About two weeks in the past, we launched TensorFlow Likelihood (TFP), exhibiting find out how to create and pattern from distributions and put them to make use of in a Variational Autoencoder (VAE) that learns its prior. At the moment, we transfer on to a unique specimen within the VAE mannequin zoo: the Vector Quantised Variational Autoencoder (VQ-VAE) described in Neural Discrete Illustration Studying (Oord, Vinyals, and Kavukcuoglu 2017). This mannequin differs from most VAEs in that its approximate posterior isn’t steady, however discrete – therefore the “quantised” within the article’s title. We’ll rapidly take a look at what this implies, after which dive straight into the code, combining Keras layers, keen execution, and TFP.

Many phenomena are finest considered, and modeled, as discrete. This holds for phonemes and lexemes in language, higher-level buildings in pictures (suppose objects as a substitute of pixels),and duties that necessitate reasoning and planning.
The latent code utilized in most VAEs, nonetheless, is steady – often it’s a multivariate Gaussian. Steady-space VAEs have been discovered very profitable in reconstructing their enter, however usually they endure from one thing referred to as posterior collapse: The decoder is so highly effective that it could create real looking output given simply any enter. This implies there isn’t a incentive to study an expressive latent house.

In VQ-VAE, nonetheless, every enter pattern will get mapped deterministically to considered one of a set of embedding vectors. Collectively, these embedding vectors represent the prior for the latent house.
As such, an embedding vector comprises much more info than a imply and a variance, and thus, is far tougher to disregard by the decoder.

The query then is: The place is that magical hat, for us to tug out significant embeddings?

From the above conceptual description, we now have two inquiries to reply. First, by what mechanism can we assign enter samples (that went via the encoder) to applicable embedding vectors?
And second: How can we study embedding vectors that truly are helpful representations – that when fed to a decoder, will end in entities perceived as belonging to the identical species?

As regards task, a tensor emitted from the encoder is solely mapped to its nearest neighbor in embedding house, utilizing Euclidean distance. The embedding vectors are then up to date utilizing exponential transferring averages. As we’ll see quickly, because of this they’re really not being discovered utilizing gradient descent – a characteristic value stating as we don’t come throughout it every single day in deep studying.

Concretely, how then ought to the loss perform and coaching course of look? It will most likely best be seen in code.

The entire code for this instance, together with utilities for mannequin saving and picture visualization, is accessible on github as a part of the Keras examples. Order of presentation right here might differ from precise execution order for expository functions, so please to truly run the code take into account making use of the instance on github.

As in all our prior posts on VAEs, we use keen execution, which presupposes the TensorFlow implementation of Keras.

As in our earlier submit on doing VAE with TFP, we’ll use Kuzushiji-MNIST(Clanuwat et al. 2018) as enter.
Now’s the time to have a look at what we ended up producing that point and place your wager: How will that evaluate towards the discrete latent house of VQ-VAE?

np <- import("numpy")
 
kuzushiji <- np$load("kmnist-train-imgs.npz")
kuzushiji <- kuzushiji$get("arr_0")

train_images <- kuzushiji %>%
  k_expand_dims() %>%
  k_cast(dtype = "float32")

train_images <- train_images %>% `/`(255)

buffer_size <- 60000
batch_size <- 64
num_examples_to_generate <- batch_size

batches_per_epoch <- buffer_size / batch_size

train_dataset <- tensor_slices_dataset(train_images) %>%
  dataset_shuffle(buffer_size) %>%
  dataset_batch(batch_size, drop_remainder = TRUE)

Hyperparameters

Along with the “standard” hyperparameters now we have in deep studying, the VQ-VAE infrastructure introduces a couple of model-specific ones. To begin with, the embedding house is of dimensionality variety of embedding vectors occasions embedding vector dimension:

# variety of embedding vectors
num_codes <- 64L
# dimensionality of the embedding vectors
code_size <- 16L

The latent house in our instance shall be of dimension one, that’s, now we have a single embedding vector representing the latent code for every enter pattern. This shall be advantageous for our dataset, nevertheless it must be famous that van den Oord et al. used far higher-dimensional latent areas on e.g. ImageNet and Cifar-10.

Encoder mannequin

The encoder makes use of convolutional layers to extract picture options. Its output is a three-D tensor of form batchsize * 1 * code_size.

activation <- "elu"
# modularizing the code just a bit bit
default_conv <- set_defaults(layer_conv_2d, listing(padding = "identical", activation = activation))

base_depth <- 32

encoder_model <- perform(title = NULL,
                          code_size) {
  
  keras_model_custom(title = title, perform(self) {
    
    self$conv1 <- default_conv(filters = base_depth, kernel_size = 5)
    self$conv2 <- default_conv(filters = base_depth, kernel_size = 5, strides = 2)
    self$conv3 <- default_conv(filters = 2 * base_depth, kernel_size = 5)
    self$conv4 <- default_conv(filters = 2 * base_depth, kernel_size = 5, strides = 2)
    self$conv5 <- default_conv(filters = 4 * latent_size, kernel_size = 7, padding = "legitimate")
    self$flatten <- layer_flatten()
    self$dense <- layer_dense(models = latent_size * code_size)
    self$reshape <- layer_reshape(target_shape = c(latent_size, code_size))
    
    perform (x, masks = NULL) {
      x %>% 
        # output form:  7 28 28 32 
        self$conv1() %>% 
        # output form:  7 14 14 32 
        self$conv2() %>% 
        # output form:  7 14 14 64 
        self$conv3() %>% 
        # output form:  7 7 7 64 
        self$conv4() %>% 
        # output form:  7 1 1 4 
        self$conv5() %>% 
        # output form:  7 4 
        self$flatten() %>% 
        # output form:  7 16 
        self$dense() %>% 
        # output form:  7 1 16
        self$reshape()
    }
  })
}

As at all times, let’s make use of the truth that we’re utilizing keen execution, and see a couple of instance outputs.

iter <- make_iterator_one_shot(train_dataset)
batch <-  iterator_get_next(iter)

encoder <- encoder_model(code_size = code_size)
encoded  <- encoder(batch)
encoded

tf.Tensor(
[[[ 0.00516277 -0.00746826  0.0268365  ... -0.012577   -0.07752544
   -0.02947626]]
...

 [[-0.04757921 -0.07282603 -0.06814402 ... -0.10861694 -0.01237121
    0.11455103]]], form=(64, 1, 16), dtype=float32)

Now, every of those 16d vectors must be mapped to the embedding vector it’s closest to. This mapping is taken care of by one other mannequin: vector_quantizer.

Vector quantizer mannequin

That is how we’ll instantiate the vector quantizer:

vector_quantizer <- vector_quantizer_model(num_codes = num_codes, code_size = code_size)

This mannequin serves two functions: First, it acts as a retailer for the embedding vectors. Second, it matches encoder output to accessible embeddings.

Right here, the present state of embeddings is saved in codebook. ema_means and ema_count are for bookkeeping functions solely (be aware how they’re set to be non-trainable). We’ll see them in use shortly.

vector_quantizer_model <- perform(title = NULL, num_codes, code_size) {
  
    keras_model_custom(title = title, perform(self) {
      
      self$num_codes <- num_codes
      self$code_size <- code_size
      self$codebook <- tf$get_variable(
        "codebook",
        form = c(num_codes, code_size), 
        dtype = tf$float32
        )
      self$ema_count <- tf$get_variable(
        title = "ema_count", form = c(num_codes),
        initializer = tf$constant_initializer(0),
        trainable = FALSE
        )
      self$ema_means = tf$get_variable(
        title = "ema_means",
        initializer = self$codebook$initialized_value(),
        trainable = FALSE
        )
      
      perform (x, masks = NULL) { 
        
        # to be stuffed in shortly ...
        
      }
    })
}

Along with the precise embeddings, in its name technique vector_quantizer holds the task logic.
First, we compute the Euclidean distance of every encoding to the vectors within the codebook (tf$norm).
We assign every encoding to the closest as by that distance embedding (tf$argmin) and one-hot-encode the assignments (tf$one_hot). Lastly, we isolate the corresponding vector by masking out all others and summing up what’s left over (multiplication adopted by tf$reduce_sum).

Concerning the axis argument used with many TensorFlow features, please take into accounts that in distinction to their k_* siblings, uncooked TensorFlow (tf$*) features anticipate axis numbering to be 0-based. We even have so as to add the L’s after the numbers to adapt to TensorFlow’s datatype necessities.

vector_quantizer_model <- perform(title = NULL, num_codes, code_size) {
  
    keras_model_custom(title = title, perform(self) {
      
      # right here now we have the above occasion fields
      
      perform (x, masks = NULL) {
    
        # form: bs * 1 * num_codes
         distances <- tf$norm(
          tf$expand_dims(x, axis = 2L) -
            tf$reshape(self$codebook, 
                       c(1L, 1L, self$num_codes, self$code_size)),
                       axis = 3L 
        )
        
        # bs * 1
        assignments <- tf$argmin(distances, axis = 2L)
        
        # bs * 1 * num_codes
        one_hot_assignments <- tf$one_hot(assignments, depth = self$num_codes)
        
        # bs * 1 * code_size
        nearest_codebook_entries <- tf$reduce_sum(
          tf$expand_dims(
            one_hot_assignments, -1L) * 
            tf$reshape(self$codebook, c(1L, 1L, self$num_codes, self$code_size)),
                       axis = 2L 
                       )
        listing(nearest_codebook_entries, one_hot_assignments)
      }
    })
  }

Now that we’ve seen how the codes are saved, let’s add performance for updating them.
As we mentioned above, they don’t seem to be discovered by way of gradient descent. As an alternative, they’re exponential transferring averages, regularly up to date by no matter new “class member” they get assigned.

So here’s a perform update_ema that can deal with this.

update_ema makes use of TensorFlow moving_averages to

• first, maintain observe of the variety of at present assigned samples per code (updated_ema_count), and
• second, compute and assign the present exponential transferring common (updated_ema_means).

moving_averages <- tf$python$coaching$moving_averages

# decay to make use of in computing exponential transferring common
decay <- 0.99

update_ema <- perform(
  vector_quantizer,
  one_hot_assignments,
  codes,
  decay) {
 
  updated_ema_count <- moving_averages$assign_moving_average(
    vector_quantizer$ema_count,
    tf$reduce_sum(one_hot_assignments, axis = c(0L, 1L)),
    decay,
    zero_debias = FALSE
  )

  updated_ema_means <- moving_averages$assign_moving_average(
    vector_quantizer$ema_means,
    # selects all assigned values (masking out the others) and sums them up over the batch
    # (shall be divided by rely later, so we get a mean)
    tf$reduce_sum(
      tf$expand_dims(codes, 2L) *
        tf$expand_dims(one_hot_assignments, 3L), axis = c(0L, 1L)),
    decay,
    zero_debias = FALSE
  )

  updated_ema_count <- updated_ema_count + 1e-5
  updated_ema_means <-  updated_ema_means / tf$expand_dims(updated_ema_count, axis = -1L)
  
  tf$assign(vector_quantizer$codebook, updated_ema_means)
}

Earlier than we take a look at the coaching loop, let’s rapidly full the scene including within the final actor, the decoder.

Decoder mannequin

The decoder is fairly customary, performing a collection of deconvolutions and eventually, returning a likelihood for every picture pixel.

default_deconv <- set_defaults(
  layer_conv_2d_transpose,
  listing(padding = "identical", activation = activation)
)

decoder_model <- perform(title = NULL,
                          input_size,
                          output_shape) {
  
  keras_model_custom(title = title, perform(self) {
    
    self$reshape1 <- layer_reshape(target_shape = c(1, 1, input_size))
    self$deconv1 <-
      default_deconv(
        filters = 2 * base_depth,
        kernel_size = 7,
        padding = "legitimate"
      )
    self$deconv2 <-
      default_deconv(filters = 2 * base_depth, kernel_size = 5)
    self$deconv3 <-
      default_deconv(
        filters = 2 * base_depth,
        kernel_size = 5,
        strides = 2
      )
    self$deconv4 <-
      default_deconv(filters = base_depth, kernel_size = 5)
    self$deconv5 <-
      default_deconv(filters = base_depth,
                     kernel_size = 5,
                     strides = 2)
    self$deconv6 <-
      default_deconv(filters = base_depth, kernel_size = 5)
    self$conv1 <-
      default_conv(filters = output_shape[3],
                   kernel_size = 5,
                   activation = "linear")
    
    perform (x, masks = NULL) {
      
      x <- x %>%
        # output form:  7 1 1 16
        self$reshape1() %>%
        # output form:  7 7 7 64
        self$deconv1() %>%
        # output form:  7 7 7 64
        self$deconv2() %>%
        # output form:  7 14 14 64
        self$deconv3() %>%
        # output form:  7 14 14 32
        self$deconv4() %>%
        # output form:  7 28 28 32
        self$deconv5() %>%
        # output form:  7 28 28 32
        self$deconv6() %>%
        # output form:  7 28 28 1
        self$conv1()
      
      tfd$Impartial(tfd$Bernoulli(logits = x),
                      reinterpreted_batch_ndims = size(output_shape))
    }
  })
}

input_shape <- c(28, 28, 1)
decoder <- decoder_model(input_size = latent_size * code_size,
                         output_shape = input_shape)

Now we’re prepared to coach. One factor we haven’t actually talked about but is the associated fee perform: Given the variations in structure (in comparison with customary VAEs), will the losses nonetheless look as anticipated (the standard add-up of reconstruction loss and KL divergence)?
We’ll see that in a second.

Coaching loop

Right here’s the optimizer we’ll use. Losses shall be calculated inline.

optimizer <- tf$prepare$AdamOptimizer(learning_rate = learning_rate)

The coaching loop, as standard, is a loop over epochs, the place every iteration is a loop over batches obtained from the dataset.
For every batch, now we have a ahead move, recorded by a gradientTape, based mostly on which we calculate the loss.
The tape will then decide the gradients of all trainable weights all through the mannequin, and the optimizer will use these gradients to replace the weights.

To date, all of this conforms to a scheme we’ve oftentimes seen earlier than. One level to notice although: On this identical loop, we additionally name update_ema to recalculate the transferring averages, as these usually are not operated on throughout backprop.
Right here is the important performance:

num_epochs <- 20

for (epoch in seq_len(num_epochs)) {
  
  iter <- make_iterator_one_shot(train_dataset)
  
  until_out_of_range({
    
    x <-  iterator_get_next(iter)
    with(tf$GradientTape(persistent = TRUE) %as% tape, {
      
      # do ahead move
      # calculate losses
      
    })
    
    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())
    
    update_ema(vector_quantizer,
               one_hot_assignments,
               codes,
               decay)

    # periodically show some generated pictures
    # see code on github 
    # visualize_images("kuzushiji", epoch, reconstructed_images, random_images)
  })
}

Now, for the precise motion. Contained in the context of the gradient tape, we first decide which encoded enter pattern will get assigned to which embedding vector.

codes <- encoder(x)
c(nearest_codebook_entries, one_hot_assignments) %<-% vector_quantizer(codes)

Now, for this task operation there isn’t a gradient. As an alternative what we will do is move the gradients from decoder enter straight via to encoder output.
Right here tf$stop_gradient exempts nearest_codebook_entries from the chain of gradients, so encoder and decoder are linked by codes:

codes_straight_through <- codes + tf$stop_gradient(nearest_codebook_entries - codes)
decoder_distribution <- decoder(codes_straight_through)

In sum, backprop will deal with the decoder’s in addition to the encoder’s weights, whereas the latent embeddings are up to date utilizing transferring averages, as we’ve seen already.

Now we’re able to sort out the losses. There are three parts:

• First, the reconstruction loss, which is simply the log likelihood of the particular enter beneath the distribution discovered by the decoder.

reconstruction_loss <- -tf$reduce_mean(decoder_distribution$log_prob(x))

• Second, now we have the dedication loss, outlined because the imply squared deviation of the encoded enter samples from the closest neighbors they’ve been assigned to: We wish the community to “commit” to a concise set of latent codes!

commitment_loss <- tf$reduce_mean(tf$sq.(codes - tf$stop_gradient(nearest_codebook_entries)))

• Lastly, now we have the standard KL diverge to a previous. As, a priori, all assignments are equally possible, this part of the loss is fixed and might oftentimes be distributed of. We’re including it right here primarily for illustrative functions.

prior_dist <- tfd$Multinomial(
  total_count = 1,
  logits = tf$zeros(c(latent_size, num_codes))
  )
prior_loss <- -tf$reduce_mean(
  tf$reduce_sum(prior_dist$log_prob(one_hot_assignments), 1L)
  )

Summing up all three parts, we arrive on the total loss:

beta <- 0.25
loss <- reconstruction_loss + beta * commitment_loss + prior_loss

Earlier than we take a look at the outcomes, let’s see what occurs inside gradientTape at a single look:

with(tf$GradientTape(persistent = TRUE) %as% tape, {
      
  codes <- encoder(x)
  c(nearest_codebook_entries, one_hot_assignments) %<-% vector_quantizer(codes)
  codes_straight_through <- codes + tf$stop_gradient(nearest_codebook_entries - codes)
  decoder_distribution <- decoder(codes_straight_through)
      
  reconstruction_loss <- -tf$reduce_mean(decoder_distribution$log_prob(x))
  commitment_loss <- tf$reduce_mean(tf$sq.(codes - tf$stop_gradient(nearest_codebook_entries)))
  prior_dist <- tfd$Multinomial(
    total_count = 1,
    logits = tf$zeros(c(latent_size, num_codes))
  )
  prior_loss <- -tf$reduce_mean(tf$reduce_sum(prior_dist$log_prob(one_hot_assignments), 1L))
  
  loss <- reconstruction_loss + beta * commitment_loss + prior_loss
})

Outcomes

And right here we go. This time, we will’t have the second “morphing view” one usually likes to show with VAEs (there simply isn’t any second latent house). As an alternative, the 2 pictures beneath are (1) letters generated from random enter and (2) reconstructed precise letters, every saved after coaching for 9 epochs.

Two issues soar to the attention: First, the generated letters are considerably sharper than their continuous-prior counterparts (from the earlier submit). And second, would you’ve been capable of inform the random picture from the reconstruction picture?

At this level, we’ve hopefully satisfied you of the ability and effectiveness of this discrete-latents method.
Nevertheless, you would possibly secretly have hoped we’d apply this to extra advanced knowledge, similar to the weather of speech we talked about within the introduction, or higher-resolution pictures as present in ImageNet.

The reality is that there’s a steady tradeoff between the variety of new and thrilling methods we will present, and the time we will spend on iterations to efficiently apply these methods to advanced datasets. Ultimately it’s you, our readers, who will put these methods to significant use on related, actual world knowledge.