Lately, we confirmed methods to generate photographs utilizing generative adversarial networks (GANs). GANs could yield wonderful outcomes, however the contract there mainly is: what you see is what you get.
Generally this can be all we would like. In different circumstances, we could also be extra concerned with truly modelling a website. We don’t simply need to generate realistic-looking samples – we would like 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 perhaps conceived as two-dimensional: In accordance with underlying emotional states, expressions fluctuate on a positive-negative scale. On the similar time, they fluctuate in depth. Now if we skilled a VAE on a set of facial expressions adequately protecting the ranges, and it did in truth “uncover” our hypothesized dimensions, we may then use it to generate previously-nonexisting incarnations of factors (faces, that’s) in latent house.

Variational autoencoders are much like probabilistic graphical fashions in that they assume a latent house 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 plot a loss operate that enables to acquire informative representations in latent house.

In a nutshell

In customary 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 house (usually, a typical 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. Options embrace (beta)-VAE(Burgess et al. 2018), Data-VAE (Zhao, Music, and Ermon 2017), and extra. The MMD-VAE(Zhao, Music, and Ermon 2017) carried out under is a subtype of Data-VAE that as an alternative of constructing every illustration in latent house as comparable as doable to the prior, coerces the respective distributions to be as shut as doable. 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 in the present day

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

Our focus will probably 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 probably be glamorous (vogue!), 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.

An ordinary variational autoencoder

Seeing we haven’t used TensorFlow keen execution for some weeks, we’ll do the mannequin in an keen approach.
Should you’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 for those who use it. A easy but full, template-like instance is accessible as a part of the Keras documentation.

Setup and information preparation

As standard, 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 information streaming.

By the way in which: No have to 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 information comes conveniently with keras, all we have to do is the standard 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’d like 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) information 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 break up into two elements, one storing the imply of the latent variables, the opposite their variance.

latent_dim <- 2

encoder_model <- operate(identify = NULL) {
  
  keras_model_custom(identify = identify, 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$break up(num_or_size_splits = 2L, axis = 1L) 
    }
  })
}

We select the latent house to be of dimension 2 – simply because that makes visualization straightforward.
With extra complicated information, you’ll in all probability profit from selecting the next dimensionality right here.

So the encoder compresses actual information 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 to 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(identify = NULL) {
  
  keras_model_custom(identify = identify, operate(self) {
    
    self$dense <- layer_dense(models = 7 * 7 * 32, activation = "relu")
    self$reshape <- layer_reshape(target_shape = c(7, 7, 32))
    self$deconv1 <-
      layer_conv_2d_transpose(
        filters = 64,
        kernel_size = 3,
        strides = 2,
        padding = "similar",
        activation = "relu"
      )
    self$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"
      )
    
    operate (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 might need anticipated. It’s because we will probably be utilizing tf$nn$sigmoid_cross_entropy_with_logits when calculating the loss.

Talking of losses, let’s examine them now.

Loss calculations

One strategy 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 instantly as loss_kullback_leibler_divergence.

Right here, we comply with a latest 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 conventional loglikelihood is packaged right into a operate so we will 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’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 house (a.ok.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 customary 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’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.

Remaining setup

Apart from the loss, we’d like an optimizer that may attempt to decrease it.

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

We instantiate our fashions …

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

and arrange checkpointing, so we will 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’ll, in sure intervals, additionally name three capabilities not reproduced right here (however accessible 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 check set in latent (2-dimensional, thus simply visualizable) house; and show_grid, that generates garments in line with 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 capabilities show earlier than any coaching: As an alternative 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 standard 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’ve got 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)} complete"
    ),
    "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 sorts 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 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 habits particularly within the clustering and morphing plots.

Information setup is similar, 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 should return the variance, so there is no such thing as a want for tf$break up. The encoder’s name technique 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 is no such thing as a reparameterization.
And since we received’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 large novelty is within the loss operate.

The loss, most imply discrepancy (MMD), is predicated 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^z-z’}{2sigma^2}]

to evaluate similarity between distributions.

The thought then is that if two distributions are equivalent, the typical similarity between samples from every distribution ought to be equivalent to the typical 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 writer’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(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 <- 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 properly that labored!

Outcomes

Once more, let’s have 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 might have hoped for.

Lastly, let’s see garments morph into each other. Right here, the sleek, steady evolutions are spectacular!
Additionally, practically 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 are usually not that large.

However right here too, the morphing appears to be like far more natural with MMD-VAE.

Conclusion

To us, this demonstrates impressively what large a distinction the associated fee operate could make when working with VAEs.
One other element open to experimentation would be the prior used for the latent house – see this speak for an outline of other priors and the “Variational Combination of Posteriors” paper (Tomczak and Welling 2017) for a preferred latest strategy.

For each value capabilities and priors, we anticipate efficient variations to develop into approach larger nonetheless once we go away the managed surroundings of (Trend) MNIST and work with real-world datasets.