Sequence Networks

Hi, in this blog I will be covering most of the sequence networks. Sequence networks can have either input as sequence or output as sequence or both(input and output) as sequences. We can sub-divide these sequence networks in the following three ways:

Vec2Seq
Seq2Vec
Seq2Seq

1. Vec2Seq(Sequence Generation):

$f_{\theta}:\mathbb{R}^{D}\to\mathbb{R}^{N_{\infty}\cdot C}$

$p(y_{1:T}|x)=\sum_{h_{1:T}}p(y_{1:T},h_{1:T}|x)=\sum_{h_{1:T}}\prod_{t=1}^{T}p(y_{t}|h_{t})p(h_{t}|h_{t-1},y_{t-1},x)$

where $h_{t}$ is hidden state and $p(h_{1}|h_{0},y_{0},x)=p(h_{1}|x)$ (initial hidden state distribution). For categorical and real-valued output, the distributions are given by:

$p(y_{t}|h_{t})=Cat(y_{t}|softmax(W_{hy}h_{t}+b_{y}))$

The above generative model is called Recurrent Neural Network(RNN).

2. Seq2Vec(Sequence Classification):

$f_{\theta}:\mathbb{R}^{TD}\to\mathbb{R}^{C}$

In classification task, the output is class label, $y \in \left\{1, \dots, C\right\}$

$p(y|x_{1:T})=Cat(y|softmax(Wh_T))$

We get better results if we let the hidden states depends on past as well as future context $\implies$ bidirectional RNN.

$h_{t}^{\rightarrow}=\varphi(W_{xh}^{\rightarrow}x_{t}+W_{hh}^{\rightarrow}h_{t-1}^{\rightarrow}+b_{h}^{\rightarrow})$

$h_{t}^{\leftarrow}=\varphi(W_{xh}^{\leftarrow}x_{t}+W_{hh}^{\leftarrow}h_{t+1}^{\leftarrow}+b_{h}^{\leftarrow})$

Then we define, hidden state at time $t$ ,

$h_{t}=[h_{t}^{\rightarrow}, h_{t}^{\leftarrow}]$ .

$p(y|x_{1:T})=Cat(y|W \cdot softmax(\bar{h}))$

where

$\bar{h}=\frac{1}{T}\sum_{t=1}^{T}h_{t}$

3. Seq2Seq(Sequence Translation):

$f_{\theta}:\mathbb{R}^{TD}\to\mathbb{R}^{T^{\prime}C}$

Aligned case

$T^{\prime}=T$ $p(y_{1:T}|x_{1:T})=\sum_{h_{1:T}}\prod_{t=1}^{T}p(y_{t}|h_{t})\mathbb{I}(h_{t}=f(h_{t-1},x_{t}))$

where the initial state,

$h_{1}=f(h_{0},x_{1})=f_{0}(x_{1})$

we can get more espressive models by stacking multiple hidden chains on top of each other. For layer $l$ at time $t$ :

$h_{t}^{l}=\varphi_{l}(W_{xh}^{l}h_{t}^{l-1}+W_{hh}^{l}h_{t-1}^{l}+b_{h}^{l})$

$o_{t}=W_{ho}h_{t}^{L}+b_{0}$

Unaligned case

$T^{\prime}\neq T$

We map a sequence of length $T$ to a sequence of length $T^{\prime}$ . In this we first encode the input to get context vector $c=f_{e}(x_{1:T})$ using the last state of an RNN(or avg pooling over a biRNN). Then generate the output sequence using an RNN decoder $y_{1:T^{\prime}}=f_{d}(c)$ (encoder-decoder architecture).

Comments

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