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<!doctype html>
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<meta name="author" content="Emre Neftci">
<title>Neural Networks and Machine Learning</title>
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<!--
<section data-markdown><textarea data-template>
<h2> </h2>
<ul>
<li/>
</ul>
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[](https://colab.research.google.com/github/surrogate-gradient-learning/pytorch-lif-autograd/blob/master/tutorial01_dcll_localerrors.ipynb)
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<section data-markdown data-vertical-align-top data-background-color=#B2BA67><textarea data-template>
<h1> Generative models </h1>
<h2> Week 6: Autoencoders</h2>
### Instructor: Prof. Emre Neftci
<center>https://canvas.eee.uci.edu/courses/21750</center>
<center>http://tinyurl.com/nmi-lab-appointments</center>
[](?print-pdf)
</textarea>
</section>
<section data-markdown><textarea data-template>
<h2> Goody Bag III Data Loading and Pre-Processing </h2>
<ul>
<li/> Loading and pre-processing the data is a very tedious process, often dwarfing the implementation of the neural network
<li /> Machine learning frameworks simplify this process. In PyTorch, these are called dataloaders.
<li /> Severla data loaders for popular vision datasets already exist (MNIST, Fashion MNIST, CIFAR10, ImageNet etc.). But often you will have to create your own dataset.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Goody III.1 ImageFolder and DatasetFolder </h2>
<ul>
<li/> If your dataset is organized as png images in a directory, in the following way:
<pre><code class="Python" data-trim data-noescape>
root/dog/xxx.png
root/dog/xxy.png
root/dog/xxz.png
root/cat/123.png
root/cat/nsdf3.png
root/cat/asd932_.png
</code></pre>
then you can use
<pre><code class="Python" data-trim data-noescape>
torchvision.datasets.ImageFolder
</code></pre>
<li />If instead of .png you have other files (not necessarily images), you can use
<pre><code class="Python" data-trim data-noescape>
torchvision.datasets.DatasetFolder
</code></pre>
This class requires as argument a "loader" function that can open the file and produce an array
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Goody III.2 Dataset and DataLoader </h2>
<ul>
<li/> The most general case is to create your own DataSet class as follows:
<pre><code class="Python" data-trim data-noescape>
from torch.utils.data import Dataset
class MyTrainingDataset(Dataset):
def __init__(self, root, transform=None, target_transform=None):
self.root = root
self.transform = transform
self.target_transform = target_transform
self.data = np.load(self.root) #replace this
def __len__(self):
return len(self.data)
def __getitem__(self, idx):
... #write here code to obtain one data sample and target
if self.transform:
sample = self.transform(sample)
if self.target_transform:
target = self.target_transform(target)
return sample, target
</code></pre>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Goody III.2 Dataset Example: The Wandering Anteater </h2>
<ul>
<li/> Let's create a dataset with the UCI anteater overlayed on grocery store pictures
</ul>loader.
https://drive.google.com/open?id=1LqWInxSS198mlqHmGLGt4dNlWbWFVr2U
</textarea></section>
<section data-markdown data-vertical-align-top><textarea data-template>
<h2> Generative vs. Discrimitive Models </h2>
<ul>
<li/> So far, we have been interested in discriminative models, i.e. learning the distrubution
$$ p(\mathbf{y}|\mathbf{x}) $$
where $\mathbf{x}$ is.
<li /> Generative models instead focus on learning the data distribution
$$ p(\mathbf{x}) $$
For example in MNIST, $p(\mathbf{x})$ may be high around vectors $\mathbf{x}$ corresponding to digits, and zero elsewhere
<li /> Generative models can be used for unsupervised training (no labels), but are more challenging to learn.
<li /> In this class, we will see different approaches for learning generative models
</ul>
</textarea>
</section>
<section data-markdown data-vertical-align-top><textarea data-template>
<h2> Approaches to Generative Modeling </h2>
<img src="images/generative_modeling_approaches.png" class=stretch />
<p class=ref> NIPS 2016 Tutorial: Generative Adversarial Networks, Ian Goodfellow 2017 </p>
</textarea>
</section>
<section data-markdown><textarea data-template>
<h2> Autoencoder </h2>
<ul>
<li /> An Autoencoder (AE) is a neural network that is trained to copy its input to its output.
<li /> AE haves two parts: an encoder $\mathbf{h} = f(\mathbf{x})$ and a decoder $\mathbf{r} = g(\mathbf{h})$
<img src="images/Autoencoder_schema.png" class=large />
<p class=ref> By <a href="//commons.wikimedia.org/w/index.php?title=User:Michela_Massi&action=edit&redlink=1" class="new" title="User:Michela Massi (page does not exist)">Michela Massi</a> - <span class="int-own-work" lang="en"></span>, <a href="https://creativecommons.org/licenses/by-sa/4.0" title="Creative Commons Attribution-Share Alike 4.0">CC BY-SA 4.0</a>, <a href="https://commons.wikimedia.org/w/index.php?curid=80177333"></a> </p>
<li /> The hidden layer activations $\mathbf{h}$ describes a <em>code</em> used to represent the input
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Autoencoder </h2>
<ul>
<li /> AEs are designed to be unable to replicate the input perfectly, <em>e.g.</em> they cannot learn $\mathbf{x} = g(f\mathbf{x})$ exactly.
<li />Instead, the capability of the AE is limited so that it learns useful properties/patterns of the data. Otherwise the network could learn $g^{-1}$
<li /> In practice, $f$ and $g$ are neural networks as we have seen so far, <em> e.g. </em> feed forward or convolutional networks.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Undercomplete AE</h2>
<ul>
<li/> In AE we are interested in $\mathbf{h}$, <em>i.e.</em> the output of the encoder.
<li/> One way to prevent the AE to learn the identitiy function is to choose a code whose dimension is smaller than the input. This is called an undercomplete AE.
<li /> Learning the undercomplete representation forces the autoencoder to capture the most salient features of the training data. This is the classical idea of "bottleneck"
<li /> The loss function is: $$L(\mathbf{x}, g(f(\mathbf{x})))$$ where $L$ is typically Mean Squared Error.
<li /> Undercomplete AEs are limited by the fact that the code dimension and network complexity must remain smaller than the input, <em>i.e.</em> their capacity must remain small. This prevents us from using more complex networks.
</ul>
[](https://drive.google.com/open?id=1doORfr4mUh-IQKwDS_p3xVTLuTrU94yz)
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Regularized AE </h2>
<ul>
<li/> Another way to limit capacity is to use regularization
<li /> Regularized AEs encourages properties other than "copying" its input, such as sparsity of the representation, smooth representations, and robustness to noise.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Example Regularized AE: Sparse AE</h2>
<ul>
<li/> A sparse autoencoder used a sparseness penalty, such as L1 norm. This forces many components in $\mathbf{h}$ to be zero.
$$ L(\mathbf{x}, g(f(\mathbf{x}))) + \lambda \sum_i |h_i| $$
<li /> There is a loose connection with log-likelihoods optimization:
$$
\begin{split}
p(\mathbf{x}) &= \sum_\mathbf{h} p(\mathbf{x},\mathbf{h})\\
p(\mathbf{x},\mathbf{h}) & = p(\mathbf{x}|\mathbf{h})p(\mathbf{h})\\
\log p(\mathbf{x},\mathbf{h}) & = \log p(\mathbf{x}|\mathbf{h}) + \log p(\mathbf{h})\\
\end{split}
$$
such that the regularizing term is a prior on $\mathbf{h}$. In this case, the prior distribution is a Laplace prior
$$
p(h_i) = \frac{\lambda}{2} \exp(\lambda |h_i|)
$$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Denoising Autoencoder </h2>
<ul>
<li/> Another way to reduce capacity is to add noise to the data (<em> dropout </em>)
<li/> A denoising autoencoder (DAE) instead minimizes
$$ L(\mathbf{x}, g(f(\mathbf{\tilde{x}}))) $$
where
$\tilde{\mathbf{x}}$ is a copy of $\mathbf{x}$ that has been corrupted by some form of noise.
<li /> The noise can be simply additive Gaussian noise
<img src="images/denoising_ae_example.png" class=small />
<li /> Denoising training forces $\mathbf{f}$ and $\mathbf{g}$ to learn the structure of the input $\mathbf{x}$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> In-class Assignment: Improve AE</h2>
- Using a technique of your choice (regularization, noise, sparsity), try to improve the reconstruction quality on test data.
- Use a visualization of your choice (PCA, ICA, t-SNE) to visualize the latent space.
- Extra: Use convolutional and transposed convulational layers to implement a convolutional AE
<pre><code class="Python" data-trim data-noescape>
nn.ConvTranspose2d(16, 1, 2, stride=2)
</code></pre>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Stochastic Encoders and Decoders </h2>
<ul>
<li /> AEs are typically described in the probabilistic domain, using probabilistic encoders $q_{enc}(\mathbf{h}|\mathbf{x})$ and decoder $p_{dec}(\mathbf{x}|\mathbf{h})$
<img src="images/goodfellow_14_2.png" class=normal />
<li /> Given a hidden code $\mathbf{h}$, we may think of the decoder as providing a conditional distribution $p_{dec}(\mathbf{x}|\mathbf{h})$. We may then
train the autoencoder by minimizing $-\log p_{dec}(\mathbf{x}|\mathbf{h})$.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Denoising Autoencoder using Stochastic Encoders and Decoders </h2>
<ul>
<li/> Images are corrupted by noise: $C(\tilde{\mathbf{x}}|\mathbf{x})$ .
<img src="images/denoising_ae_graph.png" />
<ol>
<li /> Sample $\mathbf{x}$ from training distribution
<li /> Sample a corrupted version $\tilde{\mathbf{x}}$ from $C(\tilde{\mathbf{x}}|\mathbf{x})$
<li /> Use pair $(\tilde{\mathbf{x}},\mathbf{x})$ to evaluate $L$ $$ L = \log p_{decoder}(\mathbf{x}|\mathbf{h}(\tilde{\mathbf{x}}))$$
</ol>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Learning Manifolds </h3>
<ul>
<li /> AEs exploit the idea that data concentrates around a low-dimensional manifold
<li /> The AE specializes to representing variations that are needed to construct the training samples
<li /> For example, the space of vertical translations is 1D, which defines a line (here the activations are projected on the 2 principal components)
<img src="images/manifold_learning.png" c/>
</ul>
</textarea></section>
<section data-markdown data-vertical-align-top><textarea data-template>
<h2> Generative Models </h2>
<ul>
<li /> In high dimensional spaces, many vectors $\mathbf{x}$ result in $p(\mathbf{x})$ that vanish. It is more useful to focus around points that are likely.
<li /> For example, if we wish to estimate the half of a digit, it would be very helpful to first decide which digit to draw. The digit forms the <em> latent variable </em> (code), which we called $\mathbf{z}$.
<img src="images/Samples-drawn-from-the-prediction-of-the-lower-half-of-the-MNIST-test-data-digits-based.png" class=small />
<li /> For this, we need to create a relationshop between $\mathbf{z}$ and $\mathbf{x}$. This means we aim to maximize $p(x)$ for every $\mathbf{x}$ in the dataset, <em> i.e. </em>
$$ p(\mathbf{x}) = \int_\mathbf{z} p(\mathbf{x}|\mathbf{z}) p(\mathbf{z}) \mathrm{d}\mathbf{z} $$
where $\mathbf{z}$ is a latent variable
<li /> There are two challenges: how to decide on $\mathbf{z}$, and how to perform the integral.
</ul>
</textarea>
</section>
<section data-markdown><textarea data-template>
<h2> Variational AE (VAE) </h2>
<ul>
<li /> Variational Autoencoders were proposed as approximate but fast algorithms to maximizing $p(\mathbf{x})$
<li /> VAEs assert that $\mathbf{z}$ are drawn from a simple distribution, such as a multivariate Gaussian $N(0,\mathbb{I})$.
<li /> How can this $\mathbf{z}$ be powerful enough? Given a random varaible with one distribution, it is possible to create any other distribution provided a complex enough function.
<img src="images/from_gaussian_to_arbitrary.png">
<p class=ref>Doersch, 2016, <em>Tutorial on Variational Autoencoders</em> </p>
<li /> So from independent Gaussian-distributed variables, we can obtain any latent variable distribution, which can then be mapped to $\mathbf{x}$ using a some function $f$.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Variational AE (VAE) </h2>
<ul>
<li/> VAEs use a Gaussian distribution in $p(\mathbf{x}|\mathbf{z})$,
$$ p(\mathbf{x}) = \int_\mathbf{z} N(\mathbf{x}|f_\theta(\mathbf{z}), \sigma^2 \mathbb{I}) p(\mathbf{z}) \mathrm{d}\mathbf{z} = \mathbb{E}_{z\sim p} N(\mathbf{x}|f_\theta(\mathbf{z}), \sigma^2 \mathbb{I})$$
(but any density that can be computed and differentiable with respect to the parameters $\theta$.)
<li /> Unfortunately, many samples $\mathbf{z}$ are required to estimate $p(\mathbf{x})$. VAEs solve this problem by restricting the integral to values of $\mathbf{z}$ that are likely to have produced $\mathbf{x}$. This is achieved by using a density $q(\mathbf{z}|\mathbf{x})$ to produce $\mathbf{z}$.
$$ p(\mathbf{x}) = \mathbb{E}_{\mathbf{z}\sim q} p(\mathbf{x}|\mathbf{z})$$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Variational Lower Bound </h2>
<ul>
<li /> What is the relationship between $\mathbb{E}_{z\sim q} p(\mathbf{x}|\mathbf{z})$ and $p(\mathbf{x})$?
<li /> The following relation, known as Variational lower bound is the cornerstone of all ariational Bayes:
$$
\log p(\mathbf{x}) \ge \mathbb{E}_{\mathbf{z}\sim q} \log p(\mathbf{x}|\mathbf{z}) - KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z}))
$$
<li /> By maximizing the right hand side, we optimize $\log p(\mathbf{x})$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Optimizing the Variational Lower Bound </h2>
$$
\text{ Variational Lower Bound: } \mathbb{E}_{\mathbf{z}\sim q} \log p(\mathbf{x}|\mathbf{z}) - KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z}))
$$
<ul>
<li /> Right term: we need to choose $q(\mathbf{z}|\mathbf{x})$. Typically, it is a Gaussian distribution $N(\mathbf{z}| \mu(\mathbf{x},\theta), \Sigma(\mathbf{x},\theta))$, where $\mu$ and $\Sigma$ are implemented using neural networks. With this choice, the right term can be computed in closed form.
<li /> Left term: We could evaluate it by sampling $\mathbf{z}$ and averaging them. But this is expensive. The solution is to take a "single sample average".
<li /> We now have:
$$
\log p(\mathbf{x}|\mathbf{z}) - KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z}))
$$
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Reparametrization Trick</h2>
$$
\log p(\mathbf{x}|\mathbf{z}) - KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z}))
$$
<ul>
<li /> But there is a problem: we also need to optimize $q$, which was used to <em>sample </em> $\mathbf{z}$. Stochastic gradient descent cannot deal with stochastic variables (stochastic samples are not differentiable). The trick is to sample $\mathbf{z}$ as follows:
$$ z = \mu(\mathbf{x}) + \Sigma^{\frac12}(\mathbf{x})\odot \omega $$
where $\omega \sim N(0,\mathbb{I})$. This trick only works for certain distributions!
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Variational AE (VAE) Summary </h2>
<ul>
<li /> Variational Autoencoders were proposed as approximate but fast algorithms to maximizing $p(\mathbf{x})$. With generative adversarial networks, they are the current state-of-the-art for generative modeling.
<li /> As in AEs, VAEs consist in an encoder and a decoder function, both of which are neural networks.
<li /> The code $\mathbf{z}$ is a random variable.
<ul>
<li /> Encoder $q_\phi(\mathbf{z}|\mathbf{x})$ is typically a gaussian probability density (dashed).
<li /> Decoder $p_\theta(\mathbf{x}|\mathbf{z})$ (solid) is a likelihood function of choice (Bernouilli, Multinomial, Gaussian etc.)
<img src="images/vae_graph.png" />
</ul>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Variational AE (VAE) Architecture </h2>
<ul>
<li /> $p_\theta(\mathbf{x}|\mathbf{z})$ is a neural network taking inputs $\mathbf{z}$ and producing image
<li /> $q_\phi(\mathbf{z}|\mathbf{x})$ is a neural network producing $\mu(\mathbf{x}) $ and $\mathbf{\Sigma}(\mathbf{x})$
<li /> $\mathbf{z}$ is created using the reparamerization trick $\mathbf{z} = \mu(\mathbf{x}) + \Sigma^{\frac12}(\mathbf{x})\odot \omega $
<img src="https://miro.medium.com/max/3374/1*22cSCfmktNIwH5m__u2ffA.png" />
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Variational AE (VAE) Implementation </h2>
<ul>
<li /> The loss function to evaluate is: $$ \log p(\mathbf{x}|\mathbf{z}) - KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z})) $$
<ul>
<li /> $ KL(q(\mathbf{z}||\mathbf{x})|p(\mathbf{z})) = KL(N(\mathbf{z}| \mu(\mathbf{x},\theta), \Sigma(\mathbf{x},\theta))||N(0,\mathbb{I}))$. The KL divergence between two Gaussians can be computed analytically.
<li /> $\log p(\mathbf{x}|\mathbf{z})$, i.e. what is the likelihood of the true data $\mathbf{x}$ given $\mathbf{z}$? This depends on the likelihood function $p(\mathbf{x}|\mathbf{z})$
<ul>
<li />Binary Data $\sim$ Bernouilli distribution: Binary cross entropy loss.
<li />Categorical Data $\sim$ Multinomial distribution: Cross entropy loss.
<li />Real Data $\sim$ Gaussian Density: Mean-Squared Error loss.
</ul>
</ul>
</ul>
[](https://drive.google.com/open?id=1Yh1qtakIrsukG_CYr86obSQiLhD3Gw9K)
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Sources </h2>
<ul>
<li /> Autoencoder: <a href=>Deep Learning, Goodfellow et al. 2015</a>
<li /> Variational Autoencoder: <a href=>Deep Learning, Goodfellow et al. 2015</a>
<li /> Variational Autoencoder: <a href=>Doersch et al. 2016</a>
</ul>
</textarea></section>
</div>
</div>
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{ title: 'Plugins', icon: '<i class="fa fa-external-link-alt"></i>', src: 'toc.html' },
{ title: 'About', icon: '<i class="fa fa-info"></i>', src: 'about.html' }
]
},
math: {
mathjax: 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js',
config: 'TeX-AMS_HTML-full', // See http://docs.mathjax.org/en/latest/config-files.html
// pass other options into `MathJax.Hub.Config()`
TeX: { Macros: { Dp: ["\\frac{\\partial #1}{\\partial #2}",2] }}
//TeX: { Macros: { Dp#1#2: },
},
chalkboard: {
src:'slides_2-chalkboard.json',
penWidth : 1.0,
chalkWidth : 1.5,
chalkEffect : .5,
readOnly: false,
toggleChalkboardButton: { left: "80px" },
toggleNotesButton: { left: "130px" },
transition: 100,
theme: "whiteboard",
},
menu : { titleSelector: 'h1', hideMissingTitles: true,},
keyboard: {
67: function() { RevealChalkboard.toggleNotesCanvas() }, // toggle notes canvas when 'c' is pressed
66: function() { RevealChalkboard.toggleChalkboard() }, // toggle chalkboard when 'b' is pressed
46: function() { RevealChalkboard.reset() }, // reset chalkboard data on current slide when 'BACKSPACE' is pressed
68: function() { RevealChalkboard.download() }, // downlad recorded chalkboard drawing when 'd' is pressed
88: function() { RevealChalkboard.colorNext() }, // cycle colors forward when 'x' is pressed
89: function() { RevealChalkboard.colorPrev() }, // cycle colors backward when 'y' is pressed
},
dependencies: [
{ src: '../reveal.js/lib/js/classList.js', condition: function() { return !document.body.classList; } },
{ src: 'plugin/markdown/marked.js' },
{ src: 'plugin/markdown/markdown.js' },
{ src: 'plugin/notes/notes.js', async: true },
{ src: 'plugin/highlight/highlight.js', async: true, languages: ["Python"] },
{ src: 'plugin/math/math.js', async: true },
{ src: 'external-plugins/chalkboard/chalkboard.js' },
//{ src: 'external-plugins/menu/menu.js'},
{ src: 'node_modules/reveal.js-menu/menu.js' }
]
});
</script>
<script type="text/bibliography">
@article{gregor2015draw,
title={DRAW: A recurrent neural network for image generation},
author={Gregor, Karol and Danihelka, Ivo and Graves, Alex and Rezende, Danilo Jimenez and Wierstra, Daan},
journal={arXivreprint arXiv:1502.04623},
year={2015},
url={https://arxiv.org/pdf/1502.04623.pdf}
}
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