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Posit AI Weblog: Illustration studying with MMD-VAE

Not too long ago, we confirmed generate images utilizing generative adversarial networks (GANs). GANs could yield superb outcomes, however the contract there mainly is: what you see is what you get.Typically this can be all we wish. In different instances, we could also be extra eager about truly modelling a site. We don’t simply need to generate realistic-looking samples – we wish our samples to be situated at particular coordinates in area area.

For instance, think about our area to be the area of facial expressions. Then our latent area may be conceived as two-dimensional: In accordance with underlying emotional states, expressions differ on a positive-negative scale. On the similar time, they differ in depth. Now if we educated a VAE on a set of facial expressions adequately protecting the ranges, and it did actually “uncover” our hypothesized dimensions, we may then use it to generate previously-nonexisting incarnations of factors (faces, that’s) in latent area.

Variational autoencoders are much like probabilistic graphical fashions in that they assume a latent area that’s chargeable for the observations, however unobservable. They’re much like plain autoencoders in that they compress, after which decompress once more, the enter area. In distinction to plain autoencoders although, the essential level right here is to plot a loss perform that permits to acquire informative representations in latent area.

In a nutshell

In commonplace VAEs (Kingma and Welling 2013), the target is to maximise the proof decrease sure (ELBO):

[ELBO = E[log p(x|z)] – KL(q(z)||p(z))]

In plain phrases and expressed when it comes to how we use it in follow, the primary element is the reconstruction loss we additionally see in plain (non-variational) autoencoders. The second is the Kullback-Leibler divergence between a previous imposed on the latent area (sometimes, a typical regular distribution) and the illustration of latent area as discovered from the info.

A serious criticism relating to the normal VAE loss is that it ends in uninformative latent area. Options embody (beta)-VAE(Burgess et al. 2018), Information-VAE (Zhao, Song, and Ermon 2017), and extra. The MMD-VAE(Zhao, Song, and Ermon 2017) carried out beneath is a subtype of Information-VAE that as an alternative of constructing every illustration in latent area as related as attainable to the prior, coerces the respective distributions to be as shut as attainable. Right here MMD stands for most imply discrepancy, a similarity measure for distributions primarily based on matching their respective moments. We clarify this in additional element beneath.

Our goal at the moment

On this put up, we’re first going to implement a typical VAE that strives to maximise the ELBO. Then, we examine its efficiency to that of an Information-VAE utilizing the MMD loss.

Our focus shall be on inspecting the latent areas and see if, and the way, they differ as a consequence of the optimization standards used.

The area we’re going to mannequin shall be glamorous (style!), however for the sake of manageability, confined to dimension 28 x 28: We’ll compress and reconstruct photographs from the Fashion MNIST dataset that has been developed as a drop-in to MNIST.

An ordinary variational autoencoder

Seeing we haven’t used TensorFlow keen execution for some weeks, we’ll do the mannequin in an keen means.For those who’re new to keen execution, don’t fear: As each new method, it wants some getting accustomed to, however you’ll rapidly discover that many duties are made simpler in case you use it. A easy but full, template-like instance is out there as a part of the Keras documentation.

Setup and information preparation

As normal, we begin by ensuring we’re utilizing the TensorFlow implementation of Keras and enabling keen execution. Moreover tensorflow and keras, we additionally load tfdatasets to be used in information streaming.

By the way in which: No must copy-paste any of the beneath code snippets. The 2 approaches can be found amongst our Keras examples, specifically, as eager_cvae.R and mmd_cvae.R.

The information comes conveniently with keras, all we have to do is the standard normalization and reshaping. 

style <- dataset_fashion_mnist()

c(train_images, train_labels) %<-% style$practice
c(test_images, test_labels) %<-% style$check

train_x <- train_images %>%
  `/`(255) %>%
  k_reshape(c(60000, 28, 28, 1))

test_x <- test_images %>% `/`(255) %>%
  k_reshape(c(10000, 28, 28, 1))

What do we’d like the check set for, given we’re going to practice an unsupervised (a greater time period being: semi-supervised) mannequin? We’ll use it to see how (beforehand unknown) information factors cluster collectively in latent area.

Now put together for streaming the info to keras: 

buffer_size <- 60000
batch_size <- 100
batches_per_epoch <- buffer_size / batch_size

train_dataset <- tensor_slices_dataset(train_x) %>%
  dataset_shuffle(buffer_size) %>%
  dataset_batch(batch_size)

test_dataset <- tensor_slices_dataset(test_x) %>%
  dataset_batch(10000)

Subsequent up is defining the mannequin.

Encoder-decoder mannequin

The mannequin actually is 2 fashions: the encoder and the decoder. As we’ll see shortly, in the usual model of the VAE there’s a third element in between, performing the so-called reparameterization trick.

The encoder is a custom model, comprised of two convolutional layers and a dense layer. It returns the output of the dense layer cut up into two components, one storing the imply of the latent variables, the opposite their variance. 

latent_dim <- 2

encoder_model <- perform(title = NULL) {
  
  keras_model_custom(title = title, perform(self) {
    self$conv1 <-
      layer_conv_2d(
        filters = 32,
        kernel_size = 3,
        strides = 2,
        activation = "relu"
      )
    self$conv2 <-
      layer_conv_2d(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        activation = "relu"
      )
    self$flatten <- layer_flatten()
    self$dense <- layer_dense(items = 2 * latent_dim)
    
    perform (x, masks = NULL) {
      x %>%
        self$conv1() %>%
        self$conv2() %>%
        self$flatten() %>%
        self$dense() %>%
        tf$cut up(num_or_size_splits = 2L, axis = 1L) 
    }
  })
}

We select the latent area to be of dimension 2 – simply because that makes visualization simple.With extra complicated information, you’ll most likely profit from selecting the next dimensionality right here.

So the encoder compresses actual information into estimates of imply and variance of the latent area.We then “not directly” pattern from this distribution (the so-called reparameterization trick): 

reparameterize <- perform(imply, logvar) {
  eps <- k_random_normal(form = imply$form, dtype = tf$float64)
  eps * k_exp(logvar * 0.5) + imply
}

The sampled values will function enter to the decoder, who will try and map them again to the unique area.The decoder is mainly a sequence of transposed convolutions, upsampling till we attain a decision of 28×28. 

decoder_model <- perform(title = NULL) {
  
  keras_model_custom(title = title, perform(self) {
    
    self$dense <- layer_dense(items = 7 * 7 * 32, activation = "relu")
    self$reshape <- layer_reshape(target_shape = c(7, 7, 32))
    self$deconv1 <-
      layer_conv_2d_transpose(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        padding = "similar",
        activation = "relu"
      )
    self$deconv2 <-
      layer_conv_2d_transpose(
        filters = 32,
        kernel_size = 3,
        strides = 2,
        padding = "similar",
        activation = "relu"
      )
    self$deconv3 <-
      layer_conv_2d_transpose(
        filters = 1,
        kernel_size = 3,
        strides = 1,
        padding = "similar"
      )
    
    perform (x, masks = NULL) {
      x %>%
        self$dense() %>%
        self$reshape() %>%
        self$deconv1() %>%
        self$deconv2() %>%
        self$deconv3()
    }
  })
}

Observe how the ultimate deconvolution doesn’t have the sigmoid activation you may need anticipated. It is because we shall be utilizing tf$nn$sigmoid_cross_entropy_with_logits when calculating the loss.

Talking of losses, let’s examine them now.

Loss calculations

One approach to implement the VAE loss is combining reconstruction loss (cross entropy, within the current case) and Kullback-Leibler divergence. In Keras, the latter is out there instantly as loss_kullback_leibler_divergence.

Right here, we comply with a current Google Colaboratory notebook in batch-estimating the whole ELBO as an alternative (as an alternative of simply estimating reconstruction loss and computing the KL-divergence analytically):

[ELBO batch estimate = log p(x_{batch}|z_{sampled})+log p(z)−log q(z_{sampled}|x_{batch})]

Calculation of the conventional loglikelihood is packaged right into a perform so we will reuse it throughout the coaching loop. 

normal_loglik <- perform(pattern, imply, logvar, reduce_axis = 2) {
  loglik <- k_constant(0.5, dtype = tf$float64) *
    (k_log(2 * k_constant(pi, dtype = tf$float64)) +
    logvar +
    k_exp(-logvar) * (pattern - imply) ^ 2)
  - k_sum(loglik, axis = reduce_axis)
}

Peeking forward some, throughout coaching we’ll compute the above as follows.

First, 

crossentropy_loss <- tf$nn$sigmoid_cross_entropy_with_logits(
  logits = preds,
  labels = x
)
logpx_z <- - k_sum(crossentropy_loss)

yields (log p(x|z)), the loglikelihood of the reconstructed samples given values sampled from latent area (a.ok.a. reconstruction loss).

Then, 

logpz <- normal_loglik(
  z,
  k_constant(0, dtype = tf$float64),
  k_constant(0, dtype = tf$float64)
)

offers (log p(z)), the prior loglikelihood of (z). The prior is assumed to be commonplace regular, as is most frequently the case with VAEs.

Lastly, 

logqz_x <- normal_loglik(z, imply, logvar)

vields (log q(z|x)), the loglikelihood of the samples (z) given imply and variance computed from the noticed samples (x).

From these three parts, we’ll compute the ultimate loss as 

loss <- -k_mean(logpx_z + logpz - logqz_x)

After this peaking forward, let’s rapidly end the setup so we prepare for coaching.

Last setup

Moreover the loss, we’d like an optimizer that can try to decrease it. 

optimizer <- tf$practice$AdamOptimizer(1e-4)

We instantiate our fashions … 

encoder <- encoder_model()
decoder <- decoder_model()

and arrange checkpointing, so we will later restore educated weights. 

checkpoint_dir <- "./checkpoints_cvae"
checkpoint_prefix <- file.path(checkpoint_dir, "ckpt")
checkpoint <- tf$practice$Checkpoint(
  optimizer = optimizer,
  encoder = encoder,
  decoder = decoder
)

From the coaching loop, we’ll, in sure intervals, additionally name three features not reproduced right here (however obtainable within the code example): generate_random_clothes, used to generate garments from random samples from the latent area; show_latent_space, that shows the whole check set in latent (2-dimensional, thus simply visualizable) area; and show_grid, that generates garments in keeping with enter values systematically spaced out in a grid.

Let’s begin coaching! Really, earlier than we try this, let’s take a look at what these features show earlier than any coaching: As an alternative of garments, we see random pixels. Latent area has no construction. And various kinds of garments don’t cluster collectively in latent area.

Coaching loop

We’re coaching for 50 epochs right here. For every epoch, we loop over the coaching set in batches. For every batch, we comply with the standard keen execution circulation: Contained in the context of a GradientTape, apply the mannequin and calculate the present loss; then exterior this context calculate the gradients and let the optimizer carry out backprop.

What’s particular right here is that we have now two fashions that each want their gradients calculated and weights adjusted. This may be taken care of by a single gradient tape, supplied we create it persistent.

After every epoch, we save present weights and each ten epochs, we additionally save plots for later inspection. 

num_epochs <- 50

for (epoch in seq_len(num_epochs)) {
  iter <- make_iterator_one_shot(train_dataset)
  
  total_loss <- 0
  logpx_z_total <- 0
  logpz_total <- 0
  logqz_x_total <- 0
  
  until_out_of_range({
    x <-  iterator_get_next(iter)
    
    with(tf$GradientTape(persistent = TRUE) %as% tape, {
      
      c(imply, logvar) %<-% encoder(x)
      z <- reparameterize(imply, logvar)
      preds <- decoder(z)
      
      crossentropy_loss <-
        tf$nn$sigmoid_cross_entropy_with_logits(logits = preds, labels = x)
      logpx_z <-
        - k_sum(crossentropy_loss)
      logpz <-
        normal_loglik(z,
                      k_constant(0, dtype = tf$float64),
                      k_constant(0, dtype = tf$float64)
        )
      logqz_x <- normal_loglik(z, imply, logvar)
      loss <- -k_mean(logpx_z + logpz - logqz_x)
      
    })

    total_loss <- total_loss + loss
    logpx_z_total <- tf$reduce_mean(logpx_z) + logpx_z_total
    logpz_total <- tf$reduce_mean(logpz) + logpz_total
    logqz_x_total <- tf$reduce_mean(logqz_x) + logqz_x_total
    
    encoder_gradients <- tape$gradient(loss, encoder$variables)
    decoder_gradients <- tape$gradient(loss, decoder$variables)
    
    optimizer$apply_gradients(
      purrr::transpose(list(encoder_gradients, encoder$variables)),
      global_step = tf$practice$get_or_create_global_step()
    )
    optimizer$apply_gradients(
      purrr::transpose(list(decoder_gradients, decoder$variables)),
      global_step = tf$practice$get_or_create_global_step()
    )
    
  })
  
  checkpoint$save(file_prefix = checkpoint_prefix)
  
  cat(
    glue(
      "Losses (epoch): {epoch}:",
      "  {(as.numeric(logpx_z_total)/batches_per_epoch) %>% spherical(2)} logpx_z_total,",
      "  {(as.numeric(logpz_total)/batches_per_epoch) %>% spherical(2)} logpz_total,",
      "  {(as.numeric(logqz_x_total)/batches_per_epoch) %>% spherical(2)} logqz_x_total,",
      "  {(as.numeric(total_loss)/batches_per_epoch) %>% spherical(2)} whole"
    ),
    "n"
  )
  
  if (epoch %% 10 == 0) {
    generate_random_clothes(epoch)
    show_latent_space(epoch)
    show_grid(epoch)
  }
}

Outcomes

How effectively did that work? Let’s see the sorts of garments generated after 50 epochs.

Additionally, how disentangled (or not) are the completely different lessons in latent area?

And now watch completely different garments morph into each other.

How good are these representations? That is onerous to say when there may be nothing to match with.

So let’s dive into MMD-VAE and see the way it does on the identical dataset.

MMD-VAE

MMD-VAE guarantees to generate extra informative latent options, so we might hope to see completely different habits particularly within the clustering and morphing plots.

Knowledge setup is identical, and there are solely very slight variations within the mannequin. Please take a look at the whole code for this instance, mmd_vae.R, as right here we’ll simply spotlight the variations.

Variations within the mannequin(s)

There are three variations as regards mannequin structure.

One, the encoder doesn’t need to return the variance, so there isn’t a want for tf$cut up. The encoder’s name technique now simply is 

perform (x, masks = NULL) {
  x %>%
    self$conv1() %>%
    self$conv2() %>%
    self$flatten() %>%
    self$dense() 
}

Between the encoder and the decoder, we don’t want the sampling step anymore, so there isn’t a reparameterization.And since we gained’t use tf$nn$sigmoid_cross_entropy_with_logits to compute the loss, we let the decoder apply the sigmoid within the final deconvolution layer: 

self$deconv3 <- layer_conv_2d_transpose(
  filters = 1,
  kernel_size = 3,
  strides = 1,
  padding = "similar",
  activation = "sigmoid"
)

Loss calculations

Now, as anticipated, the massive novelty is within the loss perform.

The loss, most imply discrepancy (MMD), relies on the concept two distributions are similar if and provided that all moments are similar.Concretely, MMD is estimated utilizing a kernel, such because the Gaussian kernel

[k(z,z’)=frac{e^}{2sigma^2}]

to evaluate similarity between distributions.

The thought then is that if two distributions are similar, the common similarity between samples from every distribution needs to be similar to the common similarity between combined samples from each distributions:

[MMD(p(z)||q(z))=E_{p(z),p(z’)}[k(z,z’)]+E_{q(z),q(z’)}[k(z,z’)]−2E_{p(z),q(z’)}[k(z,z’)]]The next code is a direct port of the creator’s original TensorFlow code: 

compute_kernel <- perform(x, y) {
  x_size <- k_shape(x)[1]
  y_size <- k_shape(y)[1]
  dim <- k_shape(x)[2]
  tiled_x <- k_tile(
    k_reshape(x, k_stack(list(x_size, 1, dim))),
    k_stack(list(1, y_size, 1))
  )
  tiled_y <- k_tile(
    k_reshape(y, k_stack(list(1, y_size, dim))),
    k_stack(list(x_size, 1, 1))
  )
  k_exp(-k_mean(k_square(tiled_x - tiled_y), axis = 3) /
          k_cast(dim, tf$float64))
}

compute_mmd <- perform(x, y, sigma_sqr = 1) {
  x_kernel <- compute_kernel(x, x)
  y_kernel <- compute_kernel(y, y)
  xy_kernel <- compute_kernel(x, y)
  k_mean(x_kernel) + k_mean(y_kernel) - 2 * k_mean(xy_kernel)
}

Coaching loop

The coaching loop differs from the usual VAE instance solely within the loss calculations.Listed below are the respective traces: 

 with(tf$GradientTape(persistent = TRUE) %as% tape, {
      
      imply <- encoder(x)
      preds <- decoder(imply)
      
      true_samples <- k_random_normal(
        form = c(batch_size, latent_dim),
        dtype = tf$float64
      )
      loss_mmd <- compute_mmd(true_samples, imply)
      loss_nll <- k_mean(k_square(x - preds))
      loss <- loss_nll + loss_mmd
      
    })

So we merely compute MMD loss in addition to reconstruction loss, and add them up. No sampling is concerned on this model.After all, we’re curious to see how effectively that labored!

Outcomes

Once more, let’s have a look at some generated garments first. It looks as if edges are a lot sharper right here.

The clusters too look extra properly unfold out within the two dimensions. And, they’re centered at (0,0), as we might have hoped for.

Lastly, let’s see garments morph into each other. Right here, the graceful, steady evolutions are spectacular!Additionally, practically all area is crammed with significant objects, which hasn’t been the case above.

MNIST

For curiosity’s sake, we generated the identical sorts of plots after coaching on authentic MNIST.Right here, there are hardly any variations seen in generated random digits after 50 epochs of coaching.

Additionally the variations in clustering will not be that large.

However right here too, the morphing appears rather more natural with MMD-VAE.

Conclusion

To us, this demonstrates impressively what large a distinction the fee perform could make when working with VAEs.One other element open to experimentation often is the prior used for the latent area – see this talk for an outline of other priors and the “Variational Combination of Posteriors” paper (Tomczak and Welling 2017) for a preferred current method.

For each value features and priors, we anticipate efficient variations to change into means greater nonetheless after we depart the managed surroundings of (Trend) MNIST and work with real-world datasets.

Burgess, C. P., I. Higgins, A. Pal, L. Matthey, N. Watters, G. Desjardins, and A. Lerchner. 2018. “Understanding Disentangling in Beta-VAE.” ArXiv e-Prints, April. https://arxiv.org/abs/1804.03599.

Doersch, C. 2016. “Tutorial on Variational Autoencoders.” ArXiv e-Prints, June. https://arxiv.org/abs/1606.05908.

Kingma, Diederik P., and Max Welling. 2013. “Auto-Encoding Variational Bayes.” CoRR abs/1312.6114.

Tomczak, Jakub M., and Max Welling. 2017. “VAE with a VampPrior.” CoRR abs/1705.07120.

Zhao, Shengjia, Jiaming Track, and Stefano Ermon. 2017. “InfoVAE: Data Maximizing Variational Autoencoders.” CoRR abs/1706.02262. http://arxiv.org/abs/1706.02262.

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