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

Usually, these concerned Variational Autoencoders (VAEs), whose enchantment lies in them permitting to mannequin a latent area of
underlying, impartial (ideally) components that decide the seen options. A potential draw back will be the inferior
high quality of generated samples. Generative Adversarial Networks (GANs) are one other in style strategy. Conceptually, these are
extremely enticing because 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 below PixelCNN – are usually identified for his or her good outcomes. They appear to contain
some extra alchemy although. Beneath these circumstances, what could possibly be extra welcome than a simple approach of experimenting with
them? By TensorFlow Likelihood (TFP) and its R wrapper, tfprobability, we now have
such a approach.

This publish first provides an introduction to PixelCNN, concentrating on high-level ideas (leaving the small print 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’d like (some) order

The fundamental 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 photos have been two-dimensional. So this implies we now have to impose
an order on the pixels. Generally this shall be raster scan order: row after row, from left to proper. However when coping with
coloration photos, 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 properly, with a pixel’s depth for
purple relying on simply prior pixels, these for inexperienced relying on these identical 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 approach.

Technically, then, we all know how autoregressivity is realized; intuitively, it might nonetheless appear shocking that imposing a raster
scan order “simply works” (to me, at the very least, it’s). Possibly 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 regular 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, 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 downside 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 special downside: With every successive convolutional layer consuming its
predecessor’s output, there’s a repeatedly 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
by utilizing two completely 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 all the time talked about “producing photos” in a purely generic approach. However the actual 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 additionally it is the place that feeling of magic resurfaces.
Once more, as “basic math” it’s not arduous 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})]

(In the event you’re questioning concerning the second half on the correct, 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, comparable 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 observe the unique paper, however the latter PixelCNN++ one. Initially,
pixels have been modeled as discrete values, selected by a softmax over 256 (0-255) potential values. (That this really 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 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 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 also be configurable, however from my experiments it’s finest to maintain that quantity moderately low to keep away from
producing NaNs throughout coaching.

Let’s now see an entire instance.

Finish-to-end instance

Our playground shall 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 try
the web site). As of right this moment, there are greater than a fifty million cases, from 345
completely 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 will be obtained, in tfdatasets-ready type, 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.

Making ready the info

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 practice 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 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 photos:

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
  
  photos <- 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))
  photos %>%
    purrr::array_tree(1) %>%
    purrr::map(as.raster) %>%
    purrr::iwalk(plot)
  dev.off()
}

From our twenty courses, right here’s a alternative 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
alternative – TFP’s PixelCNN distribution makes it straightforward.

Wrapping up

On this publish, we had tfprobability / TFP do all of the heavy lifting for us, and so, may deal with the underlying ideas.
Relying in your inclinations, this may be a great scenario – 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 will have a reference
implementation to start out from. So regardless of the consequence, the addition of such higher-level performance to TFP is a win for the
customers. (In the event you’re a TFP developer studying this: Sure, we’d like extra :-)).

To everybody although, thanks for studying!