About two weeks in the past, we launched TensorFlow Chance (TFP), exhibiting easy methods to create and pattern from *distributions* and put them to make use of in a Variational Autoencoder (VAE) that learns its prior. As we speak, 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 just isn’t steady, however discrete – therefore the “quantised” within the article’s title. We’ll rapidly have a look at what this implies, after which dive immediately 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 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 undergo from one thing known as *posterior collapse*: The decoder is so highly effective that it might create real looking output given simply *any* enter. This implies there isn’t any incentive to be taught an expressive latent area.

In VQ-VAE, nevertheless, every enter pattern will get mapped deterministically to one among 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 drag 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 via the encoder) to acceptable embedding vectors?

And second: How can we be taught 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 area, 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 truly not being discovered utilizing gradient descent – a function value mentioning as we don’t come throughout it on daily basis 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 entire code for this instance, together with utilities for mannequin saving and picture visualization, is obtainable 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 think about 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 put up on doing VAE with TFP, we’ll use Kuzushiji-MNIST(Clanuwat et al. 2018) as enter.

Now could be the time to have a look at what we ended up producing that point and place your guess: How will that evaluate in opposition to 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 “ordinary” hyperparameters we have now in deep studying, the VQ-VAE infrastructure introduces just a few model-specific ones. Initially, the embedding area is of dimensionality *variety of embedding vectors* occasions *embedding vector measurement*:

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

The latent area in our instance can be of measurement one, that’s, we have now a single embedding vector representing the latent code for every enter pattern. This can be tremendous for our dataset, however it ought 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 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, record(padding = "similar", 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$dense <- layer_dense(items = latent_size * code_size)
self$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 all the time, 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’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 obtainable 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`

methodology `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`

).

Relating to the `axis`

argument used with many TensorFlow features, please think about 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 <- operate(identify = NULL, num_codes, code_size) {
keras_model_custom(identify = identify, operate(self) {
# right here we have now 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
)
record(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 stated above, they aren’t 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 operate `update_ema`

that may maintain this.

`update_ema`

makes use of TensorFlow moving_averages to

- first, hold observe of the variety of at the moment 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
# (can 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 have 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 eventually, returning a likelihood for every picture pixel.

```
default_deconv <- set_defaults(
layer_conv_2d_transpose,
record(padding = "similar", 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 = 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 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 can be calculated inline.

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

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

For every batch, we have now a ahead cross, 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 similar loop, we additionally name `update_ema`

to recalculate the transferring averages, as these should 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$GradientTape(persistent = TRUE) %as% tape, {
# do ahead cross
# calculate losses
})
encoder_gradients <- tape$gradient(loss, encoder$variables)
decoder_gradients <- tape$gradient(loss, decoder$variables)
optimizer$apply_gradients(purrr::transpose(record(
encoder_gradients, encoder$variables
)),
global_step = tf$practice$get_or_create_global_step())
optimizer$apply_gradients(purrr::transpose(record(
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 task operation there isn’t any gradient. As an alternative what we will do is cross 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 maintain 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 elements:

- 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, we have now 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, we have now the same old KL diverge to a previous. As, a priori, all assignments are equally possible, this part of the loss is fixed and may oftentimes be disbursed 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 elements, we arrive on the total loss:

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

Earlier than we have 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 2nd “morphing view” one usually likes to show with VAEs (there simply is not any 2nd latent area). As an alternative, 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 put up). And second, would you’ve gotten 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.

Nonetheless, you may secretly have hoped we’d apply this to extra advanced knowledge, akin to 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 will present, and the time we will spend on iterations to efficiently apply these strategies to advanced datasets. In the long run 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.