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<!DOCTYPE html>
<html>
<head>
<title>Deep Learning on Graphs [Marc Lelarge]</title>
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<textarea id="source">
class: center, middle, title-slide
count: false
# Deep Learning on Graphs<br/>
# (1/3)
<br/><br/>
.bold[Marc Lelarge]
.bold[[www.dataflowr.com](https://www.dataflowr.com)]
---
# (1) Node embedding
## Language model
### one fixed graph, no signal. Ex: community detection
# .gray[(2) Signal processing on graphs]
## .gray[Fourier analysis on graphs]
### .gray[one fixed graph, various signals. Ex: classification of signals]
# .gray[(3) Graph embedding]
## .gray[Graph Neural Networks]
### .gray[various graphs. Ex: classification of graphs]
---
# Node embedding
## Language model
### one fixed graph, no signal. Ex: community detection
- [DeepWalk](https://arxiv.org/abs/1403.6652)
- hierarchical softmax
- [node2vec](https://snap.stanford.edu/node2vec/)
- negative sampling
---
# Language model
- goal: to estimate the likelihood of a specific sequence of words appearing in a corpus.
- formally: to maximize $p(w\_{n} \| w\_{1}, ... , w\_{n-1})$ where $w\_i$ are in the vocabulary $V$.
--
count: false
## skip-gram model
.center.width-50[]
- from the text, construct $D$ the multiset of all word and context pairs: $(w,c)$.
---
# skip-gram model
.center.width-50[]
- from the text, construct $D$ the multiset of all word and context pairs: $(w,c)$.
- goal: maximize $\prod\_{(w,c)\in D} p(c\| w ; \theta)$ in the parameter $\theta$.
- parametrization of the conditional probability thanks to a softmax:
$$
p(c\|w; \theta) = \frac{e^{u\_c \cdot u\_w}}{\sum\_{c'\in C}e^{u\_{c'} \cdot u\_w}},
$$
where $u\_c$ and $u\_w$ are vector representations for $c$ and $w$ and $C$ is the set of all available contexts.
---
# Hierarchical softmax
.center.width-30[]
- Assign the contexts to the leaves of a binary tree and use the parametrization:
$$
p(c\|w; \theta) = \prod\_{b\in \pi(c)} \sigma(h\_b \cdot u\_w),
$$
where $\pi(c)$ is the path from the root to the leaf $c$ and $\sigma$ is the sigmoid function:
$$
\sigma(x) = \frac{1}{1+e^{-x}}.
$$
- reduce computational complexity from $O(|C|)$ to $O(\log|C|)$.
---
# DeepWalk
- a sentence = a random walk on the graph
- Hierarchical softmax speed up: assign shorter paths to frequent contexts using Huffman coding.
--
count: false
<img align="left" width="500" src="images/graphs/deep_walk1.png"><img align="right" width="500" src="images/graphs/deep_walk2.png">
---
# Negative sampling
.center.width-50[]
- introduce $B = \mathbf{1}((w,c)\in D)$ with the parameterization
$$
p(b=1|w,c,\theta) = \sigma(u\_w \cdot u\_c)=\frac{1}{1+e^{-u\_w \cdot u\_c}} .
$$
- instead of maximizing $\prod\_{(w,c)\in D} p(c\| w ; \theta)$, we now maximize $\prod\_{(w,c)\in D} p(b=1 |c, w ; \theta)$
---
count: false
# Negative sampling
- introduce $B = \mathbf{1}((w,c)\in D)$ with the parameterization
$$
p(b=1|w,c,\theta) = \sigma(u\_w \cdot u\_c)=\frac{1}{1+e^{-u\_w \cdot u\_c}} .
$$
- instead of maximizing $\prod\_{(w,c)\in D} p(c\| w ; \theta)$, we now maximize $\prod\_{(w,c)\in D} p(b=1 |c, w ; \theta)$
.red[Problem, this has a trivial solution! We only have positive examples!]
--
count: false
- create negative examples $D'$ and maximize in $\theta$:
$$
\sum\_{(w,c)\in D}\log \sigma(u\_w \cdot u\_c) + \sum_{(w,c)\in D'}\log \sigma(-u\_w \cdot u\_c)
$$
- in [Mikolov et al.](http://papers.nips.cc/paper/5021-distributed-representations-of-words-andphrases) $D'$ is $k$ times larger than $D$, and for each $(w,c)\in D$, we construct $k$ samples $(w,c\_1) ... (w,c\_k)$ where $c\_i$ is drawn according to its empirical distribution raised to the $3/4$ power.
---
# node2vec
- parameterization of the skip-gram model approximated thanks to negative sampling
- notion of context obtained thanks to biased random walk.
.center.width-50[]
---
# some results
.center.width-40[]
---
# some results
In the multi-label classification setting, every node is assigned one or more labels from a finite set. During training phase, a certain fraction of nodes with all their labels is observed. The task is to predict the labels for the remaining nodes.
.center.width-50[]
--
count: false
Guess from which paper these results are taken from?
.center.width-10[]
---
# (1) Node embedding
## Language model
### one fixed graph, no signal. Ex: community detection
# .gray[(2) Signal processing on graphs]
## .gray[Fourier analysis on graphs]
### .gray[one fixed graph, various signals. Ex: classification of signals]
# .gray[(3) Graph embedding]
## .gray[Graph Neural Networks]
### .gray[various graphs. Ex: classification of graphs]
---
class: end-slide, center
count: false
The end.
.bold[[www.dataflowr.com](https://www.dataflowr.com)]
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