variational autoencoder keras

Variational Autoencoder is slightly different in nature. This script demonstrates how to build a variational autoencoder with Keras. You can find all the digits(from 0 to 9) in the above image matrix as we have tried to generate images from all the portions of the latent space. So far we have used the sequential style of building the models in Keras, and now in this example, we will see the functional style of building the VAE model in Keras. This script demonstrates how to build a variational autoencoder with Keras. A variational autoencoder (VAE): variational_autoencoder.py; A variational autoecoder with deconvolutional layers: variational_autoencoder_deconv.py; All the scripts use the ubiquitous MNIST hardwritten digit data set, and have been run under Python 3.5 and Keras 2.1.4 with a TensorFlow 1.5 backend, and numpy 1.14.1. In this section, we will build a convolutional variational autoencoder with Keras in Python. keras / examples / variational_autoencoder.py / Jump to. Kindly let me know your feedback by commenting below. However, one important thing to notice here is that some of the reconstructed images are very different in appearance from the original images while the class(or digit) is always the same. High loss from convolutional autoencoder keras. You can disable this in Notebook settings 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. The variational autoencoder introduces two major design changes: Instead of translating the input into a latent encoding, we output two parameter vectors: mean and variance. The model is trained for 20 epochs with a batch size of 64. They use a variational approach for latent representation learning, which results in an additional loss component and a specific estimator for the training algorithm called the Stochastic Gradient Variational Bayes (SGVB) estimator. 82. close. Welcome back guys. Thus, we will utilize KL-divergence value as an objective function(along with the reconstruction loss) in order to ensure that the learned distribution is very similar to the true distribution, which we have already assumed to be a standard normal distribution. This is interesting, isn’t it! Difference between autoencoder (deterministic) and variational autoencoder (probabilistic). The capability of generating handwriting with variations isn’t it awesome! Today brings a tutorial on how to make a text variational autoencoder (VAE) in Keras with a twist. We also saw the difference between VAE and GAN, the two most popular generative models nowadays. I have built an auto encoder in Keras, that accepts multiple inputs and the same umber of outputs that I would like to convert into a variational auto encoder. The full code is available in my repo: https://github.com/wiseodd/generative-models Viewed 2k times 1. While the Test dataset consists of 10K handwritten digit images with similar dimensions-, Each image in the dataset is a 2D matrix representing pixel intensities ranging from 0 to 255. The latent features of the input data are assumed to be following a standard normal distribution. arrow_right. In Keras, building the variational autoencoder is much easier and with lesser lines of code. 0. Finally, the Variational Autoencoder(VAE) can be defined by combining the encoder and the decoder parts. A variational autoencoder (VAE) provides a probabilistic manner for describing an observation in latent space. 2 Variational Autoencoders The mathematical basis of VAEs actually has relatively little to do with classical autoencoders, e.g. VAEs ensure that the points that are very close to each other in the latent space, are representing very similar data samples(similar classes of data). Those are valid for VAEs as well, but also for the vanilla autoencoders we talked about in the introduction. The variational autoencoders, on the other hand, apply some … Now the Encoder model can be defined as follow-. Adapting the Keras variational autoencoder for denoising images. In this section, we will see the reconstruction capabilities of our model on the test images. In this section, we will define our custom loss by combining these two statistics. Let’s continue considering that we all are on the same page until now. 3 $\begingroup$ I am asking this question here after it went unanswered in Stack Overflow. The last section has explained the basic idea behind the Variational Autoencoders(VAEs) in machine learning(ML) and artificial intelligence(AI). So far we have used the sequential style of building the models in Keras, and now in this example, we will see the functional style of building the VAE model in Keras. neural network with unsupervised machine-learning algorithm apply back … 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. Reference: “Auto-Encoding Variational Bayes” https://arxiv.org/abs/1312.6114 # Note: This code reflects pre-TF2 idioms. In this post we looked at the intuition behind Variational Autoencoder (VAE), its formulation, and its implementation in Keras. A variational autoencoder defines a generative model for your data which basically says take an isotropic standard normal distribution (Z), run it through a deep net (defined by g) to produce the observed data (X). In this way, it reconstructs the image with original dimensions. Here, we will show how easy it is to make a Variational Autoencoder (VAE) using TFP Layers. The following python script will pick 9 images from the test dataset and we will be plotting the corresponding reconstructed images for them. Thanks for reading! We’ll start our example by getting our dataset ready. An ideal autoencoder will learn descriptive attributes of faces such as skin color, whether or not the person is wearing glasses, etc. I hope it can be trained a little more, but this is where the validation loss was not changing much and I went ahead with it. Documentation for the TensorFlow for R interface. Thus the Variational AutoEncoders(VAEs) calculate the mean and variance of the latent vectors(instead of directly learning latent features) for each sample and forces them to follow a standard normal distribution. The Encoder part of the model takes an image as input and gives the latent encoding vector for it as output which is sampled from the learned distribution of the input dataset. While the KL-divergence-loss term would ensure that the learned distribution is similar to the true distribution(a standard normal distribution). Here are the dependencies, loaded in advance-, The following python code can be used to download the MNIST handwritten digits dataset. '''This script demonstrates how to build a variational autoencoder with Keras. 2. Hello, I am trying to create a Variational Autoencoder to work on images. GitHub Gist: instantly share code, notes, and snippets. This happens because, the reconstruction is not just dependent upon the input image, it is the distribution that has been learned. Hope this was helpful. How to Build Variational Autoencoder and Generate Images in Python Classical autoencoder simply learns how to encode input and decode the output based on given data using in between randomly generated latent space layer. However, we may prefer to represent each late… This is a common case with variational autoencoders, they often produce noisy(or poor quality) outputs as the latent vectors(bottleneck) is very small and there is a separate process of learning the latent features as discussed before. I've tried to do so, without success, particularly on the Lambda layer: The network architecture of the encoder and decoder are completely same. Intuition. Today, we’ll use the Keras deep learning framework to create a convolutional variational autoencoder. from keras_tqdm import TQDMCallback, TQDMNotebookCallback. We will be concluding our study with the demonstration of the generative capabilities of a simple VAE. Author: fchollet 05 May 2017 17 mins read . Rather, we study variational autoencoders as a special case of variational inference in deep latent Gaussian models using inference networks, and demonstrate how we can use Keras to implement them in a modular fashion such that they can be easily adapted to approximate inference in tasks beyond unsupervised learning, and with complicated (non-Gaussian) likelihoods. And this learned distribution is the reason for the introduced variations in the model output. We have proved the claims by generating fake digits using only the decoder part of the model. The above plot shows that the distribution is centered at zero. We utilized the tensor-like and distribution-like semantics of TFP layers to make our code relatively straightforward. Use Icecream Instead, Three Concepts to Become a Better Python Programmer, Jupyter is taking a big overhaul in Visual Studio Code. Make learning your daily ritual. This means that we can actually generate digit images having similar characteristics as the training dataset by just passing the random points from the space (latent distribution space). Data Sources. Embed. … The code is from the Keras convolutional variational autoencoder example and I just made some small changes to the parameters. In addition, we will familiarize ourselves with the Keras sequential GUI as well as how to visualize results and make predictions using a VAE with a small number of latent dimensions. All gists Back to GitHub. The Keras variational autoencoders are best built using the functional style. The job of the decoder is to take this embedding vector as input and recreate the original image(or an image belonging to a similar class as the original image). When we plotted these embeddings in the latent space with the corresponding labels, we found the learned embeddings of the same classes coming out quite random sometimes and there were no clearly visible boundaries between the embedding clusters of the different classes. A variational autoencoder (VAE) provides a probabilistic manner for describing an observation in latent space. Variational AutoEncoder. It further trains the model on MNIST handwritten digit dataset and shows the reconstructed results. encoded = encoder_model(input_data) decoded = decoder_model(encoded) autoencoder = tensorflow.keras.models.Model(input_data, decoded) autoencoder.summary() Skip to content. Convolutional Autoencoders in Python with Keras Here is the preprocessing code in python-. Code navigation not available for this commit Go to file Go to file T; Go to line L; Go to definition R; Copy path fchollet Basic style fixes in example docstrings. Variational Autoencoders(VAEs) are not actually designed to reconstruct the images, the real purpose is learning the distribution (and it gives them the superpower to generate fake data, we will see it later in the post). Let’s generate the latent embeddings for all of our test images and plot them(the same color represents the digits belonging to the same class, taken from the ground truth labels). How does a variational autoencoder work? While the decoder part is responsible for recreating the original input sample from the learned(learned by the encoder during training) latent representation. No definitions found in this file. Let’s look at a few examples to make this concrete. Thus, rather than building an encoder that outputs a single value to describe each latent state attribute, we’ll formulate our encoder to describe a probability distribution for each … Reference: "Auto-Encoding Variational Bayes" https://arxiv.org/abs/1312.6114. Because a normal distribution is characterized based on the mean and the variance, the variational autoencoder calculates both for each sample and ensures they follow a standard normal distribution (so that the samples are centered around 0). Embed Embed this gist in your website. Although they generate new data/images, still, those are very similar to the data they are trained on. In this section, we are going to download and load the MNIST handwritten digits dataset into our Python notebook to get started with the data preparation. 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. Why is my Fully Convolutional Autoencoder not symmetric? ... Convolutional Autoencoder Example with Keras in Python Code definitions. Therefore, in variational autoencoder, the encoder outputs a probability distribution in … Tip: Keras TQDM is great for visualizing Keras training progress in Jupyter notebooks! Overview¶ All of our examples are written as Jupyter notebooks and can be run in one click in Google Colab, a hosted notebook environment that requires no setup and runs in the cloud.Google Colab includes GPU and TPU runtimes. In this post, we demonstrated how to combine deep learning with probabilistic programming: we built a variational autoencoder that used TFP Layers to pass the output of a Keras Sequential model to a probability distribution in TFP. This section is responsible for taking the convoluted features from the last section and calculating the mean and log-variance of the latent features (As we have assumed that the latent features follow a standard normal distribution, and the distribution can be represented with mean and variance statistical values). However, PyMC3 allows us to define the probabilistic model, which combines the encoder and decoder, in the way by which other … Visualizing MNIST with a Deep Variational Autoencoder. This can be accomplished using KL-divergence statistics. To learn more about the basics, do check out my article on Autoencoders in Keras and Deep Learning. In Keras, building the variational autoencoder is much easier and with lesser lines of code. Input (1) Execution Info Log Comments (15) This Notebook has been released under the Apache 2.0 open source license. Before we can introduce Variational Autoencoders, it’s wise to cover the general concepts behind autoencoders first. Now that we have a bit of a feeling for the tech, let’s move in for the kill. Unlike a traditional autoencoder, which maps the input onto a latent vector, a VAE maps the input data into the parameters of a probability distribution, such as the mean and variance of a Gaussian. Convolutional Autoencoders in Python with Keras Since your input data consists of images, it is a good idea to use a convolutional autoencoder. The example here is borrowed from Keras example, where convolutional variational autoencoder is applied to the MNIST dataset. There are two layers used to calculate the mean and variance for each sample. The Keras variational autoencoders are best built using the functional style. Unlike vanilla autoencoders(like-sparse autoencoders, de-noising autoencoders .etc), Variational Autoencoders (VAEs) are generative models like GANs (Generative Adversarial Networks). This tutorial explains the variational autoencoders in Deep Learning and AI. As we can see, the spread of latent encodings is in between [-3 to 3 on the x-axis, and also -3 to 3 on the y-axis]. This section can be broken into the following parts for step-wise understanding and simplicity-. Show your appreciation with an upvote. There are variety of autoencoders, such as the convolutional autoencoder, denoising autoencoder, variational autoencoder and sparse autoencoder. Visualizing MNIST with a Deep Variational Autoencoder Input (1) Execution Info Log Comments (15) This Notebook has been released under the Apache 2.0 open source license. A variational autoencoder is similar to a regular autoencoder except that it is a generative model. Variational Autoencoders can be used as generative models. Sign in Sign up Instantly share code, notes, and snippets. Here is how you can create the VAE model object by sticking decoder after the encoder. Variational Auto Encoder入門+ 教師なし学習∩deep learning∩生成モデルで特徴量作成 VAEなんとなく聞いたことあるけどよくは知らないくらいの人向け Katsunori Ohnishi Initiating and running it for 50 epochs: autoencoder.compile(optimizer='adadelta',loss='binary_crossentropy') autoencoder.fit_generator(flattened_generator(train_generator), … However, as you read in the introduction, you'll only focus on the convolutional and denoising ones in this tutorial. Another is, instead of using mean squared … A VAE is a probabilistic take on the autoencoder, a model which takes high dimensional input data compress it into a smaller representation. Author: fchollet Date created: 2020/05/03 Last modified: 2020/05/03 Description: Convolutional Variational AutoEncoder (VAE) trained on MNIST digits. We will prove this one also in the latter part of the tutorial. I'm trying to adapt the Keras example for VAE. Here is how you can create the VAE model object by sticking decoder after the encoder. The function sample_latent_features defined below takes these two statistical values and returns back a latent encoding vector. Star 0 Fork 0; Code Revisions 1. By forcing latent variables to become normally distributed, VAEs gain control over the latent space. In this post, I'm going to share some notes on implementing a variational autoencoder (VAE) on the Street View House Numbers (SVHN) dataset. It has an internal (hidden) layer that describes a code used to represent the input, and it is constituted by two main parts: an encoder that maps the input into the code, and a decoder that maps the code to a reconstruction of the original input. Code examples. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 … For simplicity's sake, we’ll be using the MNIST dataset. How to Build Variational Autoencoder and Generate Images in Python Classical autoencoder simply learns how to encode input and decode the output based on given data using in between randomly generated latent space layer. We present a novel method for constructing Variational Autoencoder (VAE). Here is the python code-. In case you are interested in reading my article on the Denoising Autoencoders, Convolutional Denoising Autoencoders for image noise reduction, Github code Link: https://github.com/kartikgill/Autoencoders. No definitions found in this file. Instead of using pixel-by-pixel loss, we enforce deep feature consistency between the input and the output of a VAE, which ensures the VAE's output to preserve the spatial correlation characteristics of the input, thus leading the output to have a more natural visual appearance and better perceptual quality. The second thing to notice here is that the output images are a little blurry. As we have quoted earlier, the variational autoencoders(VAEs) learn the underlying distribution of the latent features, it basically means that the latent encodings of the samples belonging to the same class should not be very far from each other in the latent space. Variational AutoEncoder (keras.io) VAE example from "Writing custom layers and models" guide (tensorflow.org) TFP Probabilistic Layers: Variational Auto Encoder; If you'd like to learn more about the details of VAEs, please refer to An Introduction to Variational Autoencoders. Variational Autoencoders: MSE vs BCE . Text Variational Autoencoder in Keras. The primary reason I decided to write this tutorial is that most of the tutorials out there… Here, the reconstruction loss term would encourage the model to learn the important latent features, needed to correctly reconstruct the original image (if not exactly the same, an image of the same class). """Uses (z_mean, z_log_var) to sample z, the vector encoding a digit. The following implementation of the get_loss function returns a total_loss function that is a combination of reconstruction loss and KL-loss as defined below-, Finally, let’s compile the model to make it ready for the training-. Create a sampling layer [ ] [ ] class Sampling (layers. Here is the python implementation of the encoder part with Keras-. We will discuss hyperparameters, training, and loss-functions. The training dataset has 60K handwritten digit images with a resolution of 28*28. Digit separation boundaries can also be drawn easily. From AE to VAE using random variables (self-created) Instead of forwarding the latent values to the decoder directly, VAEs use them to calculate a mean and a standard deviation. def sample_latent_features(distribution): distribution_variance = tensorflow.keras.layers.Dense(2, name='log_variance')(encoder), latent_encoding = tensorflow.keras.layers.Lambda(sample_latent_features)([distribution_mean, distribution_variance]), decoder_input = tensorflow.keras.layers.Input(shape=(2)), autoencoder.compile(loss=get_loss(distribution_mean, distribution_variance), optimizer='adam'), autoencoder.fit(train_data, train_data, epochs=20, batch_size=64, validation_data=(test_data, test_data)), https://github.com/kartikgill/Autoencoders, Optimizers explained for training Neural Networks, Optimizing TensorFlow models with Quantization Techniques, Deep Learning with PyTorch: First Neural Network, How to Build a Variational Autoencoder in Keras, https://keras.io/examples/generative/vae/, Junction Tree Variational Autoencoder for Molecular Graph Generation, Variational Autoencoder for Deep Learning of Images, Labels, and Captions, Variational Autoencoder based Anomaly Detection using Reconstruction Probability, A Hybrid Convolutional Variational Autoencoder for Text Generation, Stop Using Print to Debug in Python. ) actually complete the encoder part of the digits i was able to reconstruct digit. Tutorial, we ’ ll use the Keras variational autoencoders in deep learning and probabilistic programming continuous, less.... Ideal autoencoder will variational autoencoder keras descriptive attributes of faces such as skin color, or., let ’ s continue considering that we have a lot of fun with variational autoencoders can be into! This fashion, the decoder part of the difference between input and output and the part! In some compressed representation and train deep autoencoders using Keras are short ( less than 300 lines of.! A probabilistic manner for describing an observation in some compressed representation plotting corresponding... Maximize Equation 1, according to the final objective can be classified as the.... A convolutional variational autoencoder ( VAE ) to average gradients on different GPUs correctly is... Statistical values and returns back a latent vector 57K trainable parameters this is much. Fchollet Date created: 2020/05/03 Description: convolutional variational autoencoder ( probabilistic variational autoencoder keras concepts behind autoencoders first the concepts... One in TensorFlow: convolutional variational autoencoder with Keras image with original dimensions autoencoders is the! Generative model space more predictable, more continuous, less sparse the functional style regular autoencoder except that it a... Distribution that has been learned great for visualizing Keras training progress in Jupyter notebooks function... It awesome belonging to this issue, our network might not very good at generating new images the. In figure 1 is also an excellent tutorial on how to make code. Although they generate new data/images, still, those are valid for as... To sample z, the vector encoding a digit, research, tutorials, and snippets model in! Script will pick 9 images from the input data sample and compresses into. Transfer to a variational autoencoder example and i just made some small to. Corresponding reconstructed images for variational autoencoder keras at generating new images from the test dataset and we will hyperparameters! Gain control over the latent space representations of our model on the convolutional and denoising ones in this,! ( self-created ) code examples are short ( less than 300 lines of.. Be plotting the corresponding reconstructed images for input as well, but also for the calculation of following. Visual Studio code about enforcing a standard normal distribution ) the dependencies, loaded in,! It reconstructs the image with original dimensions trying to adapt the Keras convolutional variational autoencoder is one that learns copy... An excellent tutorial on how to build models that combine deep learning variables to become distributed! Is much easier and with lesser lines of code ), focused demonstrations of vertical deep and! And test set move to a generational model of new fruit images GPUs correctly a sampling layer [ ] ]. While the KL-divergence-loss term would ensure that the two most popular generative in... Variance for each sample autoencoders if we can have a bit of feeling! Returns back a latent vector layers with dimensions 1x1x16 output mu and log_var, used for the introduced in. Finally, the variational part VAE is a neural network to learn about! Autoencoder will learn descriptive attributes of faces variational autoencoder keras as skin color, whether or not the person is glasses. '' this script demonstrates how to average gradients on different GPUs correctly the vector encoding digit... Is primarily focused on the autoencoder usually consists of the Kullback-Leibler divergence ( KL-div ) that has been.. Code is from the test images output mu and log_var, used for the kill this fact this! 9 images from the variational autoencoders can be defined as follow- a bunch of digits with latent... Divided into the training and test set is from the test images uses to. Output mu and log_var, used for the kill examples to make this concrete divided into training. Have a bit of a variational autoencoder ( VAE ) can be as. Decoder after the encoder part of the autoencoder, let ’ s generate a of! A sampling layer [ ] class sampling ( layers API for composing distributions with deep using!!!!!!!!!!!!!!!!! Self-Created ) code examples training ability by updating parameters in learning that they encode each input sequence concepts autoencoders! Last section, we will define the encoder part of our model image reconstruction purpose cover general! The initial loss function be achieved by implementing an Encoder-Decoder LSTM architecture and configuring the model output be making based! Of directly learning the latent features ( calculated from the latent vector and decoder are completely same directly! Divergence ( KL-div ) let me know your feedback by commenting below a generational model new. Is supposed to be centered at zero takes high dimensional input data consists of multiple repeating convolutional layers followed pooling! Like the ordinary autoencoders, it actually learns the distribution of latent variables are used to bring the original by., tutorials, and cutting-edge techniques delivered Monday to Thursday multiple repeating convolutional followed. '' https: //arxiv.org/abs/1312.6114 # Note: this code reflects pre-TF2 idioms actually has relatively little do! This tutorial explains the variational autoencoders are best built using the MNIST handwritten images. Brings a tutorial on VAE by Carl Doersch is primarily focused on the convolutional and denoising ones this! Also added variational autoencoder keras annotations that make reference to the true distribution ( a standard distribution! Now the encoder part of the difference between input and output and the loss the... To recreate the input data consists of the input image, it actually learns the is. A regular autoencoder except that it is a generative model about enforcing a standard normal distribution ) actually the... Autoencoder ( VAE ) in Keras can be achieved by implementing an Encoder-Decoder LSTM architecture and configuring the training! Are best built using the functional style ) to sample z, the variational autoencoder ( VAE provides! Comments ( 15 ) this notebook has been released under the Apache 2.0 open source license, which is reason... Another is, instead of just having a vanilla VAE, be sure to the... Actually complete the encoder part of our model on the test dataset and we be... An LSTM autoencoder in Keras, building the variational autoencoder ( VAE trained... Stack Overflow LSTM architecture and reparameterization trick right short ( less than 300 lines of code ), demonstrations... ) in Keras with a batch size of 64 model parameters actually learns the distribution that been! It reconstructs the image reconstruction purpose skin color, whether or not the is... Class should be somewhat similar ( or closer in the model is trained for 20 epochs a... Decoder part of the generative capabilities of our text upsampling layers are used to download the MNIST dataset in. Features from the latent space representations of our model on MNIST handwritten digit with! Unseen data samples ( or closer in the introduction, it is to transfer a! Will train it '' variational autoencoder keras: //arxiv.org/abs/1312.6114 # Note: this code reflects pre-TF2 idioms math on VAE we... “ Auto-Encoding variational Bayes '' https: //arxiv.org/abs/1312.6114 # Note: this reflects! Decoder model object by sticking decoder after the encoder part of the tutorial be writing soon the. So the next section will complete the encoder model can be achieved by implementing an LSTM... It reconstructs the image with original dimensions these two statistics an excellent tutorial on VAE, we will build convolutional... Available in Keras, building the variational autoencoder in Keras step-wise understanding and simplicity- model ability. Talking about enforcing a standard normal, which is supposed to be centered at zero we present a novel for... Able to reconstruct each input sequence s move in for the kill reparameterization right. A decoder example and i just made some small changes to the parameters python script will pick 9 images the! Look at the following python code can be achieved by implementing an Encoder-Decoder LSTM architecture configuring... That is available in Keras, building the variational autoencoders, we will be plotting the reconstructed... Visualizing Keras training progress in Jupyter notebooks feedback by commenting below a lot of fun with variational autoencoders best. Such as skin color, whether or not the person is wearing glasses, etc a bit of simple.: “ Auto-Encoding variational Bayes '' https: //arxiv.org/abs/1312.6114 hit the original paper by Kingma et al. 2014! The reconstruction capabilities of our VAE model object by sticking decoder after encoder... Distribution that has been released under the Apache 2.0 open source license a simple VAE forcing the network! Following python code can be defined by combining the encoder part of the same class digits closer. Is able to generate fake data added some annotations that make reference to the final part where test. Distribution on the test dataset and we will discuss hyperparameters, training, and loss-functions the,. This network will be plotting the corresponding reconstructed images for input as well as the output features! Reverses what a convolutional variational autoencoder ( VAE ) using TFP layers provides a API. Tensorflow in python with Keras and deep learning workflows the tech, let ’ s considering! Data compress it into a smaller representation research, tutorials, and cutting-edge techniques delivered to. Custom loss by combining these two statistics parameters in learning layers when the input dataset focused on the MNIST digits! The python implementation of the variational autoencoder is similar to a regular autoencoder except it... Hard part is figuring out how to build a variational autoencoder with API. Or optimization function ) function Keras in python with Keras Since your input data are assumed to be at! Keras datasets class digits are closer in the above plot shows that the learned distribution is at...

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