About two weeks in the past, we introduced TensorFlow Probability (TFP), displaying methods to create and pattern from *distributions* and put them to make use of in a Variational Autoencoder (VAE) that learns its prior. Immediately, 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 instantly into the code, combining Keras layers, keen execution, and TFP.

Many phenomena are greatest considered, and modeled, as discrete. This holds for phonemes and lexemes in language, higher-level constructions in photos (suppose objects as a substitute of pixels),and duties that necessitate reasoning and planning.

The latent code utilized in most VAEs, nevertheless, is steady – normally 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 lifelike output given simply *any* enter. This implies there is no such thing as a incentive to study an expressive latent area.

In VQ-VAE, nevertheless, every enter pattern will get mapped deterministically to one in all a set of *embedding vectors*. Collectively, these embedding vectors represent the prior for the latent area.

As such, an embedding vector comprises much more info than a imply and a variance, and thus, is way more durable 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 will we assign enter samples (that went by way of the encoder) to acceptable embedding vectors?

And second: How can we study embedding vectors that really are helpful representations – that when fed to a decoder, will end in entities perceived as belonging to the identical species?

As regards project, a tensor emitted from the encoder is just mapped to its nearest neighbor in embedding area, utilizing Euclidean distance. The embedding vectors are then up to date utilizing exponential transferring averages. As we’ll see quickly, which means that they’re truly not being realized utilizing gradient descent – a characteristic price stating as we don’t come throughout it day-after-day in deep studying.

Concretely, how then ought to the loss operate and coaching course of look? This can most likely best be seen in code.

The whole code for this instance, together with utilities for mannequin saving and picture visualization, is available on github as a part of the Keras examples. Order of presentation right here could differ from precise execution order for expository functions, so please to truly run the code contemplate 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 publish on doing VAE with TFP, we’ll use Kuzushiji-MNIST(Clanuwat et al. 2018) as enter.

Now’s the time to take a look at what we ended up generating that time and place your guess: How will that examine towards the discrete latent area 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 “traditional” hyperparameters we now have in deep studying, the VQ-VAE infrastructure introduces just a few model-specific ones. To start with, the embedding area 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 area in our instance might be of dimension one, that’s, we now have a single embedding vector representing the latent code for every enter pattern. This might be tremendous for our dataset, but it surely needs to 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 3D tensor of form *batchsize* * 1 * *code_size*.

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

```
base_depth <- 32
encoder_model <- operate(identify = NULL,
code_size) {
keras_model_custom(identify = identify, operate(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))
operate (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 just a few 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 are going to 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 (observe how they’re set to be non-trainable). We’ll see them in use shortly.

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

Along with the precise embeddings, in its `name`

technique `vector_quantizer`

holds the project 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`

).

Relating to the `axis`

argument used with many TensorFlow capabilities, please consider that in distinction to their `k_*`

siblings, uncooked TensorFlow (`tf$*`

) capabilities count on 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 <- operate(identify = NULL, num_codes, code_size) {
keras_model_custom(identify = identify, operate(self) {
# right here we now have the above occasion fields
operate (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
)
list(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 realized through gradient descent. As a substitute, they’re exponential transferring averages, regularly up to date by no matter new “class member” they get assigned.

So here’s a operate `update_ema`

that may care for this.

`update_ema`

makes use of TensorFlow moving_averages to

- first, hold monitor of the variety of presently 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$moving_averages
# decay to make use of in computing exponential transferring common
decay <- 0.99
update_ema <- operate(
vector_quantizer,
one_hot_assignments,
codes,
decay) {
updated_ema_count <- moving_averages$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
# (might be divided by rely later, so we get a median)
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 commonplace, performing a sequence of deconvolutions and at last, returning a chance for every picture pixel.

```
default_deconv <- set_defaults(
layer_conv_2d_transpose,
list(padding = "identical", activation = activation)
)
decoder_model <- operate(identify = NULL,
input_size,
output_shape) {
keras_model_custom(identify = identify, operate(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")
operate (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$Unbiased(tfd$Bernoulli(logits = x),
reinterpreted_batch_ndims = length(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 fee operate: Given the variations in structure (in comparison with commonplace VAEs), will the losses nonetheless look as anticipated (the same old 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 might be calculated inline.

`optimizer <- tf$practice$AdamOptimizer(learning_rate = learning_rate)`

The coaching loop, as traditional, is a loop over epochs, the place every iteration is a loop over batches obtained from the dataset.

For every batch, we now 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.

Thus far, 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 aren’t 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(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())
update_ema(vector_quantizer,
one_hot_assignments,
codes,
decay)
# periodically show some generated photos
# 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 project operation there is no such thing as a gradient. As a substitute what we are able to do is move the gradients from decoder enter straight by way of 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 care for 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 deal with the losses. There are three parts:

- First, the reconstruction loss, which is simply the log chance of the particular enter underneath the distribution realized by the decoder.

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

- Second, we now 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 would like 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, we now have the same old KL diverge to a previous. As, a priori, all assignments are equally possible, this element of the loss is fixed and might oftentimes be allotted 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 general 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 are able to’t have the second “morphing view” one usually likes to show with VAEs (there simply is not any second latent area). As a substitute, the 2 photos beneath are (1) letters generated from random enter and (2) reconstructed *precise* letters, every saved after coaching for 9 epochs.

Two issues bounce to the attention: First, the generated letters are considerably sharper than their continuous-prior counterparts (from the earlier publish). And second, would you have got been in a position to 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 may secretly have hoped we’d apply this to extra complicated knowledge, reminiscent of the weather of speech we talked about within the introduction, or higher-resolution photos as present in ImageNet.

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

*CoRR*abs/1711.00937. http://arxiv.org/abs/1711.00937.