From 37e6f9d7454c32c30fd405d7f6fc4967d994561c Mon Sep 17 00:00:00 2001 From: Hui Zhang Date: Fri, 12 Nov 2021 14:17:20 +0800 Subject: [PATCH] add ctc loss topic --- .mergify.yml | 10 +- .pre-commit-config.yaml | 4 +- docs/topic/ctc/ctc_loss.md | 336 ++++++++++++++++++ .../ctc/img/ctc_loss_alpha_definition.png | Bin 0 -> 165965 bytes docs/topic/ctc/img/ctc_loss_alpha_recurse.png | Bin 0 -> 67348 bytes .../ctc/img/ctc_loss_alpha_recurse_2.png | Bin 0 -> 51776 bytes docs/topic/ctc/img/ctc_loss_backward_1.png | Bin 0 -> 21126 bytes docs/topic/ctc/img/ctc_loss_backward_2.png | Bin 0 -> 51612 bytes docs/topic/ctc/img/ctc_loss_backward_3.png | Bin 0 -> 49412 bytes .../ctc/img/ctc_loss_backward_recurse.png | Bin 0 -> 117603 bytes docs/topic/ctc/img/ctc_loss_cat_lattice.png | Bin 0 -> 126385 bytes .../ctc/img/ctc_loss_forward_backward.png | Bin 0 -> 115339 bytes .../img/ctc_loss_forward_backward_to_loss.png | Bin 0 -> 51019 bytes docs/topic/ctc/img/ctc_loss_forward_loss.png | Bin 0 -> 32007 bytes .../ctc/img/ctc_loss_gradient_of_y_hat.png | Bin 0 -> 114190 bytes .../ctc/img/ctc_loss_gradient_with_y.png | Bin 0 -> 228463 bytes docs/topic/ctc/img/ctc_loss_prob_l_x.png | Bin 0 -> 143468 bytes docs/topic/ctc/img/ctc_loss_prob_pi_x.png | Bin 0 -> 110486 bytes docs/topic/ctc/img/ctc_loss_rescale_loss.png | Bin 0 -> 128482 bytes 19 files changed, 346 insertions(+), 4 deletions(-) create mode 100644 docs/topic/ctc/ctc_loss.md create mode 100644 docs/topic/ctc/img/ctc_loss_alpha_definition.png create mode 100644 docs/topic/ctc/img/ctc_loss_alpha_recurse.png create mode 100644 docs/topic/ctc/img/ctc_loss_alpha_recurse_2.png create mode 100644 docs/topic/ctc/img/ctc_loss_backward_1.png create mode 100644 docs/topic/ctc/img/ctc_loss_backward_2.png create mode 100644 docs/topic/ctc/img/ctc_loss_backward_3.png create mode 100644 docs/topic/ctc/img/ctc_loss_backward_recurse.png create mode 100644 docs/topic/ctc/img/ctc_loss_cat_lattice.png create mode 100644 docs/topic/ctc/img/ctc_loss_forward_backward.png create mode 100644 docs/topic/ctc/img/ctc_loss_forward_backward_to_loss.png create mode 100644 docs/topic/ctc/img/ctc_loss_forward_loss.png create mode 100644 docs/topic/ctc/img/ctc_loss_gradient_of_y_hat.png create mode 100644 docs/topic/ctc/img/ctc_loss_gradient_with_y.png create mode 100644 docs/topic/ctc/img/ctc_loss_prob_l_x.png create mode 100644 docs/topic/ctc/img/ctc_loss_prob_pi_x.png create mode 100644 docs/topic/ctc/img/ctc_loss_rescale_loss.png diff --git a/.mergify.yml b/.mergify.yml index 1e306d67..951f05c4 100644 --- a/.mergify.yml +++ b/.mergify.yml @@ -53,7 +53,7 @@ pull_request_rules: add: ["T2S"] - name: "auto add label=Audio" conditions: - - files~=^paddleaudio/ + - files~=^audio/ actions: label: add: ["Audio"] @@ -69,6 +69,12 @@ pull_request_rules: actions: label: add: ["Example"] + - name: "auto add label=Demo" + conditions: + - files~=^demos/ + actions: + label: + add: ["Demo"] - name: "auto add label=README" conditions: - files~=README.md @@ -77,7 +83,7 @@ pull_request_rules: add: ["README"] - name: "auto add label=Documentation" conditions: - - files~=^doc/ + - files~=^docs/ actions: label: add: ["Documentation"] diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml index b3fc1c77..2f80e46b 100644 --- a/.pre-commit-config.yaml +++ b/.pre-commit-config.yaml @@ -13,8 +13,8 @@ files: (?!.*paddle)^.*$ - id: end-of-file-fixer files: \.md$ - - id: trailing-whitespace - files: \.md$ + #- id: trailing-whitespace + # files: \.md$ - id: requirements-txt-fixer exclude: (?=third_party).*$ - id: check-yaml diff --git a/docs/topic/ctc/ctc_loss.md b/docs/topic/ctc/ctc_loss.md new file mode 100644 index 00000000..51ef3dc9 --- /dev/null +++ b/docs/topic/ctc/ctc_loss.md @@ -0,0 +1,336 @@ +## Derivative of CTC Loss + +关于CTC的介绍已经有很多不错的教程了,但是完整的描述CTCLoss的前向和反向过程的很少,而且有些公式推导省略和错误。本文主要关注CTC Loss的梯度是如何计算的,关于CTC的介绍这里不做过多赘述,具体参看文末参考。 + +CTC主要应用于语音和OCR中,已语音[Deepspeech2](https://arxiv.org/abs/1512.02595)模型为例,CTC的网络一般如下图所示,包含softmax和CTCLoss两部分。反向传播需要求得loss L相对于logits $u^i$​的梯度。下面先介绍CTCLoss的前向计算。 + +> 图片来源于文末参考 + +![img](./img/ctc_loss_backward_1.png) + +## 1. CTC Loss 的计算 + +CTC中path的定义与概率的计算如下: + +image-20211104200811966 + +path 是 $ L'^T$​​的元素,用 $ \pi $​​表示。 $ \textbf{x} $​​ 是输入特征,$\textbf{y}$​​ 是输出label, 都是序列。 $ L $​​ 是输出的 vocab, L‘ 是 $ L \cup {blank}$​​。 $y_{\pi_{t}}^t$​​ 表示在t时刻,$\pi_{t}$​​ label时的观察概率。其中$\pi_{t}$​​ 表示 $\pi$​​ path在t时刻的label。$\pi$​​ 是 $\textbf{y}$​​ 与 $ \textbf{x}$​​ 的一个alignment,长度是$T$​​,取值空间为$L'$​​​。path也称为alignment。 + +公式(2)解释了给定输入 $\textbf{x}$​ ,输出 $ \pi $​ path 的概率,即从时间t=1到T每个时间点的概率 $y_{\pi_{t}}^t$​ 相乘。 + +求出单条path后,就可以计算$p(l \mid x)$​ 的概率,计算如下: + +image-20211104202358513 + +这里边 $\mathcal{B}$ 就是映射, 即所有多对一的映射(many-to-one mapping )的集合。 这样就算出来对应一个真正的 label $\textbf{l}$ 的概率了,这里是求和。 求和的原因就是 aab 和 abb 都是对应成ab, 所以 aab 的概率 + abb 的概率才是生成ab的概率。 + +公式(3)解释了给定输入 $\mathbf{x}$​​​​​​ ,求输出$\mathbf{l}$​​​​​​ 的概率, 即所有集合 $\mathcal{B}^{-1} (\mathbf{l})$​​​​​​​​​​ 中 path的概率和。 + +### 1.1 CTC forward-backward 算法 + +CTC的优化采用算最大似然估计[MLE (maximum likelihood estimation)](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation), 这个和神经网络本身的训练过程是一致的。 + +这个CTC 计算过程类似HMM的 [forward-backward algorithm](https://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithm),下面就是这个算法的推导过程: + +image-20211104203040307 + +上图中的定义很清楚, 但是$ \alpha_{t-1}(s) $ and $ \alpha_{t-1}(s-1)$ 和 $\alpha_t(s)$ 的关系也不那么好看出来,下图给出了具体的关于 $\alpha_t(s)$ 的推导过程: + +image-20211108155714843 + +image-20211109153011816 + +这里的公式比较适合用下面的图来理解,$\alpha_1(1)$​​​​ 其实对应的就是下图中左上角白色的圆圈。 就是上来第一个是blank 的概率, 而 $\alpha_1(2)$​​​​是label l 的第一个字母。 这里边我们假设每个字母之间都插入了空白,即label l扩展成l',例如,l=[a, b, b, c], l'=[-, a, -, b, -, b, -, c, -]。 然后对于其他圆点,在时间是1 的情况下概率都是 0. Figure 3中横轴是时间 t,从左到右是1到T;纵轴是s(sequence),从上到下是 1 到 $\mathbf{\mid l' \mid}$​​​​. + +image-20211108155918442 + +接下来我们分析递归公式 (resursion),更多介绍可以参看 [2]. 公式6分情况考虑: + +* 第一种情况就是当前的label是blank, 或者 $\mathbf{l'}_{s}= \mathbf{l'}_{s-2}$​​​​​​​(相邻是重复字符): + + ![img](https://distill.pub/2017/ctc/assets/cost_no_skip.svg) + + 这个时候他的概率来自于过去t-1的两个label 概率, 也就是 $a_{t-1} (s)$​​ 和 $a_{t-1} (s-1)$​​​ 。 + + $ a_{t-1} (s)$​​ 就是说当前的 sequence 已经是s 了,figure 3中表现为横跳, blank -->blank(例如t=3, s=3); + + 而 $a_{t-1} (s-1) $是说明当前的字符还不够, 需要再加一个, 所以在figure 3中就是斜跳,从黑色圆圈到白色圆圈(例如,t=3, s=5)。 + + 仔细观察figure 3, 除了第一排的白色圆圈, 其他白色圆圈都有两个输入, 就是上述的两种情况。 当然判断blank 的方法也可以是判断$I'_{s-2} = I'_{s}$​. 这种情况也是说明$I'_{s}$​​​ 是blank, 因为每一个字符必须用 blank 隔开, 即使是相同字符。 + +* 第二章情况 也可以用类似逻辑得出, 只不过当前的状态s 是黑色圆圈, 有三种情况输入。 + + ![img](https://distill.pub/2017/ctc/assets/cost_regular.svg) + +最终的概率就如公式8 所示, 这个计算过程就是 CTC forward algroirthm, 基于 Fig. 3 的左边的初始条件。 + +image-20211108162544982 + +基于Fig. 3 右边的初始条件,我们还是可以计算出一个概率, 那个就是 **CTC backward**. 这里我就不详细介绍了, 直接截图。 + +image-20211108162859876 + +这样一直做乘法, 数字值越来越小,很快就会underflow。 这个时候就需要做 scaling. + +image-20211108163526616 + +算出了forward probability 和 backward probability 有什么用呢, 解释如下图。 + +image-20211108164110404 + +上图是说 forward probability and backward probability 的乘积, 代表了这个 sequence $\mathbf{l}$ t时刻,是s label 的 所有paths 的概率。 这样的话 我们就计算了 Fig. 3 中的每个圆圈的概率。为什么$\alpha_t(s)\beta_t(s)$ 中多出一个 $y^t_{\mathbf{l'_s}}$ ,这是因为它在 $\alpha$ 和 $\beta$ 中都包含该项,合并公式后就多出一项。 + +image-20211109143104052 + +$p(\mathbf{l}|\mathbf{x})$​ 可以通过任意时刻 t 的所有 s 的 foward-backward 概率计算得来。取负对数后就是单个样本的NLL(Negative Log Likelihood)。 + +### 1.2 总结 + +总结一下,根据前向概率计算CTCLoss函数,可以得出如下结论: + +1. 对于时序长度为T的输入序列x和输出序列z,前向概率: + $$ + \begin{split} + + \alpha_t(s) &= \sum_{ \underset{\pi_t=l'_s}{\pi \in \mathcal{B}^{-1}(z)} } p(\pi_{1:t}|x) \newline + + \alpha_1(1) &= y_{-}^1 ; \quad \alpha_1(2)=y^1_{l'_2}, \quad \alpha_1(s)=0, \forall s > 2 \newline + + \alpha_t(s) &= 0, \quad \forall s < |l'| - 2(T-t) - 1 ,\quad \text{or} \quad \forall s < 1 \newline + + \alpha_t(s) &= + \begin{cases} + (\alpha_{t-1}(s) + \alpha_{t-1}(s-1) ) y^t_{l'_s} & \text{if $l'_s=b$ or $l'_{s-2} = l'_s$​} \newline + (\alpha_{t-1}(s) + \alpha_{t-1}(s-1) + \alpha_{t-1}(s-2))y^t_{l'_s} & \text{otherwise}\newline + \end{cases} + + \end{split} + $$ + +2. 利用 $\alpha_t(s)$计算CTCLoss: + $$ + -ln(p(l \mid x)) = -ln(\alpha_{T}(|l'|)+\alpha_{T}(|l'|-1)) + $$ + +根据后向概率计算CTCLoss函数,可以得出如下结论: + +1. 对于时序长度为T的输入序列x和输出序列z,后向概率: + $$ + \begin{split} + + \beta_t(s) &= \sum_{ \underset{\pi_t=l'_s}{\pi \in \mathcal{B}^{-1}(z)} } p(\pi_{t:T}|x) \newline + + \beta_T(|l'|) &= y_{-}^T ; \quad \beta_T(|l'|-1)=y^T_{l'_{|l'|-1}}, \quad \beta_T(s)=0, \forall s < |l'| - 1 \newline + + \beta_t(s) &= 0, \text{$\forall s > 2t$ or $\forall s < |l'|$} \newline + + \beta_t(s) &= + \begin{cases} + (\beta_{t+1}(s) + \beta_{t+1}(s+1) ) y^t_{l'_s} & \text{if $l'_s=b$ or $l'_{s+2} = l'_s$} \newline + (\beta_{t+1}(s) + \beta_{t+1}(s+1) + \beta_{t+1}(s+2))y^t_{l'_s} & \text{otherwise}\newline + \end{cases} + + \end{split} + $$ + + 2. 利用 $\beta_t(s)$计算CTCLoss: + +$$ +-ln(p(l \mid x)) = -ln(\beta_{1}(1)+\beta_{1}(2)) \newline +$$ +根据任意时刻的前向概率和后向概率计算CTC Loss函数,得到如下结论: + +1. 对于任意时刻t,利用前向概率和后向概率计算CTCLoss: + +$$ +p(l \mid x) = \sum_{s=1}^{|l'|} \frac{\alpha_t(s)\beta_t(s)}{y_{l'_s}^t} \newline + +-ln(p(l \mid x)) = -ln( \sum_{s=1}^{|l'|} \frac{\alpha_t(s) \beta_t(s)}{y_{l'_s}^t} ) +$$ +我们已经得到CTCLoss的计算方法,接下来对其进行求导。 + +## 2. CTC梯度计算 + +### 2.1 微分公式 + +在计算梯度前,我们先回顾下基本的微分公式: +$$ +C' &= 0 \newline +x' &= 1 \newline +x^n &= n \cdot x^{n-1} \newline +(e^x)' &= e^x \newline +log(x)' &= \frac{1}{x} \newline +(u + v)' &= u' + v' \newline +(\frac{u}{v})' &= \frac{u'v-uv'}{v^2} \newline +\frac{\mathrm{d}f(g(x))}{\mathrm{d}x} &= \frac{\mathrm{d}f(g(x))}{\mathrm{d}g(x)} \cdot \frac{\mathrm{d}g(x)}{\mathrm{d}x} +$$ + +### 2.2 CTC梯度 + +最大似然估计训练就是最大化训练集中每一个分类的对数概率,即最小化Eq. 12。 + +image-20211108164206136 + +最后就是算微分了, 整个推导过程就是加法和乘法, 都可以微分。 $\mathit{O}^{ML}$关于神经网络的输出 $y^t_k$的梯度见Eq. 13。因为训练样本是相互独立的,所以可以单独考虑每个样本,公式如Eq.13。 + +下面是CTCLoss的梯度计算: + +image-20211109143622448 + +### 2.3 CTC梯度推导 + +回顾下之前的公式,便于理解后续推导过程。 +$$ +p(l \mid x) = \sum_{s=1}^{|l'|} \frac{\alpha_t(s)\beta_t(s)}{y_{l'_s}^t} \\ + +\begin{equation} +\alpha_t(s) \beta_t(s) = \sum_{ \underset{\pi_t=l'_s}{\pi \in \mathcal{B}^{-1}(l):} } y^t_{l'_s} \prod_{t=1}^T y^t_{\pi_t} +\end{equation} +$$ +其中Eq. 15的计算过程如下: +$$ +\begin{align*} +\frac{\part p( +l \mid x)}{\part y_k^t} + & = \sum_{s \in lab(z,k)} \frac{ \part \frac{ \alpha_t(s) \beta_t(s)}{y_{k}^t}}{\part y_k^t} + \newline + & = \sum_{s \in lab(z,k)} \frac{(\alpha_t(s)\beta_t(s))’y_k^t - \alpha_t(s)\beta_t(s){y_k^t}'}{{y_k^t}^2} + \newline + &= \sum_{s \in lab(z,k)} \frac{( \prod_{t'=1}^{t-1} y^{t'}_{\pi_{t'}} \cdot y_k^t \cdot y_k^t \cdot \prod_{t'=t+1}^{T} y^{t'}_{\pi_{t'}} )’ y_k^t - \alpha_t(s)\beta_t(s){y_k^t}'}{{y_k^t}^2} + \newline + &= \sum_{s \in lab(z,k)} \frac{2\alpha_t(s)\beta_t(s) - \alpha_t(s)\beta_t(s)}{{y_k^t}^2} + \newline + &= \sum_{s \in lab(z,k)} \frac{\alpha_t(s)\beta_t(s)}{{y_k^t}^2} + \newline + &= \frac{1}{{y_k^t}^2} \sum_{s \in lab(z,k)} \alpha_t(s)\beta_t(s) \tag{1} \newline +\end{align*} +$$ + + +NLL的公式推导如下: +$$ +\begin{split} +\frac{\part {ln(p(l \mid x))} }{ \part y^t_k } + &= \frac{1}{p(l \mid x)} \frac{ \part{p(l \mid x)} }{ \part y_k^t } \newline + &= \frac{1}{p(l \mid x) {y^t_k}^2 } \sum_{s \in lab(z,k)} \alpha_t(s)\beta_t(s) +\end{split} + +\tag{2} +$$ + + +已经算出了CTCLoss对于 $y_k^t$​ 的梯度,接下来我们需要计算 CTCLoss对于$u^t_k$​(logits)的梯度。套用链式法则,并替换$y^t_k$​ 为 $y^t_{k'}$​,结果如下图。图中 $k'$​ 表示vocab中的某一个token,$K$​​ 是vocab的大小。 + +![](./img/ctc_loss_backward_2.png) + +图中公式4根据链式法则得到: +$$ +- \frac{ \part ln(p(l \mid x)) }{ \part u^t_k } + = - \sum_{k'=1}^{K} \frac{ \part ln(p(l \mid x)) }{ \part y^t_{k'} } \frac{ \part y^t_{k'} }{ \part u^t_k } \tag{4} +$$ +图中公式3是softmax的梯度,参考 [4],计算过程如下: +$$ +softmax(j) = S_j = \frac{ e^{a_j} }{ \sum_{k=1}^K e^{a_k} }, \enspace \forall j \in 1 \dots K +$$ + +$$ +\begin{split} +\frac{ \part S_i }{ \part a_j} + &= \frac{ \part (\frac{ e^{ a_i } }{ \sum_k e^{ a_k } }) } { \part a_j } + \newline + &= + \begin{cases} + \frac{ e^a_i \sum - e^a_j e^a_i }{ \sum^2 } + &= \frac{ e^a_i }{ \sum } \frac{ \sum - e^a_j }{ \sum } \newline + &= S_i(1-S_j) & \text{i = j, $\sum$ stands for $\sum_{k=1}^K e^a_k$} + \newline + \frac{ 0 - e^a_j e^a_i }{ \sum^2 } + &= - \frac{ e^a_j }{ \sum } \frac{ e^a_i }{ \sum } \newline + &= -S_j S_i & \text{i $\neq$ j, $\sum$ stands for $\sum_{k=1}^K e^a_k$} + \end{cases} + \newline + &= + \begin{cases} + S_i(1 - S_j) & \text{$i = j$} + \newline + -S_j S_i = S_i (0 - S_j) & \text{$i \neq j$} + \end{cases} + \newline + &= S_i (\delta_{ij} - S_j ) +\end{split} +\tag{3} +$$ +$$ +\delta_{ij} = + \begin{cases} + 1 & \text{if i = j} \newline + 0 & \text{otherwise} + \end{cases} +$$ + + + +下图中黄色框中的部分表示公式(1),即遍历所有的vocab中的token,其结果是$p(l \mid x)$​。这是因为label $l$​ 中的token一定在vocab中,且 $s \in lab(l, k')$​ 可以是空集。当 $k'$​ 在 l 中,s 则为label中token是$k'$​的概率;当$k'$​​​不在l中,s为空,概率为0。 + +![img](./img/ctc_loss_backward_3.png) + +公式(2),(3)带入(4),并结合公式(1)的结果如上图右边,即: +$$ +\begin{split} +- \frac{ \part ln(p(l \mid x)) }{ \part u^t_k } &= + - \sum_{k'=1}^K \frac{ \part ln(p(l \mid x)) }{ \part y^t_{k'} } \frac{ \part y^t_{k'}}{ \part u^t_k } \newline + &= - \sum_{k'=1}^K \frac{ y^t_{k'}( \delta_{kk'} - y^t_k ) }{ p(l \mid x) {y^t_{k'}}^2 } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= - \sum_{k'=1}^K \frac{ \delta_{kk'} - y^t_k }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= \sum_{k'=1}^K \frac{ y^t_k - \delta_{kk'} }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= \sum_{k'=1}^K \frac{ y^t }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) - \sum_{k'=1}^K \frac{ \delta_{kk'} }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= \frac{ y^t_k }{ p(l \mid x) } ( \sum_{k'=1}^K \frac{1}{y^t_{k'}} \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) ) - \sum_{k'=1}^K \frac{ \delta_{kk'} }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= \frac{ y^t_k }{ p(l \mid x) } p(l \mid x) - \sum_{k'=1}^K \frac{ \delta_{kk'} }{ p(l \mid x) y^t_{k'} } \sum_{s \in lab(l, k') } \alpha_t(s) \beta_t(s) \newline + &= y^t_k - \frac{ 1 }{ p(l \mid x) y^t_k } \sum_{s \in lab(l, k)} \alpha_t(s) \beta_t(s) \newline +\end{split} +$$ +最终,为了通过softmax层传播CTCLoss的梯度,需要计算目标函数与 logits $u^t_k$​的偏微分,即Eq. 16: +$$ +\begin{align*} +\hat{\alpha}_t(s) & \overset{def}{=} \frac{ \alpha_t(s) }{ C_t } ,\enspace C_t \overset{def}{=} \sum_s \alpha_t(s) +\newline + +\hat{\beta}_t(s) & \overset{def}{=} \frac{ \beta_t(s) }{ D_t } ,\enspace D_t \overset{def}{=} \sum_s \beta_t(s) +\newline + +- \frac{ \part ln(p(l \mid x)) }{ \part u^t_k } &= y^t_k - \frac{1}{y^t_k \sum_{s=1}^{\mid l' \mid} \frac{ \hat{\alpha}_t(s) \hat{\beta}_t(s) }{ y^t_{l'_s} } } \sum_{s \in lab(l, k)} \hat{\alpha}_t(s) \hat{\beta}_t(s) \tag{16} +\newline + +\end{align*} +$$ + +### 2.1 总结 + +* 通过动态规划算法计算$\alpha_t(s)$ 和 $\beta_t(s)$ + +* 通过$\alpha_t(s)$​ 计算 $p(l \mid x)=\alpha_T(\mid l' \mid) + \alpha_T(\mid l' \mid -1)$​​ + +* 通过$\alpha_t(s)$ 和 $\beta_t(s)$​ 计算CTcLoss函数的导数: + $$ + \begin{split} + - \frac{ \part ln(p(l \mid x)) }{ \part u^t_k } + &= y^t_k - \frac{ 1 }{ p(l \mid x) y^t_k } \sum_{s \in lab(l, k)} \alpha_t(s) \beta_t(s) + \newline + &= y^t_k - \frac{1}{y^t_k \sum_{s=1}^{\mid l' \mid} \frac{ \hat{\alpha}_t(s) \hat{\beta}_t(s) }{ y^t_{l'_s} } } \sum_{s \in lab(l, k)} \hat{\alpha}_t(s) \hat{\beta}_t(s) + \newline + \end{split} + + \tag{16} + $$ + +## Reference + +[[1] A. Graves, S. Fernandez, F. Gomez, J. Schmidhuber. Connectionist Temporal lassification: Labeling Unsegmented Sequence Data with Recurrent Neural Networks. ICML 2006, Pittsburgh, USA, pp. 369-376.](http://www.cs.toronto.edu/~graves/icml_2006.pdf) + +[[2] Sequence ModelingWith CTC](https://distill.pub/2017/ctc/) + +[[3] NLP 之 CTC Loss 的工作原理](https://www.jianshu.com/p/e073c9d91b20) + +[[4] The Softmax function and its derivative](https://eli.thegreenplace.net/2016/the-softmax-function-and-its-derivative/) + +[[5] CTC Algorithm Explained Part 1:Training the Network(CTC算法详解之训练篇)](https://xiaodu.io/ctc-explained/) + diff --git a/docs/topic/ctc/img/ctc_loss_alpha_definition.png b/docs/topic/ctc/img/ctc_loss_alpha_definition.png new file mode 100644 index 0000000000000000000000000000000000000000..c6c35cd3b8a96ce879251e1d3889301f70cebb2f GIT binary patch literal 165965 zcmdqI1ydwJ_wES{?l6rF&M>&UySvNa?(Xi+;O@@g?hb?NAcNDmyF1&w|Gm2r_r9?o zU^}CtGCHcePG+5aPG+88MJULLBf)=z2Ll5`l9Uip0t5RZ1v-Rap+VnhMDX~7fgze$ z2n#Dn3JVh{INF(7Set->Nkk;8!)Papvqq7v0GLQGWJzxXy}}3w99$jPn8JsfhYJ!> z`Fnk4DO)Rd?5u`2Wl|)G66Dr2!CwkyJqG{V=9`^B?;s&!R|5e 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