Okay, time to get our hands dirty with some code! I’ve written an example in Theano that encodes a stream of one-hot encodings, and this is the example I’ll run through with this post.

As a quick recap of what was covered in the previous post, here’s a diagram:

I’ve talked a little bit about recursive auto-encoders a couple of posts ago. In the deep learning lingo, an auto-encoder network usually refers to an architecture that takes in an input vector, and through a series of transformations, is trained to reproduce that input in its prediction layer. The reason for doing this is to extract features that describe the input. One might think of it as a form of compression: If the network is asked to be able to reproduce an input with after passing it through hidden layers with a lot less neurons than the input layer, then some sort of compression has to happen in order for it to be able to create a good reconstruction. So let’s consider the above network. 8 inputs, 8 outputs, and 3 in the hidden layer. If we feed the network a one-hot encoding of 1 to 8 (setting only the neuron corresponding to the input to 1), and insist that that input be reconstructed at the output layer, guess what happens?

There’s been some interest in Neural Turing Machines paper, and I’ve been getting some questions regarding my implementation via e-mail and the comments section on this blog. I plan to make this a blog post where I’ll regularly come back and update with answers to some of these questions as they come up, so do check back!

I haven’t written much this past year, so I guess as a parting post for 2015, I’d talk a little bit about the poster I presented at ASRU 2015. The bulk of the stuff’s in the paper, plus I’m still kind of unsure about the legality about putting stuff that’s in the paper on this blog post, so I think I’ll talk about the other things that didn’t make it in.

This will be the first time I’m trying to present code I’ve written in an ipython notebook. The style’s different, but I think I’ll permanently switch to this method of presentation for code-intensive posts from now on. A nifty little tool that makes doing this so convenient is ipy2wp. It uses WordPress’ xml-rpc to post the HTML directly to the platform.

In any case, I’ve started working with the NUS School of Computing speech recognition group, and they’ve been using deep neural networks for classification of audio frames to phonemes. This requires a preprocessing step that aligns the audio frames to phonemes in order to reduce this to a simple classification problem.

CTC describes a way to compute the probability of a sequence of phonemes for a sequence of audio frames, accounting for all possible alignments. We can then define an objective function to maximise the probability of the phoneme sequence given the audio frame sequence from training data.

I haven’t written much this past year, so I guess as a parting post for 2015, I’d talk a little bit about the poster I presented at ASRU 2015. The bulk of the stuff’s in the paper, plus I’m still kind of unsure about the legality about putting stuff that’s in the paper on this blog post, so I think I’ll talk about the other things that didn’t make it in.

This will be the first time I’m trying to present code I’ve written in an ipython notebook. The style’s different, but I think I’ll permanently switch to this method of presentation for code-intensive posts from now on. A nifty little tool that makes doing this so convenient is ipy2wp. It uses WordPress’ xml-rpc to post the HTML directly to the platform.

In any case, I’ve started working with the NUS School of Computing speech recognition group, and they’ve been using deep neural networks for classification of audio frames to phonemes. This requires a preprocessing step that aligns the audio frames to phonemes in order to reduce this to a simple classification problem.

CTC describes a way to compute the probability of a sequence of phonemes for a sequence of audio frames, accounting for all possible alignments. We can then define an objective function to maximise the probability of the phoneme sequence given the audio frame sequence from training data.

So, as a recap, here’s a (badly drawn) diagram of the architecture:

The motivation to use a hierarchical architecture for this task was two-fold:

1. Learning a vanilla encoder-decoder type of architecture for the task would be the basic deep learning go-to model for such a task. However, the noise modeled if we perform maximum likelihood is only at the pixel level. This seems inappropriate as it implies there is one “right answer”, with some pixel colour variations, given an outer context. The hierarchical VAE models different uncertainties at different levels of abstraction, so it seems like a good fit.
2. I wanted to investigate how the hierarchical factorisation of the latent variables affect learning in such a model. It turns out certain layers overfit, or I would diagnose the problem as overfitting, and I’m unsure how to remedy those problems.

Each of the following plots are samples of the conditional VAE that I’m using for the inpainting task. As expected with results from a VAE, they’re blurry. However, the fun thing about having a hierarchy of latent variables is I can freeze all the layers except for one, and vary that just to see the type of noise it models. The pictures are generated by using the $\mu_{z_l}(z_{l-1})$ for all layers except for the $i$-th layer.

$i=1$

$$\newcommand{\expected}2{\mathbb{E}_{#1}\left[ #2 \right]}

\newcommand{\prob}[3]{{#1}_{#2} \left( #3 \right)}

\newcommand{\condprob}[4]{{#1}_{#2} \left( #3 \middle| #4 \right)}

\newcommand{\Dkl}2{D_{\mathrm{KL}}\left( #1 | #2 \right)}

\newcommand{\muvec}{\boldsymbol \mu}

\newcommand{\sigmavec}{\boldsymbol \sigma}

\newcommand{\uttid}{s}

\newcommand{\lspeakervec}{\vec{w}}

\newcommand{\lframevec}{\vec{z}}

\newcommand{\lframevect}{\lframevec_t}

\newcommand{\inframevec}{\vec{x}}

\newcommand{\inframevect}{\inframevec_t}

\newcommand{\inframeset}{\inframevec_1,\hdots,\inframevec_T}

\newcommand{\lframeset}{\lframevec_1,\hdots,\lframevec_T}

\newcommand{\model}2{\condprob{#1}{#2}{\lspeakervec,\lframeset}{\inframeset}}

\newcommand{\joint}{\prob{p}{}{\lspeakervec,\lframeset,\inframeset}}

\newcommand{\normalparams}2{\mathcal{N}(#1,#2)}

\newcommand{\normal}{\normalparams{\mathbf{0}}{\mathbf{I}}}

\newcommand{\hidden}1{\vec{h}^{(#1)}}

\newcommand{\pool}{\max}

\newcommand{\hpooled}{\hidden{\pool}}

\newcommand{\Weight}1{\mathbf{W}^{(#1)}}

\newcommand{\Bias}1{\vec{b}^{(#1)}}$$

I’ve decided to approach the inpainting problem given for our class project IFT6266 using a hierarchical variational autoencoder.

While the basic VAE only has a single latent variable, this architecture assumes the image generation process comes from a hierarchy of latent variables, each dependent on its parents. So the factorisation looks like this:

$$p(x|z_1, z_2,\dots,z_L) = p(x|z_1)p(z_1|z_2) \dots p(z_{L-1}|z_L)$$

An architecture like this was used in the PixelVAE paper, but there they use a more complex PixelCNN structure at each layer, which I am attempting to do without. In their model, the `recognition model` or the encoder is not hierarchical — the $q_\phi$ network is structured in the following way:

$$q(z_1, z_2,\dots,z_L | x) = q(z_1|x)q(z_2|x) \dots q(z_L|x)$$

There’s been some interest in Neural Turing Machines paper, and I’ve been getting some questions regarding my implementation via e-mail and the comments section on this blog. I plan to make this a blog post where I’ll regularly come back and update with answers to some of these questions as they come up, so do check back!

After much fiddling around with the instability of the training procedure, I still haven’t found a recipe that would get it to converge consistently.

I did find though, that training it on shorter sequences first, before letting it see longer ones avoids huge gradients that would make the parameters explode into NaNs. And that is a huge help. Doing that still does not guarantee convergence though, and I only get a good model at random, like this one I’ve trained here copying a sequence of length 10: