As of at the moment, deep studying’s best successes have taken place within the realm of supervised studying, requiring tons and many annotated coaching information. Nevertheless, information doesn’t (usually) include annotations or labels. Additionally, unsupervised studying is engaging due to the analogy to human cognition.

On this weblog thus far, now we have seen two main architectures for unsupervised studying: variational autoencoders and generative adversarial networks. Lesser identified, however interesting for conceptual in addition to for efficiency causes are normalizing flows (Jimenez Rezende and Mohamed 2015). On this and the following submit, we’ll introduce flows, specializing in the way to implement them utilizing TensorFlow Chance (TFP).

In distinction to earlier posts involving TFP that accessed its performance utilizing low-level $-syntax, we now make use of tfprobability, an R wrapper within the model of keras, tensorflow and tfdatasets. A be aware concerning this bundle: It’s nonetheless beneath heavy improvement and the API could change. As of this writing, wrappers don’t but exist for all TFP modules, however all TFP performance is accessible utilizing $-syntax if want be.

Density estimation and sampling

Again to unsupervised studying, and particularly pondering of variational autoencoders, what are the principle issues they offer us? One factor that’s seldom lacking from papers on generative strategies are photos of super-real-looking faces (or mattress rooms, or animals …). So evidently sampling (or: era) is a vital half. If we will pattern from a mannequin and procure real-seeming entities, this implies the mannequin has discovered one thing about how issues are distributed on this planet: it has discovered a distribution.
Within the case of variational autoencoders, there’s extra: The entities are imagined to be decided by a set of distinct, disentangled (hopefully!) latent elements. However this isn’t the belief within the case of normalizing flows, so we aren’t going to elaborate on this right here.

As a recap, how can we pattern from a VAE? We draw from (z), the latent variable, and run the decoder community on it. The outcome ought to – we hope – appear like it comes from the empirical information distribution. It shouldn’t, nevertheless, look precisely like several of the objects used to coach the VAE, or else now we have not discovered something helpful.

The second factor we could get from a VAE is an evaluation of the plausibility of particular person information, for use, for instance, in anomaly detection. Right here “plausibility” is imprecise on goal: With VAE, we don’t have a way to compute an precise density beneath the posterior.

What if we wish, or want, each: era of samples in addition to density estimation? That is the place normalizing flows are available.

Normalizing flows

A move is a sequence of differentiable, invertible mappings from information to a “good” distribution, one thing we will simply pattern from and use to calculate a density. Let’s take as instance the canonical technique to generate samples from some distribution, the exponential, say.

We begin by asking our random quantity generator for some quantity between 0 and 1:

This quantity we deal with as coming from a cumulative likelihood distribution (CDF) – from an exponential CDF, to be exact. Now that now we have a price from the CDF, all we have to do is map that “again” to a price. That mapping CDF -> worth we’re on the lookout for is simply the inverse of the CDF of an exponential distribution, the CDF being

[F(x) = 1 – e^{-lambda x}]

The inverse then is

[
F^{-1}(u) = -frac{1}{lambda} ln (1 – u)
]

which implies we could get our exponential pattern doing

lambda <- 0.5 # choose some lambda
x <- -1/lambda * log(1-u)

We see the CDF is definitely a move (or a constructing block thereof, if we image most flows as comprising a number of transformations), since

• It maps information to a uniform distribution between 0 and 1, permitting to evaluate information probability.
• Conversely, it maps a likelihood to an precise worth, thus permitting to generate samples.

From this instance, we see why a move needs to be invertible, however we don’t but see why it needs to be differentiable. This may turn out to be clear shortly, however first let’s check out how flows can be found in tfprobability.

Bijectors

TFP comes with a treasure trove of transformations, known as bijectors, starting from easy computations like exponentiation to extra complicated ones just like the discrete cosine rework.

To get began, let’s use tfprobability to generate samples from the conventional distribution.
There’s a bijector tfb_normal_cdf() that takes enter information to the interval ([0,1]). Its inverse rework then yields a random variable with the usual regular distribution:

Conversely, we will use this bijector to find out the (log) likelihood of a pattern from the conventional distribution. We’ll test towards a simple use of tfd_normal within the distributions module:

x <- 2.01
d_n <- tfd_normal(loc = 0, scale = 1) 

d_n %>% tfd_log_prob(x) %>% as.numeric() # -2.938989

To acquire that very same log likelihood from the bijector, we add two elements:

• Firstly, we run the pattern by the ahead transformation and compute log likelihood beneath the uniform distribution.
• Secondly, as we’re utilizing the uniform distribution to find out likelihood of a traditional pattern, we have to observe how likelihood modifications beneath this transformation. That is executed by calling tfb_forward_log_det_jacobian (to be additional elaborated on beneath).

b <- tfb_normal_cdf()
d_u <- tfd_uniform()

l <- d_u %>% tfd_log_prob(b %>% tfb_forward(x))
j <- b %>% tfb_forward_log_det_jacobian(x, event_ndims = 0)

(l + j) %>% as.numeric() # -2.938989

Why does this work? Let’s get some background.

Chance mass is conserved

Flows are primarily based on the precept that beneath transformation, likelihood mass is conserved. Say now we have a move from (x) to (z):
[z = f(x)]

Suppose we pattern from (z) after which, compute the inverse rework to acquire (x). We all know the likelihood of (z). What’s the likelihood that (x), the reworked pattern, lies between (x_0) and (x_0 + dx)?

This likelihood is (p(x) dx), the density instances the size of the interval. This has to equal the likelihood that (z) lies between (f(x)) and (f(x + dx)). That new interval has size (f'(x) dx), so:

[p(x) dx = p(z) f'(x) dx]

Or equivalently

[p(x) = p(z) * dz/dx]

Thus, the pattern likelihood (p(x)) is decided by the bottom likelihood (p(z)) of the reworked distribution, multiplied by how a lot the move stretches house.

The identical goes in larger dimensions: Once more, the move is in regards to the change in likelihood quantity between the (z) and (y) areas:

[p(x) = p(z) frac{vol(dz)}{vol(dx)}]

In larger dimensions, the Jacobian replaces the spinoff. Then, the change in quantity is captured by absolutely the worth of its determinant:

[p(mathbf{x}) = p(f(mathbf{x})) bigg|detfrac{partial f({mathbf{x})}}{partial{mathbf{x}}}bigg|]

In observe, we work with log possibilities, so

[log p(mathbf{x}) = log p(f(mathbf{x})) + log bigg|detfrac{partial f({mathbf{x})}}{partial{mathbf{x}}}bigg| ]

Let’s see this with one other bijector instance, tfb_affine_scalar. Under, we assemble a mini-flow that maps just a few arbitrary chosen (x) values to double their worth (scale = 2):

x <- c(0, 0.5, 1)
b <- tfb_affine_scalar(shift = 0, scale = 2)

To check densities beneath the move, we select the conventional distribution, and take a look at the log densities:

d_n <- tfd_normal(loc = 0, scale = 1)
d_n %>% tfd_log_prob(x) %>% as.numeric() # -0.9189385 -1.0439385 -1.4189385

Now apply the move and compute the brand new log densities as a sum of the log densities of the corresponding (x) values and the log determinant of the Jacobian:

z <- b %>% tfb_forward(x)

(d_n  %>% tfd_log_prob(b %>% tfb_inverse(z))) +
  (b %>% tfb_inverse_log_det_jacobian(z, event_ndims = 0)) %>%
  as.numeric() # -1.6120857 -1.7370857 -2.1120858

We see that because the values get stretched in house (we multiply by 2), the person log densities go down.
We are able to confirm the cumulative likelihood stays the identical utilizing tfd_transformed_distribution():

d_t <- tfd_transformed_distribution(distribution = d_n, bijector = b)
d_n %>% tfd_cdf(x) %>% as.numeric()  # 0.5000000 0.6914625 0.8413447

d_t %>% tfd_cdf(y) %>% as.numeric()  # 0.5000000 0.6914625 0.8413447

Thus far, the flows we noticed had been static – how does this match into the framework of neural networks?

Coaching a move

Provided that flows are bidirectional, there are two methods to consider them. Above, now we have principally harassed the inverse mapping: We wish a easy distribution we will pattern from, and which we will use to compute a density. In that line, flows are generally known as “mappings from information to noise” – noise principally being an isotropic Gaussian. Nevertheless in observe, we don’t have that “noise” but, we simply have information.
So in observe, now we have to be taught a move that does such a mapping. We do that by utilizing bijectors with trainable parameters.
We’ll see a quite simple instance right here, and go away “actual world flows” to the following submit.

The instance is predicated on half 1 of Eric Jang’s introduction to normalizing flows. The principle distinction (aside from simplification to indicate the fundamental sample) is that we’re utilizing keen execution.

We begin from a two-dimensional, isotropic Gaussian, and we need to mannequin information that’s additionally regular, however with a imply of 1 and a variance of two (in each dimensions).

library(tensorflow)
library(tfprobability)

tfe_enable_eager_execution(device_policy = "silent")

library(tfdatasets)

# the place we begin from
base_dist <- tfd_multivariate_normal_diag(loc = c(0, 0))

# the place we need to go
target_dist <- tfd_multivariate_normal_diag(loc = c(1, 1), scale_identity_multiplier = 2)

# create coaching information from the goal distribution
target_samples <- target_dist %>% tfd_sample(1000) %>% tf$solid(tf$float32)

batch_size <- 100
dataset <- tensor_slices_dataset(target_samples) %>%
  dataset_shuffle(buffer_size = dim(target_samples)[1]) %>%
  dataset_batch(batch_size)

Now we’ll construct a tiny neural community, consisting of an affine transformation and a nonlinearity.
For the previous, we will make use of tfb_affine, the multi-dimensional relative of tfb_affine_scalar.
As to nonlinearities, at the moment TFP comes with tfb_sigmoid and tfb_tanh, however we will construct our personal parameterized ReLU utilizing tfb_inline:

# alpha is a learnable parameter
bijector_leaky_relu <- operate(alpha) {
  
  tfb_inline(
    # ahead rework leaves constructive values untouched and scales detrimental ones by alpha
    forward_fn = operate(x)
      tf$the place(tf$greater_equal(x, 0), x, alpha * x),
    # inverse rework leaves constructive values untouched and scales detrimental ones by 1/alpha
    inverse_fn = operate(y)
      tf$the place(tf$greater_equal(y, 0), y, 1/alpha * y),
    # quantity change is 0 when constructive and 1/alpha when detrimental
    inverse_log_det_jacobian_fn = operate(y) {
      I <- tf$ones_like(y)
      J_inv <- tf$the place(tf$greater_equal(y, 0), I, 1/alpha * I)
      log_abs_det_J_inv <- tf$log(tf$abs(J_inv))
      tf$reduce_sum(log_abs_det_J_inv, axis = 1L)
    },
    forward_min_event_ndims = 1
  )
}

Outline the learnable variables for the affine and the PReLU layers:

d <- 2 # dimensionality
r <- 2 # rank of replace

# shift of affine bijector
shift <- tf$get_variable("shift", d)
# scale of affine bijector
L <- tf$get_variable('L', c(d * (d + 1) / 2))
# rank-r replace
V <- tf$get_variable("V", c(d, r))

# scaling issue of parameterized relu
alpha <- tf$abs(tf$get_variable('alpha', listing())) + 0.01

With keen execution, the variables have for use contained in the loss operate, so that’s the place we outline the bijectors. Our little move now could be a tfb_chain of bijectors, and we wrap it in a TransformedDistribution (tfd_transformed_distribution) that hyperlinks supply and goal distributions.

loss <- operate() {
  
 affine <- tfb_affine(
        scale_tril = tfb_fill_triangular() %>% tfb_forward(L),
        scale_perturb_factor = V,
        shift = shift
      )
 lrelu <- bijector_leaky_relu(alpha = alpha)  
 
 move <- listing(lrelu, affine) %>% tfb_chain()
 
 dist <- tfd_transformed_distribution(distribution = base_dist,
                          bijector = move)
  
 l <- -tf$reduce_mean(dist$log_prob(batch))
 # preserve observe of progress
 print(spherical(as.numeric(l), 2))
 l
}

Now we will really run the coaching!

optimizer <- tf$practice$AdamOptimizer(1e-4)

n_epochs <- 100
for (i in 1:n_epochs) {
  iter <- make_iterator_one_shot(dataset)
  until_out_of_range({
    batch <- iterator_get_next(iter)
    optimizer$reduce(loss)
  })
}

Outcomes will differ relying on random initialization, however it is best to see a gradual (if sluggish) progress. Utilizing bijectors, now we have really skilled and outlined slightly neural community.

Outlook

Undoubtedly, this move is just too easy to mannequin complicated information, nevertheless it’s instructive to have seen the fundamental ideas earlier than delving into extra complicated flows. Within the subsequent submit, we’ll take a look at autoregressive flows, once more utilizing TFP and tfprobability.