0. Note: For variational autoencoders, the encoder model is sometimes referred to as the recognition model whereas the decoder model is sometimes referred to as the generative model. 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. VAEs differ from regular autoencoders in that they do not use the encoding-decoding process to reconstruct an input. Variational Auto Encoder Explained. Multiply the sample by the square root of $\Sigma_Q$. Mahmoud_Abdelkhalek (Mahmoud Abdelkhalek) November 19, 2020, 6:33pm #1. 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), … In this section, I'll provide the practical implementation details for building such a model yourself. 2. I also explored their capacity as generative models by comparing samples generated by a variational autoencoder to those generated by generative adversarial networks. 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. : https://github.com/rstudio/keras/blob/master/vignettes/examples/eager_cvae.R # Also cf. 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. One such application is called the variational autoencoder. In order to train the variational autoencoder, we only need to add the auxillary loss in our training algorithm. The two main approaches are Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). However, there are much more interesting applications for autoencoders. 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. In the variational autoencoder, is specified as a standard Normal distribution with mean zero and variance one. 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. 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. 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. 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. It is often useful to decide the late… the tfprobability-style of coding VAEs: https://rstudio.github.io/tfprobability/ # With TF-2, you can still run … Using a general autoencoder, we don’t know anything about the coding that’s been generated by our network. Explicitly made the noise an Input layer… We could compare different encoded objects, but it’s unlikely that we’ll be able to understand what’s going on. We can have a lot of fun with variational autoencoders if we can get … The variational autoencoder solves this problem by creating a defined distribution representing the data. However, the space of angles is topologically and geometrically different from Euclidean space. in an attempt to describe an observation in some compressed representation. Source: https://github.com/rstudio/keras/blob/master/vignettes/examples/variational_autoencoder.R, This script demonstrates how to build a variational autoencoder with Keras. 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. With this reparameterization, we can now optimize the parameters of the distribution while still maintaining the ability to randomly sample from that distribution. 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: $${E_{q\left( {z|x} \right)}}\log p\left( {x|z} \right) - KL\left( {q\left( {z|x} \right)||p\left( z \right)} \right)$$. 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. However, we may prefer to represent each latent attribute as a range of possible values. 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. 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. Broadly curious. 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. 3. Developed by Daniel Falbel, JJ Allaire, FranÃ§ois Chollet, RStudio, Google. 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. Here, we've sampled a grid of values from a two-dimensional Gaussian and displayed the output of our decoder network. 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. Today we’ll be breaking down VAEs and understanding the intuition behind them. For standard autoencoders, we simply need to learn an encoding which allows us to reproduce the input. Note. Variational Autoencoders are a class of deep generative models based on variational method [3]. The data set for this example is the collection of all frames. This blog post introduces a great discussion on the topic, which I'll summarize in this section. The main benefit of a variational autoencoder is that we're capable of learning smooth latent state representations of the input data. 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. # For an example of a TF2-style modularized VAE, see e.g. In the traditional derivation of a VAE, we imagine some process that generates the data, such as a latent variable generative model. Now that we have a bit of a feeling for the tech, let’s move in for the kill. However, we can apply varitational inference to estimate this value. Having those criteria, we could then actually generate the animal by sampling from the animal kingdom. 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. Variational autoencoder VAE. Example VAE in Keras; An autoencoder is a neural network that learns to copy its input to its output. 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. # Note: This code reflects pre-TF2 idioms. Variational AutoEncoder. The following code is essentially copy-and-pasted from above, with a single term added added to the loss (autoencoder.encoder.kl). So the next step here is to transfer to a Variational AutoEncoder. 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 . This example is using MNIST handwritten digits. 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. Add $\mu_Q$ to the result. 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. Thus, values which are nearby to one another in latent space should correspond with very similar reconstructions. I am a bit unsure about the loss function in the example implementation of a VAE on GitHub. Finally, I also added some annotations that make reference to the things we discussed in this post. class Sampling(layers.Layer): """Uses (z_mean, z_log_var) to sample z, the vector encoding a digit.""" Get the latest posts delivered right to your inbox, 2 Jan 2021 – For any sampling of the latent distributions, we're expecting our decoder model to be able to accurately reconstruct the input. the tfprobability-style of coding VAEs: https://rstudio.github.io/tfprobability/. Effective testing for machine learning systems. However, we may prefer to represent each late… Convolutional Autoencoders in … Worked with the log variance for numerical stability, and used aLambda layerto transform it to thestandard deviation when necessary. Click here to download the full example code. $$Sample = \mu + \epsilon\sigma$$ Here, $$\epsilon\sigma$$ is element-wise multiplication. 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. But there’s a difference between theory and practice. 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). Using a variational autoencoder, we can describe latent attributes in probabilistic terms. Variational autoencoder: They are good at generating new images from the latent vector. Unfortunately, computing $p\left( x \right)$ is quite difficult. Good way to do it is first to decide what kind of data we want to generate, then actually generate the data. The result will have a distribution equal to $Q$. We’ve covered GANs in a recent article which you can find here. 1. latent state) which was used to generate an observation. In the example above, we've described the input image in terms of its latent attributes using a single value to describe each attribute. 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. The decoder network then subsequently takes these values and attempts to recreate the original input. In other words, there are areas in latent space which don't represent any of our observed data. A simple solution for monitoring ML systems. To revisit our graphical model, we can use $q$ to infer the possible hidden variables (ie. 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. : https://github.com/rstudio/keras/blob/master/vignettes/examples/eager_cvae.R, # Also cf. This perhaps is the most important part of a … “Variational Autoencoders ... We can sample data using the PDF above. Autoencoders are an unsupervised learning technique in which we leverage neural networks for the task of representation learning. The end goal is to move to a generational model of new fruit images. $${\cal L}\left( {x,\hat x} \right) + \sum\limits_j {KL\left( {{q_j}\left( {z|x} \right)||p\left( z \right)} \right)}$$. In this post, I'll discuss commonly used architectures for convolutional networks. 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. However, this sampling process requires some extra attention. The figure below visualizes the data generated by the decoder network of a variational autoencoder trained on the MNIST handwritten digits dataset. 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. 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. What is an Autoencoder? 3 Gaussian Process Prior Variational Autoencoder Assume we are given a set of samples (e.g., images), each coupled with different types of auxiliary In the previous section, I established the statistical motivation for a variational autoencoder structure. A variational autoencoder (VAE) provides a probabilistic manner for describing an observation in latent space. I encourage you to do the same. In other words, we’d like to compute $p\left( {z|x} \right)$. How does a variational autoencoder work? →. Example implementation of a variational autoencoder. The ability of variational autoencoders to reconstruct inputs and learn meaningful representations of data was tested on the MNIST and Freyfaces datasets. $$\min KL\left( {q\left( {z|x} \right)||p\left( {z|x} \right)} \right)$$. Stay up to date! # 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. 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: Fig.2: Each training example is represented by a tangent plane of the manifold. # For an example of a TF2-style modularized VAE, see e.g. This usually turns out to be an intractable distribution. The code is from the Keras convolutional variational autoencoder example and I just made some small changes to the parameters. This script demonstrates how to build a variational autoencoder with Keras. However, we simply cannot do this for a random sampling process. A variational autoencoder (VAE) is a type of neural network that learns to reproduce its input, and also map data to latent space. # Note: This code reflects pre-TF2 idioms. 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. 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. 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 $x$. 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. $$p\left( {z|x} \right) = \frac{{p\left( {x|z} \right)p\left( z \right)}}{{p\left( x \right)}}$$. Therefore, in variational autoencoder, the encoder outputs a probability distribution in … The true latent factor is the angle of the turntable. 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. 15 min read. So, when you select a random sample out of the distribution to be decoded, you at least know its values are around 0. The dataset contains 60,000 examples for training and 10,000 examples for testing. 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 * … A VAE can generate samples by first sampling from the latent space. In a different blog post, we studied the concept of a Variational Autoencoder (or VAE) in detail. in an attempt to describe an observation in some compressed representation. Lo and behold, we get Platypus! 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. 4. Suppose we want to generate a data. The models, which are generative, can be used to manipulate datasets by learning the distribution of this input data. And the above formula is called the reparameterization trick in VAE. Variational AutoEncoders (VAEs) Background. From the story above, our imagination is analogous to latent variable. 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. With this approach, we'll now represent each latent attribute for a given input as a probability distribution. In particular, we 1. 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. We can only see $x$, but we would like to infer the characteristics of $z$. We will go into much more detail about what that actually means for the remainder of the article. In this post, we covered the basics of amortized variational inference, lookingat variational autoencoders as a specific example. In the work, we aim to develop a through under- This effectively treats every observation as having the same characteristics; in other words, we've failed to describe the original data. Kevin Frans. 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. $$p\left( x \right) = \int {p\left( {x|z} \right)p\left( z \right)dz}$$. Specifically, we'll sample from the prior distribution ${p\left( z \right)}$ which we assumed follows a unit Gaussian distribution. An ideal autoencoder will learn descriptive attributes of faces such as skin color, whether or not the person is wearing glasses, etc. Augmented the final loss with the KL divergence term by writing an auxiliarycustom layer. 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 $\sigma$. Reference: âAuto-Encoding Variational Bayesâ https://arxiv.org/abs/1312.6114. This simple insight has led to the growth of a new class of models - disentangled variational autoencoders. More specifically, our input data is converted into an encoding vector where each dimension represents some learned attribute about the data. For instance, what single value would you assign for the smile attribute if you feed in a photo of the Mona Lisa? For example, say, we want to generate an animal. class CVAE(tf.keras.Model): """Convolutional variational autoencoder.""" Figure 6 shows a sample of the digits I was able to generate with 64 latent variables in the above Keras example. 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. position. While it’s always nice to understand neural networks in theory, it’s […] $${\cal L}\left( {x,\hat x} \right) + \beta \sum\limits_j {KL\left( {{q_j}\left( {z|x} \right)||N\left( {0,1} \right)} \right)}$$. 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. Suppose that there exists some hidden variable $z$ which generates an observation $x$. 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. 10 min read, 19 Aug 2020 – An ideal autoencoder will learn descriptive attributes of faces such as skin color, whether or not the person is wearing glasses, etc. MNIST Dataset Overview. The variational auto-encoder. 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