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This example shows how to create a variational autoencoder (VAE) in MATLAB to generate digit images. The VAE generates hand-drawn digits in the style of the MNIST data set.

VAEs differ from regular autoencoders in that they do not use the encoding-decoding process to reconstruct an input. Instead, they impose a probability distribution on the latent space, and learn the distribution so that the distribution of outputs from the decoder matches that of the observed data. Then, they sample from this distribution to generate new data.

In this example, you construct a VAE network, train it on the MNIST data set, and generate new images that closely resemble those in the data set.

Download the MNIST files from http://yann.lecun.com/exdb/mnist/ and load the MNIST data set into the workspace [1]. Call the `processImagesMNIST`

and `processLabelsMNIST`

helper functions attached to this example to load the data from the files into MATLAB arrays.

Because the VAE compares the reconstructed digits against the inputs and not against the categorical labels, you do not need to use the training labels in the MNIST data set.

trainImagesFile = 'train-images-idx3-ubyte.gz'; testImagesFile = 't10k-images-idx3-ubyte.gz'; testLabelsFile = 't10k-labels-idx1-ubyte.gz'; XTrain = processImagesMNIST(trainImagesFile);

Read MNIST image data... Number of images in the dataset: 60000 ...

numTrainImages = size(XTrain,4); XTest = processImagesMNIST(testImagesFile);

Read MNIST image data... Number of images in the dataset: 10000 ...

YTest = processLabelsMNIST(testLabelsFile);

Read MNIST label data... Number of labels in the dataset: 10000 ...

Autoencoders have two parts: the encoder and the decoder. The encoder takes an image input and outputs a compressed representation (the encoding), which is a vector of size `latentDim`

, equal to 20 in this example. The decoder takes the compressed representation, decodes it, and recreates the original image.

To make calculations more numerically stable, increase the range of possible values from [0,1] to [-inf, 0] by making the network learn from the logarithm of the variances. Define two vectors of size `latent_dim`

: one for the means $\mu $ and one for the logarithm of the variances $\mathrm{log}\left({\sigma}^{2}\right)$. Then use these two vectors to create the distribution to sample from.

Use 2-D convolutions followed by a fully connected layer to downsample from the 28-by-28-by-1 MNIST image to the encoding in the latent space. Then, use transposed 2-D convolutions to scale up the 1-by-1-by-20 encoding back into a 28-by-28-by-1 image.

latentDim = 20; imageSize = [28 28 1]; encoderLG = layerGraph([ imageInputLayer(imageSize,'Name','input_encoder','Normalization','none') convolution2dLayer(3, 32, 'Padding','same', 'Stride', 2, 'Name', 'conv1') reluLayer('Name','relu1') convolution2dLayer(3, 64, 'Padding','same', 'Stride', 2, 'Name', 'conv2') reluLayer('Name','relu2') fullyConnectedLayer(2 * latentDim, 'Name', 'fc_encoder') ]); decoderLG = layerGraph([ imageInputLayer([1 1 latentDim],'Name','i','Normalization','none') transposedConv2dLayer(7, 64, 'Cropping', 'same', 'Stride', 7, 'Name', 'transpose1') reluLayer('Name','relu1') transposedConv2dLayer(3, 64, 'Cropping', 'same', 'Stride', 2, 'Name', 'transpose2') reluLayer('Name','relu2') transposedConv2dLayer(3, 32, 'Cropping', 'same', 'Stride', 2, 'Name', 'transpose3') reluLayer('Name','relu3') transposedConv2dLayer(3, 1, 'Cropping', 'same', 'Name', 'transpose4') ]);

To train both networks with a custom training loop and enable automatic differentiation, convert the layer graphs to `dlnetwork`

objects.

encoderNet = dlnetwork(encoderLG); decoderNet = dlnetwork(decoderLG);

The helper function modelGradients takes in the encoder and decoder `dlnetwork`

objects and a mini-batch of input data `X`

, and returns the gradients of the loss with respect to the learnable parameters in the networks. This helper function is defined at the end of this example.

The function performs this process in two steps: sampling and loss. The sampling step samples the mean and the variance vectors to create the final encoding to be passed to the decoder network. However, because backpropagation through a random sampling operation is not possible, you must use the *reparameterization trick*. This trick moves the random sampling operation to an auxiliary variable $\epsilon $, which is then shifted by the mean ${\mu}_{\mathit{i}}$ and scaled by the standard deviation ${\sigma}_{\mathit{i}}$. The idea is that sampling from $\mathit{N}\left({\mu}_{\mathit{i}},{\sigma}_{\mathit{i}}^{2}\right)$ is the same as sampling from ${\mu}_{\mathit{i}}+\epsilon \cdot {\sigma}_{\mathit{i}}$, where $\epsilon \sim \mathit{N}\left(0,1\right)$. The following figure depicts this idea graphically.

The loss step passes the encoding generated by the sampling step through the decoder network, and determines the loss, which is then used to compute the gradients. The loss in VAEs, also called the evidence lower bound (ELBO) loss, is defined as a sum of two separate loss terms:

$$\mathrm{ELBO}\text{\hspace{0.17em}}\mathrm{loss}=\mathrm{reconstruction}\text{\hspace{0.17em}}\mathrm{loss}+\mathrm{KL}\text{\hspace{0.17em}}\mathrm{loss}$$.

The *reconstruction loss* measures how close the decoder output is to the original input by using the mean-squared error (MSE):

$$\mathrm{reconstruction}\text{\hspace{0.17em}}\mathrm{loss}=\mathrm{MSE}\left(\mathrm{decoder}\text{\hspace{0.17em}}\mathrm{output},\mathrm{original}\text{\hspace{0.17em}}\mathrm{image}\right)$$.

The *KL loss*, or Kullback–Leibler divergence, measures the difference between two probability distributions. Minimizing the KL loss in this case means ensuring that the learned means and variances are as close as possible to those of the target (normal) distribution. For a latent dimension of size $\mathit{n}$, the KL loss is obtained as

$\mathrm{KL}\text{\hspace{0.17em}}\mathrm{loss}=-0.5\cdot \sum _{\mathit{i}=1}^{\mathit{n}}\left(1+\mathrm{log}\left({\sigma}_{\mathit{i}}^{2}\right)-{\mu}_{\mathit{i}}^{2}-{\sigma}_{\mathit{i}}^{2}\right)$.

The practical effect of including a KL loss term is to pack the clusters learned due to the reconstruction loss tightly around the center of the latent space, forming a continuous space to sample from.

Train on a GPU if one is available (requires Parallel Computing Toolbox™).

`executionEnvironment = "auto";`

Set the training options for the network. When using the Adam optimizer, you need to initialize for each network the trailing average gradient and the trailing average gradient-square decay rates with empty arrays`.`

numEpochs = 50; miniBatchSize = 512; lr = 1e-3; numIterations = floor(numTrainImages/miniBatchSize); iteration = 0; avgGradientsEncoder = []; avgGradientsSquaredEncoder = []; avgGradientsDecoder = []; avgGradientsSquaredDecoder = [];

Train the model using a custom training loop.

For each iteration in an epoch:

Obtain the next mini-batch from the training set.

Convert the mini-batch to a

`dlarray`

object, making sure to specify the dimension labels`'SSCB'`

(spatial, spatial, channel, batch).For GPU training, convert the

`dlarray`

to a`gpuArray`

object.Evaluate the model gradients using the

`dlfeval`

and`modelGradients`

functions.Update the network learnables and the average gradients for both networks, using the

`adamupdate`

function.

At the end of each epoch, pass the test set images through the autoencoder, and display the loss and the training time for that epoch.

for epoch = 1:numEpochs tic; for i = 1:numIterations iteration = iteration + 1; idx = (i-1)*miniBatchSize+1:i*miniBatchSize; XBatch = XTrain(:,:,:,idx); XBatch = dlarray(single(XBatch), 'SSCB'); if (executionEnvironment == "auto" && canUseGPU) || executionEnvironment == "gpu" XBatch = gpuArray(XBatch); end [infGrad, genGrad] = dlfeval(... @modelGradients, encoderNet, decoderNet, XBatch); [decoderNet.Learnables, avgGradientsDecoder, avgGradientsSquaredDecoder] = ... adamupdate(decoderNet.Learnables, ... genGrad, avgGradientsDecoder, avgGradientsSquaredDecoder, iteration, lr); [encoderNet.Learnables, avgGradientsEncoder, avgGradientsSquaredEncoder] = ... adamupdate(encoderNet.Learnables, ... infGrad, avgGradientsEncoder, avgGradientsSquaredEncoder, iteration, lr); end elapsedTime = toc; [z, zMean, zLogvar] = sampling(encoderNet, XTest); xPred = sigmoid(forward(decoderNet, z)); elbo = ELBOloss(XTest, xPred, zMean, zLogvar); disp("Epoch : "+epoch+" Test ELBO loss = "+gather(extractdata(elbo))+... ". Time taken for epoch = "+ elapsedTime + "s") end

Epoch : 1 Test ELBO loss = 28.0145. Time taken for epoch = 28.0573s Epoch : 2 Test ELBO loss = 24.8995. Time taken for epoch = 8.797s Epoch : 3 Test ELBO loss = 23.2756. Time taken for epoch = 8.8824s Epoch : 4 Test ELBO loss = 21.151. Time taken for epoch = 8.5979s Epoch : 5 Test ELBO loss = 20.5335. Time taken for epoch = 8.8472s Epoch : 6 Test ELBO loss = 20.232. Time taken for epoch = 8.5068s Epoch : 7 Test ELBO loss = 19.9988. Time taken for epoch = 8.4356s Epoch : 8 Test ELBO loss = 19.8955. Time taken for epoch = 8.4015s Epoch : 9 Test ELBO loss = 19.7991. Time taken for epoch = 8.8089s Epoch : 10 Test ELBO loss = 19.6773. Time taken for epoch = 8.4269s Epoch : 11 Test ELBO loss = 19.5181. Time taken for epoch = 8.5771s Epoch : 12 Test ELBO loss = 19.4532. Time taken for epoch = 8.4227s Epoch : 13 Test ELBO loss = 19.3771. Time taken for epoch = 8.5807s Epoch : 14 Test ELBO loss = 19.2893. Time taken for epoch = 8.574s Epoch : 15 Test ELBO loss = 19.1641. Time taken for epoch = 8.6434s Epoch : 16 Test ELBO loss = 19.2175. Time taken for epoch = 8.8641s Epoch : 17 Test ELBO loss = 19.158. Time taken for epoch = 9.1083s Epoch : 18 Test ELBO loss = 19.085. Time taken for epoch = 8.6674s Epoch : 19 Test ELBO loss = 19.1169. Time taken for epoch = 8.6357s Epoch : 20 Test ELBO loss = 19.0791. Time taken for epoch = 8.5512s Epoch : 21 Test ELBO loss = 19.0395. Time taken for epoch = 8.4674s Epoch : 22 Test ELBO loss = 18.9556. Time taken for epoch = 8.3943s Epoch : 23 Test ELBO loss = 18.9469. Time taken for epoch = 10.2924s Epoch : 24 Test ELBO loss = 18.924. Time taken for epoch = 9.8302s Epoch : 25 Test ELBO loss = 18.9124. Time taken for epoch = 9.9603s Epoch : 26 Test ELBO loss = 18.9595. Time taken for epoch = 10.9887s Epoch : 27 Test ELBO loss = 18.9256. Time taken for epoch = 10.1402s Epoch : 28 Test ELBO loss = 18.8708. Time taken for epoch = 9.9109s Epoch : 29 Test ELBO loss = 18.8602. Time taken for epoch = 10.3075s Epoch : 30 Test ELBO loss = 18.8563. Time taken for epoch = 10.474s Epoch : 31 Test ELBO loss = 18.8127. Time taken for epoch = 9.8779s Epoch : 32 Test ELBO loss = 18.7989. Time taken for epoch = 9.6963s Epoch : 33 Test ELBO loss = 18.8. Time taken for epoch = 9.8848s Epoch : 34 Test ELBO loss = 18.8095. Time taken for epoch = 10.3168s Epoch : 35 Test ELBO loss = 18.7601. Time taken for epoch = 10.8058s Epoch : 36 Test ELBO loss = 18.7469. Time taken for epoch = 9.9365s Epoch : 37 Test ELBO loss = 18.7049. Time taken for epoch = 10.0343s Epoch : 38 Test ELBO loss = 18.7084. Time taken for epoch = 10.3214s Epoch : 39 Test ELBO loss = 18.6858. Time taken for epoch = 10.3985s Epoch : 40 Test ELBO loss = 18.7284. Time taken for epoch = 10.9685s Epoch : 41 Test ELBO loss = 18.6574. Time taken for epoch = 10.5241s Epoch : 42 Test ELBO loss = 18.6388. Time taken for epoch = 10.2392s Epoch : 43 Test ELBO loss = 18.7133. Time taken for epoch = 9.8177s Epoch : 44 Test ELBO loss = 18.6846. Time taken for epoch = 9.6858s Epoch : 45 Test ELBO loss = 18.6001. Time taken for epoch = 9.5588s Epoch : 46 Test ELBO loss = 18.5897. Time taken for epoch = 10.4554s Epoch : 47 Test ELBO loss = 18.6184. Time taken for epoch = 10.0317s Epoch : 48 Test ELBO loss = 18.6389. Time taken for epoch = 10.311s Epoch : 49 Test ELBO loss = 18.5918. Time taken for epoch = 10.4506s Epoch : 50 Test ELBO loss = 18.5081. Time taken for epoch = 9.9671s

To visualize and interpret the results, use the helper Visualization functions. These helper functions are defined at the end of this example.

The `VisualizeReconstruction`

function shows a randomly chosen digit from each class accompanied by its reconstruction after passing through the autoencoder.

The `VisualizeLatentSpace`

function takes the mean and the variance encodings (each of dimension 20) generated after passing the test images through the encoder network, and performs principal component analysis (PCA) on the matrix containing the encodings for each of the images. You can then visualize the latent space defined by the means and the variances in the two dimensions characterized by the two first principal components.

The `Generate`

function initializes new encodings sampled from a normal distribution, and outputs the images generated when these encodings pass through the decoder network.

visualizeReconstruction(XTest, YTest, encoderNet, decoderNet)

visualizeLatentSpace(XTest, YTest, encoderNet)

generate(decoderNet, latentDim)

Variational autoencoders are only one of the many available models used to perform generative tasks. They work well on data sets where the images are small and have clearly defined features (such as MNIST). For more complex data sets with larger images, generative adversarial networks (GANs) tend to perform better and generate images with less noise. For an example showing how to implement GANs to generate 64-by-64 RGB images, see Train Generative Adversarial Network (GAN).

LeCun, Y., C. Cortes, and C. J. C. Burges. "The MNIST Database of Handwritten Digits." http://yann.lecun.com/exdb/mnist/.

The `modelGradients`

function takes the encoder and decoder `dlnetwork`

objects and a mini-batch of input data `X`

, and returns the gradients of the loss with respect to the learnable parameters in the networks. The function performs three operations:

Obtain the encodings by calling the

`sampling`

function on the mini-batch of images that passes through the encoder network.Obtain the loss by passing the encodings through the decoder network and calling the

`ELBOloss`

function.Compute the gradients of the loss with respect to the learnable parameters of both networks by calling the

`dlgradient`

function.

function [infGrad, genGrad] = modelGradients(encoderNet, decoderNet, x) [z, zMean, zLogvar] = sampling(encoderNet, x); xPred = sigmoid(forward(decoderNet, z)); loss = ELBOloss(x, xPred, zMean, zLogvar); [genGrad, infGrad] = dlgradient(loss, decoderNet.Learnables, ... encoderNet.Learnables); end

The `sampling`

function obtains encodings from input images. Initially, it passes a mini-batch of images through the encoder network and splits the output of size `(2*latentDim)*miniBatchSize`

into a matrix of means and a matrix of variances, each of size `latentDim*batchSize`

. Then, it uses these matrices to implement the reparameterization trick and to compute the encoding. Finally, it converts this encoding to a `dlarray`

object in SSCB format.

function [zSampled, zMean, zLogvar] = sampling(encoderNet, x) compressed = forward(encoderNet, x); d = size(compressed,1)/2; zMean = compressed(1:d,:); zLogvar = compressed(1+d:end,:); sz = size(zMean); epsilon = randn(sz); sigma = exp(.5 * zLogvar); z = epsilon .* sigma + zMean; z = reshape(z, [1,1,sz]); zSampled = dlarray(z, 'SSCB'); end

The `ELBOloss`

function takes the encodings of the means and the variances returned by the `sampling `

function, and uses them to compute the ELBO loss.

function elbo = ELBOloss(x, xPred, zMean, zLogvar) squares = 0.5*(xPred-x).^2; reconstructionLoss = sum(squares, [1,2,3]); KL = -.5 * sum(1 + zLogvar - zMean.^2 - exp(zLogvar), 1); elbo = mean(reconstructionLoss + KL); end

The `VisualizeReconstruction`

function randomly chooses two images for each digit of the MNIST data set, passes them through the VAE, and plots the reconstruction side by side with the original input. Note that to plot the information contained inside a `dlarray `

object, you need to extract it first using the `extractdata`

and `gather`

functions.

function visualizeReconstruction(XTest,YTest, encoderNet, decoderNet) f = figure; figure(f) title("Example ground truth image vs. reconstructed image") for i = 1:2 for c=0:9 idx = iRandomIdxOfClass(YTest,c); X = XTest(:,:,:,idx); [z, ~, ~] = sampling(encoderNet, X); XPred = sigmoid(forward(decoderNet, z)); X = gather(extractdata(X)); XPred = gather(extractdata(XPred)); comparison = [X, ones(size(X,1),1), XPred]; subplot(4,5,(i-1)*10+c+1), imshow(comparison,[]), end end end function idx = iRandomIdxOfClass(T,c) idx = T == categorical(c); idx = find(idx); idx = idx(randi(numel(idx),1)); end

The `VisualizeLatentSpace`

function visualizes the latent space defined by the mean and the variance matrices that form the output of the encoder network, and locates the clusters formed by the latent space representations of each digit.

The function starts by extracting the mean and the variance matrices from the `dlarray`

objects. Because transposing a matrix with channel/batch dimensions (C and B) is not possible, the function calls `stripdims`

before transposing the matrices. Then, it carries out a principal component analysis (PCA) on both matrices. To visualize the latent space in two dimensions, the function keeps the first two principal components and plots them against each other. Finally, the function colors the digit classes so that you can observe clusters.

function visualizeLatentSpace(XTest, YTest, encoderNet) [~, zMean, zLogvar] = sampling(encoderNet, XTest); zMean = stripdims(zMean)'; zMean = gather(extractdata(zMean)); zLogvar = stripdims(zLogvar)'; zLogvar = gather(extractdata(zLogvar)); [~,scoreMean] = pca(zMean); [~,scoreLogvar] = pca(zLogvar); c = parula(10); f1 = figure; figure(f1) title("Latent space") ah = subplot(1,2,1); scatter(scoreMean(:,2),scoreMean(:,1),[],c(double(YTest),:)); ah.YDir = 'reverse'; axis equal xlabel("Z_m_u(2)") ylabel("Z_m_u(1)") cb = colorbar; cb.Ticks = 0:(1/9):1; cb.TickLabels = string(0:9); ah = subplot(1,2,2); scatter(scoreLogvar(:,2),scoreLogvar(:,1),[],c(double(YTest),:)); ah.YDir = 'reverse'; xlabel("Z_v_a_r(2)") ylabel("Z_v_a_r(1)") cb = colorbar; cb.Ticks = 0:(1/9):1; cb.TickLabels = string(0:9); axis equal end

The `generate`

function tests the generative capabilities of the VAE. It initializes a `dlarray`

object containing 25 randomly generated encodings, passes them through the decoder network, and plots the outputs.

function generate(decoderNet, latentDim) randomNoise = dlarray(randn(1,1,latentDim,25),'SSCB'); generatedImage = sigmoid(predict(decoderNet, randomNoise)); generatedImage = extractdata(generatedImage); f3 = figure; figure(f3) imshow(imtile(generatedImage, "ThumbnailSize", [100,100])) title("Generated samples of digits") drawnow end

`adamupdate`

| `dlarray`

| `dlfeval`

| `dlgradient`

| `dlnetwork`

| `layerGraph`

| `sigmoid`