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

Usually, these concerned Variational Autoencoders (VAEs), whose attraction lies in them permitting to mannequin a latent area of
underlying, impartial (ideally) elements that decide the seen options. A doable draw back might be the inferior
high quality of generated samples. Generative Adversarial Networks (GANs) are one other widespread method. Conceptually, these are
extremely engaging attributable to their game-theoretic framing. Nonetheless, they are often tough to coach. PixelCNN variants, on the
different hand – we’ll subsume all of them right here below PixelCNN – are typically recognized for his or her good outcomes. They appear to contain
some extra alchemy although. Underneath these circumstances, what might be extra welcome than a simple means of experimenting with
them? Via TensorFlow Likelihood (TFP) and its R wrapper, tfprobability, we now have
such a means.

This put up 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 ideas

Autoregressivity, or: We want (some) order

The essential concept 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 have been two-dimensional. So this implies we’ve got 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 crimson, inexperienced,
and blue. The unique PixelCNN paper(Oord, Kalchbrenner, and Kavukcuoglu 2016) carried by way of autoregressivity right here as effectively, with a pixel’s depth for
crimson relying on simply prior pixels, these for inexperienced relying on these similar prior pixels however moreover, the present worth
for crimson, and people for blue relying on the prior pixels in addition to the present values for crimson 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 applied in TFP, PixelCNN++(Salimans et al. 2017) , introduces a simplification; it factorizes the joint
distribution in a much less compute-intensive means.

Technically, then, we all know how autoregressivity is realized; intuitively, it could nonetheless appear stunning that imposing a raster
scan order “simply works” (to me, no less than, 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 motive – as traditional in picture processing, convolutional layers (or blocks thereof) are
concerned. However – is it not the very nature of a convolution that it computes a median of some kinds, trying, 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 simpler to unravel 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 unique 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
situated 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
through the use of two totally different convolutional stacks, one continuing from prime to backside, the opposite from left to proper.

Conditioning, or: Present me a kitten

Thus far, we’ve all the time talked about “producing pictures” in a purely generic means. However the true attraction lies in creating
samples of some specified sort – one of many courses we’ve been coaching on, or orthogonal data fed into the community.
That is the place PixelCNN turns into Conditional PixelCNN(Oord et al. 2016), and it’s also the place that feeling of magic resurfaces.
Once more, as “basic math” it’s not exhausting 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})]

(For those who’re questioning in regards to the second half on the suitable, after the Hadamard product signal – we received’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, akin 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 really made?

Logistic combination chance , or: No pixel is an island

Once more, that is the place the TFP implementation doesn’t comply with the unique paper, however the latter PixelCNN++ one. Initially,
pixels have been modeled as discrete values, selected by a softmax over 256 (0-255) doable values. (That this really labored
looks like 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 combination 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, known as ResNet layers as a result of 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 certainly to experiment with, however there are just a few extra you possibly can
take a look at within the documentation. The variety of logistic
distributions within the combination can also be configurable, however from my experiments it’s greatest to maintain that quantity fairly 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 folks to attract some object in at most twenty seconds, utilizing the mouse. (To see for your self, simply take a look at
the web site). As of immediately, there are greater than a fifty million cases, from 345
totally different courses.

Firstly, these information have been chosen to take a break from MNIST and its variants. However similar to these (and lots of extra!),
QuickDraw might 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 elements. So to anchor judgment, when displaying generated samples we all the time present eight precise drawings
with them.

Getting 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 courses. 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
courses <- c(26, 29, 43, 49, 50,
             125, 134, 172, 218, 225,
             246, 255, 258, 271, 295,
             296, 308, 320, 322, 323
)

classes_tensor <- tf$forged(courses, tf$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$picture <- tf$forged(document$picture, tf$float32) 
  document$label <- tf$forged(document$label, tf$float32)
  listing(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 = listing(),
  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 = listing())
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 rapidly, however enhancements from later epochs have been smaller.

mannequin <- keras_model(inputs = listing(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 pretend pictures:

for (i in courses) {
  
  real_images <- train_ds %>%
    dataset_filter(
      perform(document) document$label == tf$forged(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 courses, right here’s a selection of six, every exhibiting actual drawings within the prime row, and pretend ones beneath.

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

Wrapping up

On this put up, 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 an excellent state of affairs – you don’t lose sight of the forest for the timber. On the
different hand: Must you discover that altering the offered parameters doesn’t obtain what you need, you may have a reference
implementation to start out from. So regardless of the end result, the addition of such higher-level performance to TFP is a win for the
customers. (For those who’re a TFP developer studying this: Sure, we’d like extra :-)).

To everybody although, thanks for studying!