We’ve seen fairly just a few examples of unsupervised studying (or self-supervised studying, to decide on the extra appropriate however much less
well-liked time period) on this weblog.

Usually, these concerned Variational Autoencoders (VAEs), whose enchantment lies in them permitting to mannequin a latent house of
underlying, unbiased (ideally) components that decide the seen options. A doable draw back could be the inferior
high quality of generated samples. Generative Adversarial Networks (GANs) are one other well-liked strategy. Conceptually, these are
extremely enticing on account of their game-theoretic framing. Nevertheless, they are often tough to coach. PixelCNN variants, on the
different hand – we’ll subsume all of them right here beneath PixelCNN – are typically identified for his or her good outcomes. They appear to contain
some extra alchemy although. Underneath these circumstances, what could possibly be extra welcome than a straightforward method of experimenting with
them? By TensorFlow Chance (TFP) and its R wrapper, tfprobability, we now have
such a method.

This publish first provides an introduction to PixelCNN, concentrating on high-level ideas (leaving the main points for the curious
to look them up within the respective papers). We’ll then present an instance of utilizing tfprobability to experiment with the TFP
implementation.

PixelCNN rules

Autoregressivity, or: We’d like (some) order

The essential thought in PixelCNN is autoregressivity. Every pixel is modeled as relying on all prior pixels. Formally:

[p(mathbf{x}) = prod_{i}p(x_i|x_0, x_1, …, x_{i-1})]

Now wait a second – what even are prior pixels? Final I noticed one pictures had been two-dimensional. So this implies now we have to impose
an order on the pixels. Generally this will likely be raster scan order: row after row, from left to proper. However when coping with
coloration pictures, there’s one thing else: At every place, we even have three depth values, one for every of purple, inexperienced,
and blue. The unique PixelCNN paper(Oord, Kalchbrenner, and Kavukcuoglu 2016) carried by means of autoregressivity right here as nicely, with a pixel’s depth for
purple relying on simply prior pixels, these for inexperienced relying on these similar prior pixels however moreover, the present worth
for purple, and people for blue relying on the prior pixels in addition to the present values for purple and inexperienced.

[p(x_i|mathbf{x}<i) = p(x_{i,R}|mathbf{x}<i) p(x_{i,G}|mathbf{x}<i, x_{i,R}) p(x_{i,B}|mathbf{x}<i, x_{i,R}, x_{i,G})]

Right here, the variant carried out in TFP, PixelCNN++(Salimans et al. 2017) , introduces a simplification; it factorizes the joint
distribution in a much less compute-intensive method.

Technically, then, we all know how autoregressivity is realized; intuitively, it could nonetheless appear shocking that imposing a raster
scan order “simply works” (to me, a minimum of, it’s). Perhaps that is a type of factors the place compute energy efficiently
compensates for lack of an equal of a cognitive prior.

Masking, or: The place to not look

Now, PixelCNN ends in “CNN” for a purpose – as normal in picture processing, convolutional layers (or blocks thereof) are
concerned. However – is it not the very nature of a convolution that it computes a mean of some kinds, wanting, for every
output pixel, not simply on the corresponding enter but in addition, at its spatial (or temporal) environment? How does that rhyme
with the look-at-just-prior-pixels technique?

Surprisingly, this drawback is less complicated to resolve than it sounds. When making use of the convolutional kernel, simply multiply with a
masks that zeroes out any “forbidden pixels” – like on this instance for a 5×5 kernel, the place we’re about to compute the
convolved worth for row 3, column 3:

[left[begin{array}
{rrr}
1 & 1 & 1 & 1 & 1
1 & 1 & 1 & 1 & 1
1 & 1 & 1 & 0 & 0
0 & 0 & 0 & 0 & 0
0 & 0 & 0 & 0 & 0
end{array}right]
]

This makes the algorithm sincere, however introduces a special drawback: With every successive convolutional layer consuming its
predecessor’s output, there’s a constantly rising blind spot (so-called in analogy to the blind spot on the retina, however
positioned within the prime proper) of pixels which might be by no means seen by the algorithm. Van den Oord et al. (2016)(Oord et al. 2016) repair this
by utilizing two totally different convolutional stacks, one continuing from prime to backside, the opposite from left to proper.

Conditioning, or: Present me a kitten

To this point, we’ve at all times talked about “producing pictures” in a purely generic method. However the true attraction lies in creating
samples of some specified sort – one of many lessons we’ve been coaching on, or orthogonal info fed into the community.
That is the place PixelCNN turns into Conditional PixelCNN(Oord et al. 2016), and it is usually the place that feeling of magic resurfaces.
Once more, as “normal math” it’s not laborious to conceive. Right here, (mathbf{h}) is the extra enter we’re conditioning on:

[p(mathbf{x}| mathbf{h}) = prod_{i}p(x_i|x_0, x_1, …, x_{i-1}, mathbf{h})]

However how does this translate into neural community operations? It’s simply one other matrix multiplication ((V^T mathbf{h})) added
to the convolutional outputs ((W mathbf{x})).

[mathbf{y} = tanh(W_{k,f} mathbf{x} + V^T_{k,f} mathbf{h}) odot sigma(W_{k,g} mathbf{x} + V^T_{k,g} mathbf{h})]

(If you happen to’re questioning concerning the second half on the best, after the Hadamard product signal – we gained’t go into particulars, however in a
nutshell, it’s one other modification launched by (Oord et al. 2016), a switch of the “gating” precept from recurrent neural
networks, corresponding to GRUs and LSTMs, to the convolutional setting.)

So we see what goes into the choice of a pixel worth to pattern. However how is that call truly made?

Logistic combination chance , or: No pixel is an island

Once more, that is the place the TFP implementation doesn’t observe the unique paper, however the latter PixelCNN++ one. Initially,
pixels had been modeled as discrete values, selected by a softmax over 256 (0-255) doable values. (That this truly labored
looks as if one other occasion of deep studying magic. Think about: On this mannequin, 254 is as removed from 255 as it’s from 0.)

In distinction, PixelCNN++ assumes an underlying steady distribution of coloration depth, and rounds to the closest integer.
That underlying distribution is a mix of logistic distributions, thus permitting for multimodality:

[nu sim sum_{i} pi_i logistic(mu_i, sigma_i)]

Total structure and the PixelCNN distribution

Total, PixelCNN++, as described in (Salimans et al. 2017), consists of six blocks. The blocks collectively make up a UNet-like
construction, successively downsizing the enter after which, upsampling once more:

In TFP’s PixelCNN distribution, the variety of blocks is configurable as num_hierarchies, the default being 3.

Every block consists of a customizable variety of layers, referred to as ResNet layers because of the residual connection (seen on the
proper) complementing the convolutional operations within the horizontal stack:

In TFP, the variety of these layers per block is configurable as num_resnet.

num_resnet and num_hierarchies are the parameters you’re almost definitely to experiment with, however there are just a few extra you possibly can
try within the documentation. The variety of logistic
distributions within the combination can be configurable, however from my experiments it’s finest to maintain that quantity quite low to keep away from
producing NaNs throughout coaching.

Let’s now see a whole instance.

Finish-to-end instance

Our playground will likely be QuickDraw, a dataset – nonetheless rising –
obtained by asking individuals to attract some object in at most twenty seconds, utilizing the mouse. (To see for your self, simply try
the web site). As of at the moment, there are greater than a fifty million situations, from 345
totally different lessons.

At the beginning, these information had been chosen to take a break from MNIST and its variants. However identical to these (and lots of extra!),
QuickDraw could be obtained, in tfdatasets-ready kind, by way of tfds, the R wrapper to
TensorFlow datasets. In distinction to the MNIST “household” although, the “actual samples” are themselves extremely irregular, and sometimes
even lacking important components. So to anchor judgment, when displaying generated samples we at all times present eight precise drawings
with them.

Making ready the information

The dataset being gigantic, we instruct tfds to load the primary 500,000 drawings “solely.”

To hurry up coaching additional, we then zoom in on twenty lessons. This successfully leaves us with ~ 1,100 – 1,500 drawings per
class.

# bee, bicycle, broccoli, butterfly, cactus,
# frog, guitar, lightning, penguin, pizza,
# rollerskates, sea turtle, sheep, snowflake, solar,
# swan, The Eiffel Tower, tractor, practice, tree
lessons <- c(26, 29, 43, 49, 50,
             125, 134, 172, 218, 225,
             246, 255, 258, 271, 295,
             296, 308, 320, 322, 323
)

classes_tensor <- tf$solid(lessons, tf$int64)

train_ds <- train_ds %>%
  dataset_filter(
    perform(document) tf$reduce_any(tf$equal(classes_tensor, document$label), -1L)
  )

The PixelCNN distribution expects values within the vary from 0 to 255 – no normalization required. Preprocessing then consists
of simply casting pixels and labels every to float:

preprocess <- perform(document) {
  document$picture <- tf$solid(document$picture, tf$float32) 
  document$label <- tf$solid(document$label, tf$float32)
  record(tuple(document$picture, document$label))
}

batch_size <- 32

practice <- train_ds %>%
  dataset_map(preprocess) %>%
  dataset_shuffle(10000) %>%
  dataset_batch(batch_size)

Creating the mannequin

We now use tfd_pixel_cnn to outline what would be the
loglikelihood utilized by the mannequin.

dist <- tfd_pixel_cnn(
  image_shape = c(28, 28, 1),
  conditional_shape = record(),
  num_resnet = 5,
  num_hierarchies = 3,
  num_filters = 128,
  num_logistic_mix = 5,
  dropout_p =.5
)

image_input <- layer_input(form = c(28, 28, 1))
label_input <- layer_input(form = record())
log_prob <- dist %>% tfd_log_prob(image_input, conditional_input = label_input)

This tradition loglikelihood is added as a loss to the mannequin, after which, the mannequin is compiled with simply an optimizer
specification solely. Throughout coaching, loss first decreased shortly, however enhancements from later epochs had been smaller.

mannequin <- keras_model(inputs = record(image_input, label_input), outputs = log_prob)
mannequin$add_loss(-tf$reduce_mean(log_prob))
mannequin$compile(optimizer = optimizer_adam(lr = .001))

mannequin %>% match(practice, epochs = 10)

To collectively show actual and faux pictures:

for (i in lessons) {
  
  real_images <- train_ds %>%
    dataset_filter(
      perform(document) document$label == tf$solid(i, tf$int64)
    ) %>% 
    dataset_take(8) %>%
    dataset_batch(8)
  it <- as_iterator(real_images)
  real_images <- iter_next(it)
  real_images <- real_images$picture %>% as.array()
  real_images <- real_images[ , , , 1]/255
  
  generated_images <- dist %>% tfd_sample(8, conditional_input = i)
  generated_images <- generated_images %>% as.array()
  generated_images <- generated_images[ , , , 1]/255
  
  pictures <- abind::abind(real_images, generated_images, alongside = 1)
  png(paste0("draw_", i, ".png"), width = 8 * 28 * 10, top = 2 * 28 * 10)
  par(mfrow = c(2, 8), mar = c(0, 0, 0, 0))
  pictures %>%
    purrr::array_tree(1) %>%
    purrr::map(as.raster) %>%
    purrr::iwalk(plot)
  dev.off()
}

From our twenty lessons, right here’s a alternative of six, every displaying actual drawings within the prime row, and faux ones under.

We most likely wouldn’t confuse the primary and second rows, however then, the precise human drawings exhibit huge variation, too.
And nobody ever mentioned PixelCNN was an structure for idea studying. Be at liberty to mess around with different datasets of your
alternative – TFP’s PixelCNN distribution makes it simple.

Wrapping up

On this publish, we had tfprobability / TFP do all of the heavy lifting for us, and so, might give attention to the underlying ideas.
Relying in your inclinations, this may be a perfect scenario – you don’t lose sight of the forest for the timber. On the
different hand: Do you have to discover that altering the supplied parameters doesn’t obtain what you need, you’ve got a reference
implementation to start out from. So regardless of the final result, the addition of such higher-level performance to TFP is a win for the
customers. (If you happen to’re a TFP developer studying this: Sure, we’d like extra :-)).

To everybody although, thanks for studying!