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 `NaN`

s 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`

:

### 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!

*CoRR*abs/1601.06759. http://arxiv.org/abs/1601.06759.

*CoRR*abs/1606.05328. http://arxiv.org/abs/1606.05328.

Salimans, Tim, Andrej Karpathy, Xi Chen, and Diederik P. Kingma. 2017. “PixelCNN++: A PixelCNN Implementation with Discretized Logistic Combination Probability and Different Modifications.” In *ICLR*.