# Experimenting with autoregressive flows in TensorFlow Chance

Within the first part of this mini-series on autoregressive flow models, we checked out *bijectors* in TensorFlow Chance (TFP), and noticed the best way to use them for sampling and density estimation. We singled out the *affine bijector* to show the mechanics of circulation development: We begin from a distribution that’s simple to pattern from, and that permits for simple calculation of its density. Then, we connect some variety of invertible transformations, optimizing for data-likelihood beneath the ultimate reworked distribution. The effectivity of that (log)chance calculation is the place normalizing flows excel: Loglikelihood beneath the (unknown) goal distribution is obtained as a sum of the density beneath the bottom distribution of the inverse-transformed information plus absolutely the log determinant of the inverse Jacobian.

Now, an affine circulation will seldom be highly effective sufficient to mannequin nonlinear, complicated transformations. In constrast, autoregressive fashions have proven substantive success in density estimation in addition to pattern technology. Mixed with extra concerned architectures, function engineering, and in depth compute, the idea of autoregressivity has powered – and is powering – state-of-the-art architectures in areas corresponding to picture, speech and video modeling.

This publish can be involved with the constructing blocks of autoregressive flows in TFP. Whereas we received’t precisely be constructing state-of-the-art fashions, we’ll attempt to perceive and play with some main substances, hopefully enabling the reader to do her personal experiments on her personal information.

This publish has three elements: First, we’ll take a look at autoregressivity and its implementation in TFP. Then, we attempt to (roughly) reproduce one of many experiments within the “MAF paper” (*Masked Autoregressive Flows for Distribution Estimation* (Papamakarios, Pavlakou, and Murray 2017)) – primarily a proof of idea. Lastly, for the third time on this weblog, we come again to the duty of analysing audio information, with blended outcomes.

## Autoregressivity and masking

In distribution estimation, autoregressivity enters the scene by way of the chain rule of likelihood that decomposes a joint density right into a product of conditional densities:

[

p(mathbf{x}) = prod_{i}p(mathbf{x}_i|mathbf{x}_{1:i−1})

]

In follow, which means autoregressive fashions must impose an order on the variables – an order which could or may not “make sense.” Approaches right here embrace selecting orderings at random and/or utilizing totally different orderings for every layer.

Whereas in recurrent neural networks, autoregressivity is conserved as a result of recurrence relation inherent in state updating, it’s not clear a priori how autoregressivity is to be achieved in a densely linked structure. A computationally environment friendly resolution was proposed in *MADE: Masked Autoencoder for Distribution Estimation*(Germain et al. 2015): Ranging from a densely linked layer, masks out all connections that shouldn’t be allowed, i.e., all connections from enter function (i) to mentioned layer’s activations (1 … i-1). Or expressed in a different way, activation (i) could also be linked to enter options (1 … i-1) solely. Then when including extra layers, care have to be taken to make sure that all required connections are masked in order that on the finish, output (i) will solely ever have seen inputs (1 … i-1).

Thus masked autoregressive flows are a fusion of two main approaches – autoregressive fashions (which needn’t be flows) and flows (which needn’t be autoregressive). In TFP these are offered by `MaskedAutoregressiveFlow`

, for use as a bijector in a `TransformedDistribution`

.

Whereas the documentation exhibits the best way to use this bijector, the step from theoretical understanding to coding a “black field” could seem huge. In case you’re something just like the creator, right here you would possibly really feel the urge to “look beneath the hood” and confirm that issues actually are the best way you’re assuming. So let’s give in to curiosity and permit ourselves a bit of escapade into the supply code.

Peeking forward, that is how we’ll assemble a masked autoregressive circulation in TFP (once more utilizing the nonetheless new-ish R bindings offered by tfprobability):

```
library(tfprobability)
maf <- tfb_masked_autoregressive_flow(
shift_and_log_scale_fn = tfb_masked_autoregressive_default_template(
hidden_layers = list(num_hidden, num_hidden),
activation = tf$nn$tanh)
)
```

Pulling aside the related entities right here, `tfb_masked_autoregressive_flow`

is a bijector, with the standard strategies `tfb_forward()`

, `tfb_inverse()`

, `tfb_forward_log_det_jacobian()`

and `tfb_inverse_log_det_jacobian()`

.

The default `shift_and_log_scale_fn`

, `tfb_masked_autoregressive_default_template`

, constructs a bit of neural community of its personal, with a configurable variety of hidden models per layer, a configurable activation operate and optionally, different configurable parameters to be handed to the underlying `dense`

layers. It’s these dense layers that must respect the autoregressive property. Can we check out how that is performed? Sure we will, offered we’re not afraid of a bit of Python.

`masked_autoregressive_default_template`

(now leaving out the `tfb_`

as we’ve entered Python-land) makes use of `masked_dense`

to do what you’d suppose a thus-named operate may be doing: assemble a dense layer that has a part of the load matrix masked out. How? We’ll see after a number of Python setup statements.

current form on master), and when potential, simplified for higher readability, accommodating simply the specifics of the chosen instance – a toy matrix of form 2×3:

Papamakarios, Pavlakou, and Murray 2017) utilized masked autoregressive flows (in addition to single-layer-*MADE*(Germain et al. 2015) and Actual NVP (Dinh, Sohl-Dickstein, and Bengio 2016)) to various datasets, together with MNIST, CIFAR-10 and several other datasets from the UCI Machine Learning Repository.

We choose one of many UCI datasets: Gas sensors for home activity monitoring. On this dataset, the MAF authors obtained one of the best outcomes utilizing a MAF with 10 flows, so that is what we are going to attempt.

Gathering info from the paper, we all know that

- information was included from the file
*ethylene_CO.txt*solely; - discrete columns had been eradicated, in addition to all columns with correlations > .98; and
- the remaining 8 columns had been standardised (z-transformed).

Concerning the neural community structure, we collect that

- every of the ten MAF layers was adopted by a batchnorm;
- as to function order, the primary MAF layer used the variable order that got here with the dataset; then each consecutive layer reversed it;
- particularly for this dataset and versus all different UCI datasets,
*tanh*was used for activation as a substitute of*relu*; - the Adam optimizer was used, with a studying charge of 1e-4;
- there have been two hidden layers for every MAF, with 100 models every;
- coaching went on till no enchancment occurred for 30 consecutive epochs on the validation set; and
- the bottom distribution was a multivariate Gaussian.

That is all helpful info for our try to estimate this dataset, however the important bit is that this. In case you knew the dataset already, you may need been questioning how the authors would take care of the dimensionality of the information: It’s a time collection, and the MADE structure explored above introduces autoregressivity between options, not time steps. So how is the extra temporal autoregressivity to be dealt with? The reply is: The time dimension is actually eliminated. Within the authors’ phrases,

[…] it’s a time collection however was handled as if every instance had been an i.i.d. pattern from the marginal distribution.

This undoubtedly is helpful info for our current modeling try, nevertheless it additionally tells us one thing else: We would must look past MADE layers for precise time collection modeling.

Now although let’s take a look at this instance of utilizing MAF for multivariate modeling, with no time or spatial dimension to be taken into consideration.

Following the hints the authors gave us, that is what we do.

```
Observations: 4,208,261
Variables: 19
$ X1 <dbl> 0.00, 0.01, 0.01, 0.03, 0.04, 0.05, 0.06, 0.07, 0.07, 0.09,...
$ X2 <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
$ X3 <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
$ X4 <dbl> -50.85, -49.40, -40.04, -47.14, -33.58, -48.59, -48.27, -47.14,...
$ X5 <dbl> -1.95, -5.53, -16.09, -10.57, -20.79, -11.54, -9.11, -4.56,...
$ X6 <dbl> -41.82, -42.78, -27.59, -32.28, -33.25, -36.16, -31.31, -16.57,...
$ X7 <dbl> 1.30, 0.49, 0.00, 4.40, 6.03, 6.03, 5.37, 4.40, 23.98, 2.77,...
$ X8 <dbl> -4.07, 3.58, -7.16, -11.22, 3.42, 0.33, -7.97, -2.28, -2.12,...
$ X9 <dbl> -28.73, -34.55, -42.14, -37.94, -34.22, -29.05, -30.34, -24.35,...
$ X10 <dbl> -13.49, -9.59, -12.52, -7.16, -14.46, -16.74, -8.62, -13.17,...
$ X11 <dbl> -3.25, 5.37, -5.86, -1.14, 8.31, -1.14, 7.00, -6.34, -0.81,...
$ X12 <dbl> 55139.95, 54395.77, 53960.02, 53047.71, 52700.28, 51910.52,...
$ X13 <dbl> 50669.50, 50046.91, 49299.30, 48907.00, 48330.96, 47609.00,...
$ X14 <dbl> 9626.26, 9433.20, 9324.40, 9170.64, 9073.64, 8982.88, 8860.51,...
$ X15 <dbl> 9762.62, 9591.21, 9449.81, 9305.58, 9163.47, 9021.08, 8966.48,...
$ X16 <dbl> 24544.02, 24137.13, 23628.90, 23101.66, 22689.54, 22159.12,...
$ X17 <dbl> 21420.68, 20930.33, 20504.94, 20101.42, 19694.07, 19332.57,...
$ X18 <dbl> 7650.61, 7498.79, 7369.67, 7285.13, 7156.74, 7067.61, 6976.13,...
$ X19 <dbl> 6928.42, 6800.66, 6697.47, 6578.52, 6468.32, 6385.31, 6300.97,...
```

```
# A tibble: 4,208,261 x 8
X4 X5 X8 X9 X13 X16 X17 X18
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 -50.8 -1.95 -4.07 -28.7 50670. 24544. 21421. 7651.
2 -49.4 -5.53 3.58 -34.6 50047. 24137. 20930. 7499.
3 -40.0 -16.1 -7.16 -42.1 49299. 23629. 20505. 7370.
4 -47.1 -10.6 -11.2 -37.9 48907 23102. 20101. 7285.
5 -33.6 -20.8 3.42 -34.2 48331. 22690. 19694. 7157.
6 -48.6 -11.5 0.33 -29.0 47609 22159. 19333. 7068.
7 -48.3 -9.11 -7.97 -30.3 47047. 21932. 19028. 6976.
8 -47.1 -4.56 -2.28 -24.4 46758. 21504. 18780. 6900.
9 -42.3 -2.77 -2.12 -27.6 46197. 21125. 18439. 6827.
10 -44.6 3.58 -0.65 -35.5 45652. 20836. 18209. 6790.
# … with 4,208,251 extra rows
```

Now arrange the information technology course of:

```
# train-test cut up
n_rows <- nrow(df2) # 4208261
train_ids <- sample(1:n_rows, 0.5 * n_rows)
x_train <- df2[train_ids, ]
x_test <- df2[-train_ids, ]
# create datasets
batch_size <- 100
train_dataset <- tf$forged(x_train, tf$float32) %>%
tensor_slices_dataset %>%
dataset_batch(batch_size)
test_dataset <- tf$forged(x_test, tf$float32) %>%
tensor_slices_dataset %>%
dataset_batch(nrow(x_test))
```

To assemble the circulation, the very first thing wanted is the bottom distribution.

Now for the circulation, by default constructed with batchnorm and permutation of function order.

```
num_hidden <- 100
dim <- ncol(df2)
use_batchnorm <- TRUE
use_permute <- TRUE
num_mafs <-10
num_layers <- 3 * num_mafs
bijectors <- vector(mode = "checklist", size = num_layers)
for (i in seq(1, num_layers, by = 3)) {
maf <- tfb_masked_autoregressive_flow(
shift_and_log_scale_fn = tfb_masked_autoregressive_default_template(
hidden_layers = list(num_hidden, num_hidden),
activation = tf$nn$tanh))
bijectors[[i]] <- maf
if (use_batchnorm)
bijectors[[i + 1]] <- tfb_batch_normalization()
if (use_permute)
bijectors[[i + 2]] <- tfb_permute((ncol(df2) - 1):0)
}
if (use_permute) bijectors <- bijectors[-num_layers]
circulation <- bijectors %>%
discard(is.null) %>%
# tfb_chain expects arguments in reverse order of software
rev() %>%
tfb_chain()
target_dist <- tfd_transformed_distribution(
distribution = base_dist,
bijector = circulation
)
```

And configuring the optimizer:

`optimizer <- tf$prepare$AdamOptimizer(1e-4)`

Underneath that isotropic Gaussian we selected as a base distribution, how doubtless are the information?

```
base_loglik <- base_dist %>%
tfd_log_prob(x_train) %>%
tf$reduce_mean()
base_loglik %>% as.numeric() # -11.33871
base_loglik_test <- base_dist %>%
tfd_log_prob(x_test) %>%
tf$reduce_mean()
base_loglik_test %>% as.numeric() # -11.36431
```

And, simply as a fast sanity test: What’s the loglikelihood of the information beneath the reworked distribution *earlier than any coaching*?

```
target_loglik_pre <-
target_dist %>% tfd_log_prob(x_train) %>% tf$reduce_mean()
target_loglik_pre %>% as.numeric() # -11.22097
target_loglik_pre_test <-
target_dist %>% tfd_log_prob(x_test) %>% tf$reduce_mean()
target_loglik_pre_test %>% as.numeric() # -11.36431
```

The values match – good. Right here now’s the coaching loop. Being impatient, we already preserve checking the loglikelihood on the (full) take a look at set to see if we’re making any progress.

```
n_epochs <- 10
for (i in 1:n_epochs) {
agg_loglik <- 0
num_batches <- 0
iter <- make_iterator_one_shot(train_dataset)
until_out_of_range({
batch <- iterator_get_next(iter)
loss <-
operate()
- tf$reduce_mean(target_dist %>% tfd_log_prob(batch))
optimizer$decrease(loss)
loglik <- tf$reduce_mean(target_dist %>% tfd_log_prob(batch))
agg_loglik <- agg_loglik + loglik
num_batches <- num_batches + 1
test_iter <- make_iterator_one_shot(test_dataset)
test_batch <- iterator_get_next(test_iter)
loglik_test_current <- target_dist %>% tfd_log_prob(test_batch) %>% tf$reduce_mean()
if (num_batches %% 100 == 1)
cat(
"Epoch ",
i,
": ",
"Batch ",
num_batches,
": ",
(agg_loglik %>% as.numeric()) / num_batches,
" --- take a look at: ",
loglik_test_current %>% as.numeric(),
"n"
)
})
}
```

With each coaching and take a look at units amounting to over 2 million information every, we didn’t have the persistence to run this mannequin *till no enchancment occurred for 30 consecutive epochs on the validation set* (just like the authors did). Nevertheless, the image we get from one full epoch’s run is fairly clear: The setup appears to work fairly okay.

```
Epoch 1 : Batch 1: -8.212026 --- take a look at: -10.09264
Epoch 1 : Batch 1001: 2.222953 --- take a look at: 1.894102
Epoch 1 : Batch 2001: 2.810996 --- take a look at: 2.147804
Epoch 1 : Batch 3001: 3.136733 --- take a look at: 3.673271
Epoch 1 : Batch 4001: 3.335549 --- take a look at: 4.298822
Epoch 1 : Batch 5001: 3.474280 --- take a look at: 4.502975
Epoch 1 : Batch 6001: 3.606634 --- take a look at: 4.612468
Epoch 1 : Batch 7001: 3.695355 --- take a look at: 4.146113
Epoch 1 : Batch 8001: 3.767195 --- take a look at: 3.770533
Epoch 1 : Batch 9001: 3.837641 --- take a look at: 4.819314
Epoch 1 : Batch 10001: 3.908756 --- take a look at: 4.909763
Epoch 1 : Batch 11001: 3.972645 --- take a look at: 3.234356
Epoch 1 : Batch 12001: 4.020613 --- take a look at: 5.064850
Epoch 1 : Batch 13001: 4.067531 --- take a look at: 4.916662
Epoch 1 : Batch 14001: 4.108388 --- take a look at: 4.857317
Epoch 1 : Batch 15001: 4.147848 --- take a look at: 5.146242
Epoch 1 : Batch 16001: 4.177426 --- take a look at: 4.929565
Epoch 1 : Batch 17001: 4.209732 --- take a look at: 4.840716
Epoch 1 : Batch 18001: 4.239204 --- take a look at: 5.222693
Epoch 1 : Batch 19001: 4.264639 --- take a look at: 5.279918
Epoch 1 : Batch 20001: 4.291542 --- take a look at: 5.29119
Epoch 1 : Batch 21001: 4.314462 --- take a look at: 4.872157
Epoch 2 : Batch 1: 5.212013 --- take a look at: 4.969406
```

With these coaching outcomes, we regard the proof of idea as mainly profitable. Nevertheless, from our experiments we additionally must say that the selection of hyperparameters appears to matter a *lot*. For instance, use of the `relu`

activation operate as a substitute of `tanh`

resulted within the community mainly studying nothing. (As per the authors, `relu`

labored high-quality on different datasets that had been z-transformed in simply the identical approach.)

*Batch normalization* right here was compulsory – and this would possibly go for flows typically. The permutation bijectors, then again, didn’t make a lot of a distinction on this dataset. Total the impression is that for flows, we’d both want a “bag of tips” (like is often mentioned about GANs), or extra concerned architectures (see “Outlook” under).

Lastly, we wind up with an experiment, coming again to our favourite audio information, already featured in two posts: Simple Audio Classification with Keras and Audio classification with Keras: Looking closer at the non-deep learning parts.

## Analysing audio information with MAF

The dataset in question consists of recordings of 30 phrases, pronounced by various totally different audio system. In these earlier posts, a convnet was educated to map spectrograms to these 30 courses. Now as a substitute we wish to attempt one thing totally different: Prepare an MAF on one of many courses – the phrase “zero,” say – and see if we will use the educated community to mark “non-zero” phrases as much less doubtless: carry out *anomaly detection*, in a approach. Spoiler alert: The outcomes weren’t too encouraging, and if you’re considering a process like this, you would possibly wish to take into account a special structure (once more, see “Outlook” under).

Nonetheless, we rapidly relate what was performed, as this process is a pleasant instance of dealing with information the place options range over a couple of axis.

Preprocessing begins as within the aforementioned earlier posts. Right here although, we explicitly use keen execution, and should generally hard-code recognized values to maintain the code snippets brief.

Audio classification with Keras: Looking closer at the non-deep learning parts, we’d like to coach the community on spectrograms as a substitute of the uncooked time area information.

Utilizing the identical settings for `frame_length`

and `frame_step`

of the Quick Time period Fourier Rework as in that publish, we’d arrive at information formed `variety of frames x variety of FFT coefficients`

. To make this work with the `masked_dense()`

employed in `tfb_masked_autoregressive_flow()`

, the information would then must be flattened, yielding a formidable 25186 options within the joint distribution.

With the structure outlined as above within the GAS instance, this result in the community not making a lot progress. Neither did leaving the information in time area kind, with 16000 options within the joint distribution. Thus, we determined to work with the FFT coefficients computed over the whole window as a substitute, leading to 257 joint options.

```
batch_size <- 100
sampling_rate <- 16000L
data_generator <- operate(df,
batch_size) {
ds <- tensor_slices_dataset(df)
ds <- ds %>%
dataset_map(operate(obs) {
wav <-
decode_wav()(tf$read_file(tf$reshape(obs$fname, list())))
samples <- wav$audio[ ,1]
# some wave information have fewer than 16000 samples
padding <- list(list(0L, sampling_rate - tf$form(samples)[1]))
padded <- tf$pad(samples, padding)
stft_out <- stft()(padded, 16000L, 1L, 512L)
magnitude_spectrograms <- tf$abs(stft_out) %>% tf$squeeze()
})
ds %>% dataset_batch(batch_size)
}
ds_train <- data_generator(df_train, batch_size)
batch <- ds_train %>%
make_iterator_one_shot() %>%
iterator_get_next()
dim(batch) # 100 x 257
```

Coaching then proceeded as on the GAS dataset.

```
# outline MAF
base_dist <-
tfd_multivariate_normal_diag(loc = rep(0, dim(batch)[2]))
num_hidden <- 512
use_batchnorm <- TRUE
use_permute <- TRUE
num_mafs <- 10
num_layers <- 3 * num_mafs
# retailer bijectors in a listing
bijectors <- vector(mode = "checklist", size = num_layers)
# fill checklist, optionally including batchnorm and permute bijectors
for (i in seq(1, num_layers, by = 3)) {
maf <- tfb_masked_autoregressive_flow(
shift_and_log_scale_fn = tfb_masked_autoregressive_default_template(
hidden_layers = list(num_hidden, num_hidden),
activation = tf$nn$tanh,
))
bijectors[[i]] <- maf
if (use_batchnorm)
bijectors[[i + 1]] <- tfb_batch_normalization()
if (use_permute)
bijectors[[i + 2]] <- tfb_permute((dim(batch)[2] - 1):0)
}
if (use_permute) bijectors <- bijectors[-num_layers]
circulation <- bijectors %>%
# probably clear out empty parts (if no batchnorm or no permute)
discard(is.null) %>%
rev() %>%
tfb_chain()
target_dist <- tfd_transformed_distribution(distribution = base_dist,
bijector = circulation)
optimizer <- tf$prepare$AdamOptimizer(1e-3)
# prepare MAF
n_epochs <- 100
for (i in 1:n_epochs) {
agg_loglik <- 0
num_batches <- 0
iter <- make_iterator_one_shot(ds_train)
until_out_of_range({
batch <- iterator_get_next(iter)
loss <-
operate()
- tf$reduce_mean(target_dist %>% tfd_log_prob(batch))
optimizer$decrease(loss)
loglik <- tf$reduce_mean(target_dist %>% tfd_log_prob(batch))
agg_loglik <- agg_loglik + loglik
num_batches <- num_batches + 1
loglik_test_current <-
target_dist %>% tfd_log_prob(ds_test) %>% tf$reduce_mean()
if (num_batches %% 20 == 1)
cat(
"Epoch ",
i,
": ",
"Batch ",
num_batches,
": ",
((agg_loglik %>% as.numeric()) / num_batches) %>% round(1),
" --- take a look at: ",
loglik_test_current %>% as.numeric() %>% round(1),
"n"
)
})
}
```

Throughout coaching, we additionally monitored loglikelihoods on three totally different courses, *cat*, *chook* and *wow*. Listed below are the loglikelihoods from the primary 10 epochs. “Batch” refers back to the present coaching batch (first batch within the epoch), all different values refer to finish datasets (the whole take a look at set and the three units chosen for comparability).

```
epoch | batch | take a look at | "cat" | "chook" | "wow" |
--------|----------|----------|----------|-----------|----------|
1 | 1443.5 | 1455.2 | 1398.8 | 1434.2 | 1546.0 |
2 | 1935.0 | 2027.0 | 1941.2 | 1952.3 | 2008.1 |
3 | 2004.9 | 2073.1 | 2003.5 | 2000.2 | 2072.1 |
4 | 2063.5 | 2131.7 | 2056.0 | 2061.0 | 2116.4 |
5 | 2120.5 | 2172.6 | 2096.2 | 2085.6 | 2150.1 |
6 | 2151.3 | 2206.4 | 2127.5 | 2110.2 | 2180.6 |
7 | 2174.4 | 2224.8 | 2142.9 | 2163.2 | 2195.8 |
8 | 2203.2 | 2250.8 | 2172.0 | 2061.0 | 2221.8 |
9 | 2224.6 | 2270.2 | 2186.6 | 2193.7 | 2241.8 |
10 | 2236.4 | 2274.3 | 2191.4 | 2199.7 | 2243.8 |
```

Whereas this doesn’t look too unhealthy, an entire comparability in opposition to all twenty-nine non-target courses had “zero” outperformed by seven different courses, with the remaining twenty-two decrease in loglikelihood. We don’t have a mannequin for anomaly detection, as but.

## Outlook

As already alluded to a number of instances, for information with temporal and/or spatial orderings extra developed architectures could show helpful. The very profitable *PixelCNN* household relies on masked convolutions, with more moderen developments bringing additional refinements (e.g. *Gated PixelCNN* (Oord et al. 2016), *PixelCNN++* (Salimans et al. 2017). **Consideration**, too, could also be masked and thus rendered autoregressive, as employed within the hybrid *PixelSNAIL* (Chen et al. 2017) and the – not surprisingly given its title – transformer-based *ImageTransformer* (Parmar et al. 2018).

To conclude, – whereas this publish was within the intersection of flows and autoregressivity – and final not least the use therein of TFP bijectors – an upcoming one would possibly dive deeper into autoregressive fashions particularly… and who is aware of, maybe come again to the audio information for a fourth time.

*CoRR*abs/1712.09763. http://arxiv.org/abs/1712.09763.

*CoRR*abs/1605.08803. http://arxiv.org/abs/1605.08803.

*CoRR*abs/1502.03509. http://arxiv.org/abs/1502.03509.

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

*arXiv e-Prints*, Might, arXiv:1705.07057. https://arxiv.org/abs/1705.07057.

*CoRR*abs/1802.05751. http://arxiv.org/abs/1802.05751.

*CoRR*abs/1701.05517. http://arxiv.org/abs/1701.05517.