Recently, I started at Mozilla Research. I am really excited to be a part of a small but great team working hard to solve important ML problems. And everything is open-sourced. We license things to make open-sourced. Oxymoron by first sight isn't it. But I like it !!

Before my presence, our team already released the best known open-sourced STT (Speech to Text) implementation based on Tensorflow. The next step is to improve the current Baidu's Deep Speech architecture and also implement a new TTS (Text to Speech) solution that complements the whole conversational AI agent. So after these two projects, anyone around the world will be able to create his own Alexa without any commercial attachment. Which is the real way to democratize AI, at least I believe it is?

Up until now, I worked on a variety of data types and ML problems, except audio. Now it is time to learn it. And the first thing to do is a comprehensive literature review (like a boss). Here I like to share the top-notch DL architectures dealing with TTS (Text to Speech). I also invite you to our Github repository hosting PyTorch implementation of the first version implementation. (We switched to PyTorch for obvious reasons). It is a work in progress and please feel free to comment and contribute.

Below I like to share my pinpoint summary of the well-known TTS papers which are by no means complete but useful to highlight important aspects of these papers. Let's start.

Phonemes: units of sounds, we pronounce as we speak. Necessary since very similar words in the letter might be pronounced very differently (e.g. "Rough" "Though")

Vocoder: part of the system decoding from features to audio signals. Wave is used in Deep Voice at that stage.

Fundamental Frequency - F0: lowest frequency of a periodic waveform describing the pitch of the sound.

Autoregressive Model: Specifies a model depending linearly on its own outputs and on a parameter set which can be approximated.

Query, Key, Value: Key is used by the attention module to compute attention weights. Value is the vector stipulated by the attention weights to compute the module output. A query vector is the hidden state of the decoder.

Grapheme: Cool way to say character.

Error Modes: Sub-optimal status for the attention block where it is not able to escape.

Monotonic Attention: Use only a limited scope of nodes close in time to the output step. It improves performance for TTS since there is a certain relation btw the output at time t and the input at time t. However, it is not that reasonable for translation problem since words orders might not be the same. https://arxiv.org/pdf/1704.00784.pdf

MOS: Mean Opinion Score. Crowd-source the evaluation process with native speakers. It is not easy to measure, especially for a layman.

Context vector: Output of an attention module which summarizes multiple time-step outputs of the encoder.

Teacher Forcing: Providing model's expected output at time t as input at time t+1. It is controlled ground-truth feedback as a teacher does to a student.

Casual convolution: Convolution which does not foresee the future units given the reference time step T which we like to predict next. In practice, it is implemented by setting right padding orientation to normal convolution layers.

Deep Voice (25 Feb 2017)

Text to phonemes. Deterministically computed with a dictionary. Or Seq2Seq model to deal with the unseen words.

Online Hard Example Mining (OHEM) is a way to pick hard examples with reduced computation cost to improve your network performance on borderline cases which generalize to the general performance. It is mostly used for Object Detection. Suppose you like to train a car detector and you have positive (with car) and negative images (with no car). Now you like to train your network. In practice, you find yourself in many negatives as oppose to relatively much small positives. To this end, it is clever to pick a subset of negatives that are the most informative for your network. Hard Example Mining is the way to go to this.

In general, to pick a subset of negatives, first you train your network for couple of iterations, then you run your network all along your negative instances then you pick the ones with the greater loss values. However, it is very computationally toilsome since you have possibly millions of images to process, and sub-optimal for your optimization since you freeze your network while picking your hard instances that are not all being used for the next couple of iterations. That is, you assume here all hard negatives you pick are useful for all the next iterations until the next selection. Which is an imperfect assumption especially for large datasets.

Okay, what Online means in this regard. OHEM solves these two aforementioned problems by performing hard example selection batch-wise. Given a batch sized K, it performs regular forward propagation and computes per instance losses. Then, it finds M<K hard examples in the batch with high loss values and it only back-propagates the loss computed over the selected instances. Smart hah ? 🙂

It reduces computation by running hand to hand with your regular optimization cycle. It also unties the assumption of the foreseen usefulness by picking hard examples per iteration so thus we now really pick the hard examples for each iteration.

If you like to test yourself, here is PyTorch OHEM implementation that I offer you to use a bit of grain of salt.

ReLU is defined as a way to train an ensemble of exponential number of linear models due to its zeroing effect. Each iteration means a random set of active units hence, combinations of different linear models. They discuss, relying on the given observation, it might be useful to remove non-linearities for some layers and letting them to learn combination of linearities as the whole layer.

Another argument as poised, some representations are hard to approximate by a stack of non-linear layers. as shown by He et al. 2016. To this end, letting linearities for a subset of layers might ameliorate the condition.

The way they apply EraseReLU is removing the last ReLU layer of each "module". "Module" here is defined depending on the model architecture as shown above.

Experiments show that EraseReLU increases the performance of networks and its effect is larger for deeper networks. It is also more resilient to over-fitting for deep networks. The loss curves also show faster convergence for EraseReLU and the difference more obvious for larger datasets.

My 2 cents: Results are not that different on ImageNet but still better to the favor of EraseReLU. Then it might be the case of lucky shoot since there is no confidence interval or variance given for the trainings.

Faster convergence makes sense with the help of second guessing after the paper. Since there are more active units possible it entails to propagate more gradients. However, all such comments assumes that error signals are always positive. Which is very unlikely. Therefore, more open valves might cause more chaotic back-propagation signal.

Yet it is very simple idea, it shows faster convergence, better results and a good investgation of ReLU function. It think it is useful and can take its position in my next training session.

Disclaimer: This is written hastily in 10 mins. If you think something wrong or even worse let me know :).

There are numerous on-line and off-line technical resources about deep learning. Everyday people publish new papers and write new things. However, it is rare to see resources teaching practical concerns for structuring a deep learning projects; from top to bottom, from problem to solution. People know fancy technicalities but even some experienced people feel lost in the details, once they need to structure their own project.

Andrew Ng’s new initiative deeplearning.ai rushes to help at this stage. deeplearning.ai is a collection of courses teaching deep learning with all the necessary details (great place to start deep learning !!) and one of the courses called “Structuring Machine Learning Project” particularly teaches the design of a deep learning solution intuitively with real-life examples. It is not possible to give a single pill ruling the all but this course establishes a good basis to think about a DL project. And apparently, I still have things to learn from Andrew Ng. after all my years of experience. Note that, I also started my ML career with his famous Coursera ML course :). Thank you Mr. Ng.

Below, I try to plot a diagram summarizing what is mentioned in the course. However be aware that it is not a verbatim depiction and probably includes some of my own measures. In the end, It was a good exercise for me, and hopefully, is a good reference for you.

Please go and check the videos, if the diagram does not make sense at all and ping me if you have something that you don’t like.

Lately, we (TwentyBN) took a part in Activity Net trimmed action recognition challenge. The dataset is called Kinetics and recently released. It is a collection of 10 second YouTube videos. Each video has a single label among 400 different action classes. The dataset released by DeepMind with a baseline 61% Top-1 and 81.3% Top-5. For baseline models please refer to their dataset paper. But, it took 2 months for people to briskly hoist the bar high above.

As you might see above, we have the best Top-5 accuracy with 97% which is ~16% improvement on top of the baseline. The average of Top-1 and Top-5 decides the leader-board which places us to 3rd place. Yet, it is a great result for us where we could dabble only 2 weeks with limited juice. Team matters here!! Thx to my mates Raghav Goyal and Valentin Haenel for being great.

Here, I like to succinctly describe our novel network architecture. It has the best single network performance. (We plan to share a more detailed description in a separate Medium soon.) Namely, it is called BesNet due to a cheap cryptographic reason :). BesNet yields 74% Top-1 with only RGB . It is half-size of the baseline network described in the DeepMind paper.

In detail, BesNet is devised on top of ResNet-50 architecture. Distinctly, BesNet performs 3D convolutions that are able to learn both spatiotemporal features. In a better extent, BesNet takes not a single frame, but a set of frames from a video. It convolves pixels between consecutive frames as wells as single frame pixels. Each ResNet-50 module buckled with 1x1 + 3x3 +1x1 filters in order. Each such module followed by a residual connection coming from preceding module. It uses ReLU activation followed by a Batch-Normalization for each layer. In order to convert Resnet-50 to BesNet, we inflate 1x1 filters to 3x1x1 filters and 3x3 filters to 1x3x3 filters where the ordering of the dimensions is sequence x height x width. After convolution layers, an average pooling layer aggregates spatial dimension as in the normal ResNet. Subsequently, a max pooling layer aggregates temporal dimensions. A fully-connected layer used for predictions.

BesNet is initialized with ImageNet weights. In order to convert 2D filter weights to 3D filter weights, we replicate 2D filters along an additional dimension and then normalize the weights by the replication factor. This normalization keeps the activation values stable despite the architectural change. For example, a 1x1 filter is converted to 3x1x1 by copying the 1x1 filter 3 times along the third dimension and weights are divided by 3 at the end.

In BesNet, 3x1x1 filters are responsible for temporal and spatial cross-channel regularities. 1x3x3 filters pay into only spatial properties of individual feature maps. This orientation excites several observations. First off, it decouples temporal and spatial computations. It learns specialized layers for each of the temporal and spatial dimensions. The idea also entertained by the pooling layers. We decomposed spatial and temporal dimension over average and max pooling layers respectively. BesNet reduces the spatial dimensions along the convolutional layers yet it keeps the size of temporal dimension constant. This makes BesNet flexible to handle videos with different number of frames. Hence, given a video with K frames, BesNet keeps the temporal dimension as K until the pooling layers. Thereafter, max pooling layer aggregates K temporal channels into one. In a practical sense, this is easy with a dynamic computational graph library. Pytorch is a bliss here !! (Sorry TF, You're so crusty.)

BesNet has a peculiar use of dilation in 3x1x1 layers which defines the real novel aspect of our architecture. BesNet uses dilation only on temporal dimension and it picks a random dilation factor per 3x1x1 layer for each mini-batch. It sets padding parameters in accordance to keep the temporal dimension unchanged. At the test time, each layer computes outputs for each possible dilation factor, then takes the average of the output feature maps. Random dilation enables the network to learn complex temporal relations. It also regularizes the network in the temporal domain. In practice, it reduces the effect of FPS used for casting videos into frames.

We discuss that for Kinetics, it is important to learn long range relations between frames. Videos are long and they have only a single label. So the network needs to learn the general context of the video. In that sense, small motions that are observed by a normal 3D convolution are not that important. Random dilation pays into this. It augments the contextual temporal window of the network.

Our experiments with only frame futures support our hypothesis here. We extracted frame features with ResNet-50 and train an MLP after pooling the features. It gets 65% accuracy. It is better than DeepMind's baseline network with 3D convolution layers. That shows us contextual information means more than motion learned by 3D layers.

Motion information might be complementary but not the core. It is then verified by the random dilation. BesNet with no dilation results 70% , dilation 2 68% and the random dilation 74% accuracy. This stands to be a simple empirical proof backing our claim here.

Random dilation is really easy to implement with Pytorch. Just take normal Conv class and overwrite its forward pass by randomizing dilation parameter. If you like to try out before we release fell free.

I try to give a very sketchy description of BesNet here by no means complete. Please ping me if you have any question. We plan to study BesNet a little more and share it in the near future in legit formats. We also plan to share a finer description of our challenge approach with some open-source enjoyment.

Please note that BesNet is a work in progress. Anyways, feedbacks are always warmly welcome. Best :).

One of the main problems of neural networks is to tame layer activations so that one is able to obtain stable gradients to learn faster without any confining factor. Batch Normalization shows us that keeping values with mean 0 and variance 1 seems to work things. However, albeit indisputable effectiveness of BN, it adds more layers and computations to your model that you'd not like to have in the best case.

ELU (Exponential Linear Unit) is a activation function aiming to tame neural networks on the fly by a slight modification of activation function. It keeps the positive values as it is and exponentially skew negative values.

ELU does its job good enough, if you like to evade the cost of Bath Normalization, however its effectiveness does not rely on a theoretical proof beside empirical satisfaction. And finding a good is just a guess.

Self-Normalizing Neural Networks takes things to next level. In short, it describes a new activation function SELU (Scaled Exponential Linear Units), a new initialization scheme and a new dropout variant as a repercussion,

The main topic here is to keep network activation in a certain basin defined by a mean and a variance values. These can be any values of your choice but for the paper it is mean 0 and variance 1 (similar to notion of Batch Normalization). The question afterward is to modifying ELU function by some scaling factors to keep the activations with that mean and variance on the fly. They find these scaling values by a long theoretical justification. Stating that, scaling factors of ELU are supposed to be defined as such any passing value of ELU should be contracted to define mean and variance. (This is just verbal definition by no means complete. Please refer to paper to be more into theory side. )

Above, the scaling factors are shown as and . After long run of computations these values appears to be 1.6732632423543772848170429916717 and 1.0507009873554804934193349852946 relatively. Nevertheless, do not forget that these scaling factors are targeting specifically mean 0 and variance 1. Any change preludes to change these values as well.

Initialization is also another important part of the whole method. The aim here is to start with the right values. They suggest to sample weights from a Gaussian distribution with mean 0 and variance where n is number of weights.

It is known with a well credence that Dropout does not play well with Batch Normalization since it smarting network activations in a purely random manner. This method seems even more brittle to dropout effect. As a cure, they propose Alpha Dropout. It randomly sets inputs to saturatied negative value of SELU which is . Then an affine transformation is applied to it with and values computed relative to dropout rate, targeted mean and variance.It randomizes network without degrading network properties.

In a practical point of view, SELU seems promising by reducing the computation time relative to RELU+BN for normalizing the network. In the paper they does not provide any vision based baseline such a MNIST, CIFAR and they only pounce on Fully-Connected models. I am still curios to see its performance vis-a-vis on these benchmarks agains Bath Normalization. I plan to give it a shoot in near future.

One tickle in my mind after reading the paper is the obsession of mean 0 and variance 1 for not only this paper but also the other normalization techniques. In deed, these values are just relative so why 0 and 1 but not 0 and 4. If you have a answer to this please ping me below.

The whole heading of the paper is "The Shattered Gradients Problem: If resnets are the answer, then what is the question?". It is really interesting work with all its findings about gradient dynamics of neural networks. It also examines Batch Normalization (BN) and Residual Networks (Resnet) under this problem.

The problem, dubbed "Shattered Gradients", described as gradient feedbacks resembling random noise for nearby data points. White noise gradients (random value around 0 with some unknown variance) are not useful for training and they stall the network. What we expect to see is Brownian noise (next value is obtained with a small change on the last value) from a working model. Deep neural networks are more prone to white noise gradients. However, latest advances like BN and Resnet are described to be more resilient to random gradients even in deep networks.

White noise gradients undermines the effectiveness of networks because they violates gradient based learning methods which expects similar gradient feedbacks for data points close by in the vector space. Once you have white noise gradient for such close points, the model is not able to capture data manifold through these learning algorithms. Brownian updates yields more correlation on updates and this preludes effective learning.

For normal networks, they give a empirical evidence that the correlation of network updates decreases with the order where L is number of layers. Decreasing correlation means more white noise gradient feedbacks.

One important reason of white noise feedbacks is to be co-activations of network units. From a working model, we expect to have units receptive to different structures in the given data. Therefore, for each different instance, different subset of units should be active for effective information flux. They observe that as activation goes through layers, co-activation rate goes higher. BN layers prevents this by keeping the co-activation rate 1/4 (1/4 units are active per layer).

Beside the co-activation rate, how dispersed units activation is another important question. Thus, similar instances need to activate similar subset of units and activation should be distributes to other subsets as we change the data structure. This stage is where the skip-connections get into the play. Their observation is skip-connections improve networks in that respect. This can be observed at below figure.

The effectiveness of skip-connections increases with Beta scaling introduced by InceptionV4 architecture. It is scaling residual connections by a constant value before summing up with the current layer activation.

A small discussion

This is a very intriguing paper to me as being one of the scarse works investigating network dynamics instead of blind updates on architectures for racing accuracy values.

Resnet is known to be train hundreds of layers which was not possible before. Now, with this work, we have another scientific argument explaining its effectiveness. I also like to point Veit et al. (2016) demystifying Resnet as an ensemble of many shallow networks. When we combine both of these papers, it makes total sense to me how Resnets are useful for training very deep networks. If shattered gradient effect, as stated here, increasing with number of layers with the order 2^L then it is impossible to train hundred layers with an ad-hoc network. Corollary, since Resnet behaves like a ensemble of shallow networks this effects is rehabilitated. We are able to see it empirically in this paper and it is complimentary in that sense.

Note: This hastily written paper note might include any kind of error. Please let me know if you find one. Best 🙂

Quora recently announced the first public dataset that they ever released. It includes 404351 question pairs with a label column indicating if they are duplicate or not. In this post, I like to investigate this dataset and at least propose a baseline method with deep learning.

Beside the proposed method, it includes some examples showing how to use Pandas, Gensim, Spacy and Keras. For the full code you check Github.

Data Quirks

There are 255045 negative (non-duplicate) and 149306 positive (duplicate) instances. This induces a class imbalance however when you consider the nature of the problem, it seems reasonable to keep the same data bias with your ML model since negative instances are more expectable in a real-life scenario.

When we analyze the data, the shortest question is 1 character long (which is stupid and useless for the task) and the longest question is 1169 character (which is a long, complicated love affair question). I see that if any of the pairs is shorter than 10 characters, they do not make sense thus, I remove such pairs. The average length is 59 and std is 32.

There are two other columns "q1id" and "q2id" but I really do not know how they are useful since the same question used in different rows has different ids.

Some labels are not true, especially for the duplicate ones. In anyways, I decided to rely on the labels and defer pruning due to hard manual effort.

Proposed Method

Converting Questions into Vectors

Here, I plan to use Word2Vec to convert each question into a semantic vector then I stack a Siamese network to detect if the pair is duplicate.

Word2Vec is a general term used for similar algorithms that embed words into a vector space with 300 dimensions in general. These vectors capture semantics and even analogies between different words. The famous example is ;

king - man + woman = queen.

Word2Vec vectors can be used for may useful applications. You can compute semantic word similarity, classify documents or input these vectors to Recurrent Neural Networks for more advance applications.

There are two well-known algorithms in this domain. One is Google's network architecture which learns representation by trying to predict surrounding words of a target word given certain window size. GLOVE is the another methos which relies on co-occurrence matrices. GLOVE is easy to train and it is flexible to add new words out-side of your vocabulary. You might like visit this tutorial to learn more and check this brilliant use-case Sense2Vec.

We still need a way to combine word vectors for singleton question representation. One simple alternative is taking the mean of all word vectors of each question. This is simple but really effective way for document classification and I expect it to work for this problem too. In addition, it is possible to enhance mean vector representation by using TF-IDF scores defined for each word. We apply weighted average of word vectors by using these scores. It emphasizes importance of discriminating words and avoid useless, frequent words which are shared by many questions.

Siamese Network

I described Siamese network in a previous post. In short, it is a two way network architecture which takes two inputs from the both side. It projects data into a space in which similar items are contracted and dissimilar ones are dispersed over the learned space. It is computationally efficient since networks are sharing parameters.

Implementation

Let's load the training data first.

For this particular problem, I train my own GLOVE model by using Gensim.

The above code trains a GLOVE model and saves it. It generates 300 dimensional vectors for words. Hyper parameters would be chosen better but it is just a baseline to see a initial performance. However, as I'll show this model gives performance below than my expectation. I believe, this is because our questions are short and does not induce a semantic structure that GLOVE is able to learn a salient model.

Due to the performance issue and the observation above, I decide to use a pre-trained GLOVE model which comes free with Spacy. It is trained on Wikipedia and therefore, it is stronger in terms of word semantics. This is how we use Spacy for this purpose.

Before going further, I really like Spacy. It is really fast and it does everything you need for NLP in a flash of time by hiding many intrinsic details. It deserves a good remuneration. Similar to Gensim model, it also provides 300 dimensional embedding vectors.

The result I get from Spacy vectors is above Gensim model I trained. It is a better choice to go further with TF-IDF scoring. For TF-IDF, I used scikit-learn (heaven of ML). It provides TfIdfVectorizer which does everything you need.

After we find TF-IDF scores, we convert each question to a weighted average of word2vec vectors by these scores. The below code does this for just "question1" column.

Now, we are ready to create training data for Siamese network. Basically, I've just fetch the labels and covert mean word2vec vectors to numpy format. I split the data into train and test set too.

In this stage, we need to define Siamese network structure. I use Keras for its simplicity. Below, it is the whole script that I used for the definition of the model.

I share here the best performing network with residual connections. It is a 3 layers network using Euclidean distance as the measure of instance similarity. It has Batch Normalization per layer. It is particularly important since BN layers enhance the performance considerably. I believe, they are able to normalize the final feature vectors and Euclidean distance performances better in this normalized space.

I tried Cosine distance which is more concordant to Word2Vec vectors theoretically but cannot handle to obtain better results. I also tried to normalize data into unit variance or L2 norm but nothing gives better results than the original feature values.

Let's train the network with the prepared data. I used the same model and hyper-parameters for all configurations. It is always possible to optimize these but hitherto I am able to give promising baseline results.

Results

In this section, I like to share test set accuracy values obtained by different model and feature extraction settings. We expect to see improvement over 0.63 since when we set all the labels as 0, it is the accuracy we get.

These are the best results I obtain with varying GLOVE models. they all use the same network and hyper-parameters after I find the best on the last configuration depicted below.

Gensim (my model) + Siamese: 0.69

Spacy + Siamese : 0.72

Spacy + TD-IDF + Siamese : 0.79

We can also investigate the effect of different model architectures. These are the values following the best word2vec model shown above.

Adam works quite well for this problem compared to SGD with learning rate scheduling. Batch Normalization also yields a good improvement. I tried to introduce Dropout between layers in different orders (before ReLU, after BN etc.), the best I obtain is 0.75. Concatenation of different layers improves the performance by 1 percent as the final gain.

In conclusion, here I tried to present a solution to this unique problem by composing different aspects of deep learning. We start with Word2Vec and combine it with TF-IDF and then use Siamese network to find duplicates. Results are not perfect and akin to different optimizations. However, it is just a small try to see the power of deep learning in this domain. I hope you find it useful :).

Updates

Switching last layer to FC layer improves performance to 0.84.

By using bidirectional RNN and 1D convolutional layers together as feature extractors improves performance to 0.91. Maybe I'll explain details with another post.

In simple terms, dilated convolution is just a convolution applied to input with defined gaps. With this definitions, given our input is an 2D image, dilation rate k=1 is normal convolution and k=2 means skipping one pixel per input and k=4 means skipping 3 pixels. The best to see the figures below with the same k values.

The figure below shows dilated convolution on 2D data. Red dots are the inputs to a filter which is 3x3 in this example, and green area is the receptive field captured by each of these inputs. Receptive field is the implicit area captured on the initial input by each input (unit) to the next layer .

Dilated convolution is a way of increasing receptive view (global view) of the network exponentially and linear parameter accretion. With this purpose, it finds usage in applications cares more about integrating knowledge of the wider context with less cost.

One general use is image segmentation where each pixel is labelled by its corresponding class. In this case, the network output needs to be in the same size of the input image. Straight forward way to do is to apply convolution then add deconvolution layers to upsample[1]. However, it introduces many more parameters to learn. Instead, dilated convolution is applied to keep the output resolutions high and it avoids the need of upsampling [2][3].

Dilated convolution is applied in domains beside vision as well. One good example is WaveNet[4] text-to-speech solution and ByteNet learn time text translation. They both use dilated convolution in order to capture global view of the input with less parameters.

In short, dilated convolution is a simple but effective idea and you might consider it in two cases;

Detection of fine-details by processing inputs in higher resolutions.

Broader view of the input to capture more contextual information.

Faster run-time with less parameters

[1] Long, J., Shelhamer, E., & Darrell, T. (2014). Fully Convolutional Networks for Semantic Segmentation. Retrieved from http://arxiv.org/abs/1411.4038v1

[2]Chen, L.-C., Papandreou, G., Kokkinos, I., Murphy, K., & Yuille, A. L. (2014). Semantic Image Segmentation with Deep Convolutional Nets and Fully Connected CRFs. Iclr, 1–14. Retrieved from http://arxiv.org/abs/1412.7062

[3]Yu, F., & Koltun, V. (2016). Multi-Scale Context Aggregation by Dilated Convolutions. Iclr, 1–9. http://doi.org/10.16373/j.cnki.ahr.150049

[4]Oord, A. van den, Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., … Kavukcuoglu, K. (2016). WaveNet: A Generative Model for Raw Audio, 1–15. Retrieved from http://arxiv.org/abs/1609.03499

[5]Kalchbrenner, N., Espeholt, L., Simonyan, K., Oord, A. van den, Graves, A., & Kavukcuoglu, K. (2016). Neural Machine Translation in Linear Time. Arxiv, 1–11. Retrieved from http://arxiv.org/abs/1610.10099