Posit AI Weblog: Illustration studying with MMD-VAE

Not too long ago, we confirmed tips on how to generate photographs utilizing generative adversarial networks (GANs). GANs could yield wonderful outcomes, however the contract there principally is: what you see is what you get.
Generally this can be all we would like. In different instances, we could also be extra all for truly modelling a website. We don’t simply wish to generate realistic-looking samples – we would like our samples to be positioned at particular coordinates in area area.

For instance, think about our area to be the area of facial expressions. Then our latent area is likely to be conceived as two-dimensional: In accordance with underlying emotional states, expressions fluctuate on a positive-negative scale. On the identical time, they fluctuate in depth. Now if we skilled a VAE on a set of facial expressions adequately overlaying the ranges, and it did in actual fact “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 liable 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 plan 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 certain (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 (usually, a regular regular distribution) and the illustration of latent area as realized from the info.

A significant criticism concerning the standard VAE loss is that it ends in uninformative latent area. Alternate options embody (beta)-VAE(Burgess et al. 2018), Information-VAE (Zhao, Music, and Ermon 2017), and extra. The MMD-VAE(Zhao, Music, and Ermon 2017) applied 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 based mostly on matching their respective moments. We clarify this in additional element beneath.

Our goal right now

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

Our focus will likely 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 will likely be glamorous (vogue!), however for the sake of manageability, confined to measurement 28 x 28: We’ll compress and reconstruct photographs from the Trend MNIST dataset that has been developed as a drop-in to MNIST.

A regular variational autoencoder

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

Setup and knowledge preparation

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

By the best way: No have to 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 info comes conveniently with keras, all we have to do is the same old normalization and reshaping.

vogue <- dataset_fashion_mnist()

c(train_images, train_labels) %<-% vogue$prepare
c(test_images, test_labels) %<-% vogue$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 want the check set for, given we’re going to prepare an unsupervised (a greater time period being: semi-supervised) mannequin? We’ll use it to see how (beforehand unknown) knowledge 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 customized mannequin, comprised of two convolutional layers and a dense layer. It returns the output of the dense layer break up into two elements, 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(models = 2 * latent_dim)
    
    perform (x, masks = NULL) {
      x %>%
        self$conv1() %>%
        self$conv2() %>%
        self$flatten() %>%
        self$dense() %>%
        tf$break up(num_or_size_splits = 2L, axis = 1L) 
    }
  })
}

We select the latent area to be of dimension 2 – simply because that makes visualization straightforward.
With extra advanced knowledge, you’ll in all probability profit from selecting a better dimensionality right here.

So the encoder compresses actual knowledge 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 to map them again to the unique area.
The decoder is principally 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$dense <- layer_dense(models = 7 * 7 * 32, activation = "relu")
    self$reshape <- layer_reshape(target_shape = c(7, 7, 32))
    self$deconv1 <-
      layer_conv_2d_transpose(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        padding = "identical",
        activation = "relu"
      )
    self$deconv2 <-
      layer_conv_2d_transpose(
        filters = 32,
        kernel_size = 3,
        strides = 2,
        padding = "identical",
        activation = "relu"
      )
    self$deconv3 <-
      layer_conv_2d_transpose(
        filters = 1,
        kernel_size = 3,
        strides = 1,
        padding = "identical"
      )
    
    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 will likely be utilizing tf$nn$sigmoid_cross_entropy_with_logits when calculating the loss.

Talking of losses, let’s examine them now.

Loss calculations

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

Right here, we observe a current Google Colaboratory pocket book in batch-estimating the entire 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 traditional loglikelihood is packaged right into a perform so we are able to reuse it in the course of 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 are going to 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 are going to compute the ultimate loss as

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

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

Last setup

Apart from the loss, we want an optimizer that can try to decrease it.

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

We instantiate our fashions …

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

and arrange checkpointing, so we are able to later restore skilled weights.

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

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

Let’s begin coaching! Truly, 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 observe the same old keen execution stream: Contained in the context of a GradientTape, apply the mannequin and calculate the present loss; then outdoors this context calculate the gradients and let the optimizer carry out backprop.

What’s particular right here is that we now have 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$apply_gradients(
      purrr::transpose(checklist(encoder_gradients, encoder$variables)),
      global_step = tf$prepare$get_or_create_global_step()
    )
    optimizer$apply_gradients(
      purrr::transpose(checklist(decoder_gradients, decoder$variables)),
      global_step = tf$prepare$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 properly did that work? Let’s see the varieties of garments generated after 50 epochs.

Additionally, how disentangled (or not) are the completely different courses 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’s nothing to check 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 conduct particularly within the clustering and morphing plots.

Information setup is similar, and there are solely very slight variations within the mannequin. Please take a look at the entire 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 any want for tf$break up. The encoder’s name methodology 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 any 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 = "identical",
  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 equivalent if and provided that all moments are equivalent.
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 concept then is that if two distributions are equivalent, the common similarity between samples from every distribution ought to be equivalent to the common similarity between blended 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 authentic 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(checklist(x_size, 1, dim))),
    k_stack(checklist(1, y_size, 1))
  )
  tiled_y <- k_tile(
    k_reshape(y, k_stack(checklist(1, y_size, dim))),
    k_stack(checklist(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 here 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 properly that labored!

Outcomes

Once more, let’s take 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 sleek, steady evolutions are spectacular!
Additionally, practically all area is stuffed 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 are usually not that large.

However right here too, the morphing seems far more natural with MMD-VAE.

Conclusion

To us, this demonstrates impressively what large a distinction the associated 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 discuss for an summary of different priors and the “Variational Combination of Posteriors” paper (Tomczak and Welling 2017) for a preferred current method.

For each price features and priors, we count on efficient variations to turn into manner greater nonetheless once we depart the managed atmosphere of (Trend) MNIST and work with real-world datasets.