Posit AI Weblog: Simple PixelCNN with tfprobability

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

Typically, 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 might be the inferior
high quality of generated samples. Generative Adversarial Networks (GANs) are one other well-liked strategy. Conceptually, these are
extremely engaging as a consequence of their game-theoretic framing. Nevertheless, they are often troublesome to coach. PixelCNN variants, on the
different hand – we’ll subsume all of them right here underneath PixelCNN – are usually recognized for his or her good outcomes. They appear to contain
some extra alchemy although. Below these circumstances, what could possibly be extra welcome than a straightforward approach of experimenting with
them? Via TensorFlow Likelihood (TFP) and its R wrapper, tfprobability, we now have
such a approach.

This put up first offers 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 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 photos have been two-dimensional. So this implies we now 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
colour photos, 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 means 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 approach.

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). Possibly that is a kind 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 cause – 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 additionally, at its spatial (or temporal) environment? How does that rhyme
with the look-at-just-prior-pixels technique?

Surprisingly, this downside is simpler 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 trustworthy, however introduces a distinct downside: 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 are 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

Up to now, we’ve at all times 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 info 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 “common 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})]

(Should you’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, similar 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 probability , 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) doable values. (That this truly 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 colour 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 most certainly to experiment with, however there are a number of 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 finest to maintain that quantity relatively low to keep away from
producing NaNs throughout coaching.

Let’s now see an entire 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 website). As of at this time, there are greater than a fifty million situations, from 345
completely different courses.

At the beginning, 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 type, through tfds, the R wrapper to
TensorFlow datasets. In distinction to the MNIST “household” although, the “actual samples” are themselves extremely irregular, and infrequently
even lacking important components. So to anchor judgment, when displaying generated samples we at all times present eight precise drawings
with them.

Getting 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$solid(courses, tf$int64)

train_ds <- train_ds %>%
  dataset_filter(
    operate(file) tf$reduce_any(tf$equal(classes_tensor, file$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 <- operate(file) {
  file$picture <- tf$solid(file$picture, tf$float32) 
  file$label <- tf$solid(file$label, tf$float32)
  list(tuple(file$picture, file$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 = list(),
  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 = list())
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 rapidly, however enhancements from later epochs have been smaller.

mannequin <- keras_model(inputs = list(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 photos:

for (i in courses) {
  
  real_images <- train_ds %>%
    dataset_filter(
      operate(file) file$label == tf$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
  
  photos <- abind::abind(real_images, generated_images, alongside = 1)
  png(paste0("draw_", i, ".png"), width = 8 * 28 * 10, peak = 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 displaying actual drawings within the prime row, and faux ones under.

We in all probability 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 straightforward.

Wrapping up

On this put up, we had tfprobability / TFP do all of the heavy lifting for us, and so, might deal with the underlying ideas.
Relying in your inclinations, this may be a great state of affairs – you don’t lose sight of the forest for the bushes. On the
different hand: Must you discover that altering the supplied parameters doesn’t obtain what you need, you’ve 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. (Should you’re a TFP developer studying this: Sure, we’d like extra :-)).

To everybody although, thanks for studying!