➩ Speech recognition.

➩ Text to speech.

➩ Machine translation.

➩ Image captioning (combines with convolutional networks).

➩ Handwritting recognition (online recognition: input is a sequence of pen positions, not an image).

➩ Time series prediction (for example, stock price prediction).

➩ Robot control (and other dynamic control tasks).

➩ \(a_{t+1} = f_{a}\left(a_{t}, x_{t+1}\right)\) and \(y_{t+1} = f_{o}\left(a_{t+1}\right)\)

➩ \(a_{t+2} = f_{a}\left(a_{t+1}, x_{t+2}\right)\) and \(y_{t+2} = f_{o}\left(a_{t+2}\right)\)

➩ \(a_{t+3} = f_{a}\left(a_{t+2}, x_{t+3}\right)\) and \(y_{t+3} = f_{o}\left(a_{t+3}\right)\)

➩ ...

➩ Each item can be a sequence with different number of elements \(t_{i}\), therefore, each item has different number of activation units \(a_{i,t}\), \(t = 1, 2, ..., t_{i}\).

➩ There can be either one output unit at the end of each item \(o = g\left(w^{\left(o\right)} \cdot a_{t_{i}} + b^{\left(o\right)}\right)\), or \(t_{i}\) output units one for each activation unit \(o_{t} = g\left(w^{\left(o\right)} \cdot a_{t} + b^{\left(o\right)}\right)\).

➩ Convolutional layers share weights over different regions of an image.

➩ Recurrent layers share weights over different times (positions in a sequence).

➩ In the case with one output unit at the end, \(\dfrac{\partial C_{i}}{\partial w^{\left(o\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial w^{\left(o\right)}}\), and \(\dfrac{\partial C_{i}}{\partial w^{\left(a\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial w^{\left(a\right)}} + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} \dfrac{\partial a_{t_{i} - 1}}{\partial w^{\left(a\right)}} + ... + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} ... \dfrac{\partial a_{2}}{\partial a_{1}} \dfrac{\partial a_{1}}{\partial w^{\left(a\right)}}\), and \(\dfrac{\partial C_{i}}{\partial w^{\left(x\right)}} = \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial w^{\left(x\right)}} + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} \dfrac{\partial a_{t_{i} - 1}}{\partial w^{\left(x\right)}} + ... + \dfrac{\partial C_{i}}{\partial o_{i}} \dfrac{\partial o_{i}}{\partial a_{t_{i}}} \dfrac{\partial a_{t_{i}}}{\partial a_{t_{i} - 1}} ... \dfrac{\partial a_{2}}{\partial a_{1}} \dfrac{\partial a_{1}}{\partial w^{\left(x\right)}}\).

➩ The case with one output unit for each activation unit is similar.

➩ Technically adding small weights will not lead to vanishing gradient like multiplying small weights, so hidden units can be added together.

➩ The (long term) memory is updated by \(a^{\left(c\right)}_{t} = a^{\left(f\right)}_{t} \times a^{\left(c\right)}_{t-1} + a^{\left(i\right)}_{t} \times a^{\left(g\right)}_{t}\), where \(a^{\left(c\right)}\) is called the cell unit, \(a^{\left(f\right)}\) is the forget gate and controls how much memory to forget, \(a^{\left(i\right)}\) is the input gate and controls how much information to add to memory, \(a^{\left(g\right)}\) is the new values added to memory.

➩ The (short term memory) state is updated by \(a^{\left(h\right)} = a^{\left(o\right)} \times g\left(a^{\left(c\right)}_{t}\right)\), where \(a^{\left(h\right)}\) is the usual recurrent unit called hidden state, \(a^{\left(o\right)}\) is the output gate and controls how much information from the memory to reflect in the next state.

➩ Each of the gates are computed based on the hidden state and the input features (or the previous layer hidden states if there are multiple LSTM layers): \(a^{\left(f\right)}_{t} = g\left(w^{\left(f\right)} \cdot x_{t} + w^{\left(F\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(f\right)}\right)\), \(a^{\left(i\right)}_{t} = g\left(w^{\left(i\right)} \cdot x_{t} + w^{\left(I\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(i\right)}\right)\), \(a^{\left(g\right)}_{t} = g\left(w^{\left(g\right)} \cdot x_{t} + w^{\left(G\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(g\right)}\right)\), \(a^{\left(o\right)}_{t} = g\left(w^{\left(o\right)} \cdot x_{t} + w^{\left(O\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(o\right)}\right)\).

➩ The memory is also updated through addition: \(a^{\left(h\right)}_{t} = \left(1 - a^{\left(z\right)}_{t}\right) \times a^{\left(h\right)}_{t-1} + a^{\left(z\right)}_{t} \times a^{\left(g\right)}_{t}\), where \(a^{\left(z\right)}\) is the update gate, and \(a^{\left(r\right)}\) is the reset gate.

➩ Each of the gates are computed in a similar way: \(a^{\left(z\right)}_{t} = g\left(w^{\left(z\right)} \cdot x_{t} + w^{\left(Z\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(z\right)}\right)\), \(a^{\left(r\right)}_{t} = g\left(w^{\left(r\right)} \cdot x_{t} + w^{\left(R\right)} \cdot a^{\left(h\right)}_{t-1} + b^{\left(r\right)}\right)\), \(a^{\left(g\right)}_{t} = g\left(w^{\left(g\right)} \cdot x_{t} + w^{\left(G\right)} \cdot a^{\left(r\right)}_{t} \times a^{\left(h\right)}_{t-1} + b^{\left(g\right)}\right)\).