RStudio AI Weblog: Illustration studying with MMD-VAE

Lately, we confirmed find out how to generate photographs utilizing generative adversarial networks (GANs). GANs could yield superb outcomes, however the contract there mainly is: what you see is what you get. Generally this can be all we wish. In different instances, we could also be extra curious about truly modelling a site. We don’t simply need to generate realistic-looking samples – we wish our samples to be positioned at particular coordinates in area house.

For instance, think about our area to be the house of facial expressions. Then our latent house 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 protecting the ranges, and it did actually “uncover” our hypothesized dimensions, we might then use it to generate previously-nonexisting incarnations of factors (faces, that’s) in latent house.

Variational autoencoders are just like probabilistic graphical fashions in that they assume a latent house that’s chargeable for the observations, however unobservable. They’re just 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 operate that enables to acquire informative representations in latent house.

In a nutshell

In normal 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 by way of how we use it in apply, 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 house (sometimes, a regular regular distribution) and the illustration of latent house as discovered from the info.

A serious criticism concerning the normal VAE loss is that it ends in uninformative latent house. Alternate options embody (beta)-VAE(Burgess et al. 2018), Information-VAE (Zhao, Track, and Ermon 2017), and extra. The MMD-VAE(Zhao, Track, and Ermon 2017) applied under is a subtype of Information-VAE that as a substitute of creating every illustration in latent house as comparable 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 under.

Our goal immediately

On this submit, 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 might 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 might be glamorous (trend!), however for the sake of manageability, confined to dimension 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 means. If you happen to’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 in the event you use it. A easy but full, template-like instance is accessible 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. In addition to tensorflow and keras, we additionally load tfdatasets to be used in knowledge streaming.

By the best way: No must copy-paste any of the under 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.

trend <- dataset_fashion_mnist()

c(train_images, train_labels) %<-% trend$prepare
c(test_images, test_labels) %<-% trend$take a look at

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 take a look at 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 house.

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 cut up into two elements, one storing the imply of the latent variables, the opposite their variance.

latent_dim <- 2

encoder_model <- operate(title = NULL) {
  
  keras_model_custom(title = title, operate(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)
    
    operate (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 house to be of dimension 2 – simply because that makes visualization simple. With extra complicated knowledge, you’ll in all probability profit from selecting the next dimensionality right here.

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

reparameterize <- operate(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 house. The decoder is mainly a sequence of transposed convolutions, upsampling till we attain a decision of 28×28.

decoder_model <- operate(title = NULL) {
  
  keras_model_custom(title = title, operate(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"
      )
    
    operate (x, masks = NULL) {
      x %>%
        self$dense() %>%
        self$reshape() %>%
        self$deconv1() %>%
        self$deconv2() %>%
        self$deconv3()
    }
  })
}

Notice how the ultimate deconvolution doesn’t have the sigmoid activation you might need anticipated. It’s because we might 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 accessible immediately as loss_kullback_leibler_divergence.

Right here, we comply with a current Google Colaboratory pocket book in batch-estimating the entire ELBO as a substitute (as a substitute 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 operate so we are able to reuse it throughout the coaching loop.

normal_loglik <- operate(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 house (a.okay.a. reconstruction loss).

Then,

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

provides (log p(z)), the prior loglikelihood of (z). The prior is assumed to be normal 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 elements, 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

In addition to the loss, we’d like 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 house; show_latent_space, that shows the entire take a look at set in latent (2-dimensional, thus simply visualizable) house; and show_grid, that generates garments in response to enter values systematically spaced out in a grid.

Let’s begin coaching! Really, earlier than we do this, let’s take a look at what these features show earlier than any coaching: As a substitute of garments, we see random pixels. Latent house has no construction. And several types of garments don’t cluster collectively in latent house.

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 same old keen execution movement: 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$apply_gradients(
      purrr::transpose(listing(encoder_gradients, encoder$variables)),
      global_step = tf$prepare$get_or_create_global_step()
    )
    optimizer$apply_gradients(
      purrr::transpose(listing(decoder_gradients, decoder$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)} complete"
    ),
    "n"
  )
  
  if (epoch %% 10 == 0) {
    generate_random_clothes(epoch)
    show_latent_space(epoch)
    show_grid(epoch)
  }
}

Outcomes

How nicely 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 house?

And now watch completely different garments morph into each other.

How good are these representations? That is exhausting to say when there’s 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’d hope to see completely different habits particularly within the clustering and morphing plots.

Information setup is identical, and there are solely very slight variations within the mannequin. Please try 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$cut up. The encoder’s name methodology now simply is

operate (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 large novelty is within the loss operate.

The loss, most imply discrepancy (MMD), is predicated on the concept two distributions are an identical if and provided that all moments are an identical. 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 an identical, the typical similarity between samples from every distribution ought to be an identical to the typical 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 <- operate(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(listing(x_size, 1, dim))),
    k_stack(listing(1, y_size, 1))
  )
  tiled_y <- k_tile(
    k_reshape(y, k_stack(listing(1, y_size, dim))),
    k_stack(listing(x_size, 1, 1))
  )
  k_exp(-k_mean(k_square(tiled_x - tiled_y), axis = 3) /
          k_cast(dim, tf$float64))
}

compute_mmd <- operate(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 nicely that labored!

Outcomes

Once more, let’s take a look at some generated garments first. It looks like 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’d have hoped for.

Lastly, let’s see garments morph into each other. Right here, the graceful, steady evolutions are spectacular! Additionally, almost all house is full of 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 massive.

However right here too, the morphing seems to be way more natural with MMD-VAE.

Conclusion

To us, this demonstrates impressively what massive a distinction the price operate could make when working with VAEs. One other element open to experimentation could be the prior used for the latent house – see this speak for an outline of different priors and the “Variational Combination of Posteriors” paper (Tomczak and Welling 2017) for a well-liked current method.

For each price features and priors, we count on efficient variations to turn out to be means greater nonetheless once we go away the managed setting of (Trend) MNIST and work with real-world datasets.