Answers.Body: <p><strong>Connectionist Temporal Classification (CTC)</strong> can be useful for sequence modeling problems, like <strong>speech recognition</strong> and <strong>handwritten recognition</strong>, where the input and output sequences might have different sizes, so there's <strong>the problem of aligning the sequences</strong>. For instance, in speech recognition, not all sounds in speech correspond to a character, so how do we know if a sound should be converted to one of the possible chars? Moreover, we assume that we don't have a training dataset where the input and output sequences are aligned. The creation/labeling of such a dataset would be quite consuming. That's why CTC was introduced.</p>
  <p>So, mathematically, the speech/handwritten recognition task can be written as</p>
  <p><span class="math-container">$$
  Y^* = \text{argmax}_Y p(Y \mid X),
  $$</span>
  where <span class="math-container">$Y^* = [y_1, \dots, y_M]$</span> is the ideal output sequence for <span class="math-container">$X = [x_1, \dots, x_T]$</span>. Note that <span class="math-container">$T$</span> may be different from <span class="math-container">$M$</span>.</p>
  <p>Now, let's say that you have a speech <span class="math-container">$X = [x_1, \dots, x_N]$</span> and the output should be the sequence <span class="math-container">$Y^* = [h, e, l, l, o]$</span>. One naive approach to solve this problem would be, for each input <span class="math-container">$x_i$</span>, we would predict the most likely char, so we could end up with an output sequence like <span class="math-container">$\hat{Y} = [h, e, e, l, l, l, o]$</span> (when <span class="math-container">$T = 7$</span> and <span class="math-container">$M = 5$</span>), then we could remove all duplicates, so we would end up with <span class="math-container">$\hat{Y}' = [h, e, l, o]$</span>. However, this is not the correct approach, because, as you can see, there are words where the same letter appears twice in a row.</p>
  <p>To solve this problem, we can introduce a <strong>special character</strong>, which we can denote by <span class="math-container">$\epsilon$</span>. So, in this case, the idea is that <span class="math-container">$\epsilon$</span> should be predicted around exactly two (and not more or less) &quot;l&quot;. Once the sequence is predicted, we can remove <span class="math-container">$\epsilon$</span> and, hopefully, we have a valid word, rather than a word like &quot;helo&quot; or &quot;hellllo&quot;.</p>
  <p>The idea of CTC is, for each <span class="math-container">$x_i$</span>, the neural network produces a probability distribution over the possible chars, which we can denote by <span class="math-container">$p_t(a_t \mid X)$</span>. In the example above, a probability distribution over <span class="math-container">$\{h, e, l, o, \epsilon \}$</span>. So, for example, the probability vector for <span class="math-container">$x_1$</span> could be <span class="math-container">$p_1(a_1 \mid X) = [0.6, 0.1, 0.1, 0.1, 0.1]$</span>. Given all probability vectors <span class="math-container">$p_t(a_t \mid X)$</span>, for <span class="math-container">$t=1, \dots, T$</span>, we can compute the probability of an alignment (a specific output sequence). Then we <a href="https://en.wikipedia.org/wiki/Marginal_distribution" rel="nofollow noreferrer">marginalize</a> over the set of alignments.</p>
  <p>To reflect these ideas, the <strong>CTC loss function</strong> is defined as follows</p>
  <p><span class="math-container">$$
  p(Y \mid X)=\underbrace{\sum_{A \in \mathcal{A}_{X, Y}} }_{\text{Marginalization}\\\text{over set of}\\ \text{alignments}} \underbrace{\prod_{t=1}^{T} p_{t}\left(a_{t} \mid X\right)}_{\text{Probability}\\\text{for a single}\\\text{alignment}}
  $$</span>
  We can then use an RNN to model <span class="math-container">$p_t(a_t \mid X)$</span> given that <a href="https://ai.stackexchange.com/a/18199/2444">RNNs are good for sequence prediction</a>.</p>
  <p>Now, another problem is that there can be many alignments, so the computation of the loss may be expensive if done naively. To solve this problem, you can use a <strong>dynamic programming</strong> algorithm, the <strong>CTC forward-backward algorithm</strong>. The details of how this is done, how the <strong>gradient</strong> of the CTC loss is computed, how <strong>inference</strong> is done in this context (including the details of <strong>beam search</strong>), and other details can be found in this nice article <a href="https://distill.pub/2017/ctc/" rel="nofollow noreferrer">Sequence Modeling With CTC</a> (2017) by Awni Hannun, which this answer is based on.</p>
  <p>You can also read the original paper <a href="http://www.cs.toronto.edu/%7Egraves/icml_2006.pdf" rel="nofollow noreferrer">Connectionist Temporal Classification: Labeling Unsegmented Sequence Data with Recurrent Neural Networks</a> (2006), by Alex Graves et al., mentioned in the linked <a href="https://www.tensorflow.org/api_docs/python/tf/nn/ctc_loss" rel="nofollow noreferrer">TensorFlow documentation</a>, which presents and explains the CTC loss and the CTC forward-backward algorithm (in section 4.1).</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-07-01T19:11:07.657Z
Answers.LastActivityDate: 2021-12-23T15:51:48.273Z
Answers.LastEditDate: 2021-12-23T15:51:48.273Z
Answers.UserDisplayName: nbro
Body: <p>I am trying to implement CTC loss in TensorFlow, but <a href="https://www.tensorflow.org/api_docs/python/tf/nn/ctc_loss" rel="nofollow noreferrer">their documentation</a> is pretty limited. So I am not sure how to approach the problem. I found <a href="https://github.com/mohammadpz/CTC-Connectionist-Temporal-Classification/blob/master/ctc_cost.py" rel="nofollow noreferrer">a good example in Theano</a>.</p>
  <p>Are any other resources that explain the CTC loss?</p>
  <p>I am also trying to understand how its forward-backward algorithm works and what the beam decoder in the case of the CTC loss is.</p>
  
Tags: <reference-request><tensorflow><recurrent-neural-networks><speech-recognition><ctc-loss>
Title: How does the CTC loss work?
UserDisplayName: user26787
ThreadText: # How does the CTC loss work?
  
  (Asked by: username_0 on 2019-07-01)
  
  I am trying to implement CTC loss in TensorFlow, but [their documentation](https://www.tensorflow.org/api_docs/python/tf/nn/ctc_loss) is pretty limited. So I am not sure how to approach the problem. I found [a good example in Theano](https://github.com/mohammadpz/CTC-Connectionist-Temporal-Classification/blob/master/ctc_cost.py).
  
  Are any other resources that explain the CTC loss?
  
  I am also trying to understand how its forward-backward algorithm works and what the beam decoder in the case of the CTC loss is.
  
  **Tags:** reference-request, tensorflow, recurrent-neural-networks, speech-recognition, ctc-loss
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2019-07-01. Score: 3
  
  **Connectionist Temporal Classification (CTC)** can be useful for sequence modeling problems, like **speech recognition** and **handwritten recognition**, where the input and output sequences might have different sizes, so there's **the problem of aligning the sequences**. For instance, in speech recognition, not all sounds in speech correspond to a character, so how do we know if a sound should be converted to one of the possible chars? Moreover, we assume that we don't have a training dataset where the input and output sequences are aligned. The creation/labeling of such a dataset would be quite consuming. That's why CTC was introduced.
  
  So, mathematically, the speech/handwritten recognition task can be written as
  
  $$
  Y^\* = \text{argmax}\_Y p(Y \mid X),
  $$
  where $Y^\* = [y\_1, \dots, y\_M]$ is the ideal output sequence for $X = [x\_1, \dots, x\_T]$. Note that $T$ may be different from $M$.
  
  Now, let's say that you have a speech $X = [x\_1, \dots, x\_N]$ and the output should be the sequence $Y^\* = [h, e, l, l, o]$. One naive approach to solve this problem would be, for each input $x\_i$, we would predict the most likely char, so we could end up with an output sequence like $\hat{Y} = [h, e, e, l, l, l, o]$ (when $T = 7$ and $M = 5$), then we could remove all duplicates, so we would end up with $\hat{Y}' = [h, e, l, o]$. However, this is not the correct approach, because, as you can see, there are words where the same letter appears twice in a row.
  
  To solve this problem, we can introduce a **special character**, which we can denote by $\epsilon$. So, in this case, the idea is that $\epsilon$ should be predicted around exactly two (and not more or less) "l". Once the sequence is predicted, we can remove $\epsilon$ and, hopefully, we have a valid word, rather than a word like "helo" or "hellllo".
  
  The idea of CTC is, for each $x\_i$, the neural network produces a probability distribution over the possible chars, which we can denote by $p\_t(a\_t \mid X)$. In the example above, a probability distribution over $\{h, e, l, o, \epsilon \}$. So, for example, the probability vector for $x\_1$ could be $p\_1(a\_1 \mid X) = [0.6, 0.1, 0.1, 0.1, 0.1]$. Given all probability vectors $p\_t(a\_t \mid X)$, for $t=1, \dots, T$, we can compute the probability of an alignment (a specific output sequence). Then we [marginalize](https://en.wikipedia.org/wiki/Marginal_distribution) over the set of alignments.
  
  To reflect these ideas, the **CTC loss function** is defined as follows
  
  $$
  p(Y \mid X)=\underbrace{\sum\_{A \in \mathcal{A}\_{X, Y}} }\_{\text{Marginalization}\\\text{over set of}\\ \text{alignments}} \underbrace{\prod\_{t=1}^{T} p\_{t}\left(a\_{t} \mid X\right)}\_{\text{Probability}\\\text{for a single}\\\text{alignment}}
  $$
  We can then use an RNN to model $p\_t(a\_t \mid X)$ given that [RNNs are good for sequence prediction](https://ai.stackexchange.com/a/18199/2444).
  
  Now, another problem is that there can be many alignments, so the computation of the loss may be expensive if done naively. To solve this problem, you can use a **dynamic programming** algorithm, the **CTC forward-backward algorithm**. The details of how this is done, how the **gradient** of the CTC loss is computed, how **inference** is done in this context (including the details of **beam search**), and other details can be found in this nice article [Sequence Modeling With CTC](https://distill.pub/2017/ctc/) (2017) by Awni Hannun, which this answer is based on.
  
  You can also read the original paper [Connectionist Temporal Classification: Labeling Unsegmented Sequence Data with Recurrent Neural Networks](http://www.cs.toronto.edu/%7Egraves/icml_2006.pdf) (2006), by Alex Graves et al., mentioned in the linked [TensorFlow documentation](https://www.tensorflow.org/api_docs/python/tf/nn/ctc_loss), which presents and explains the CTC loss and the CTC forward-backward algorithm (in section 4.1).
  
  
  ---
  
  