Suppose that we have $M_t$ contents from NxM and $w_t$ is weightings over the N locations which where emitted by read head at time t. All weightings are normalized and we have these properties:
$$\sum_i w_t(i) = 1, \quad 0 \le w_t(i) \le 1  \quad \forall i$$

The $r_t$ that is returned by the read is defined as followes:
$$ r_t \leftarrow \sum_i w_t(i)M_t(i) $$

San Diego State University

NTMs have two basic components, namely, controller and memory bank. Controller uses input and output vectors to interact with outside environment. This is similar to what we have in typical neural network architectures. Additionally, unlike other neural network architectures, it also interacts with the memory bank using read and write operations. 
NTMs can be used with gradient descent and this process would be quite straightforward as each component is differentiable.



NTM Architecture

Reading (NTM Architecture)

Each write is decomposed into two parts: **erase** and **add**.

We have erase vector $e_t$  (its M elements lie in range (0, 1))and $w_t$ which is emitted by write head, memory vectors $M_{t-1}(i)$ are changed as it is written in the given formula:
$$M_{t}^{\sim}(i) \leftarrow M_{t-1}(i)[1 - w_t(i)e_t]$$
As it is clear from the formula if at the location both weight and erase are equal to one than we reset them to 0. In  other cases we don't change the memory. 

After erase we also have add vector $a_t$ that is added to our memory:
$$M_t(i) \leftarrow M_{t}^{\sim} + w_t(i)a_t$$


Writing (NTM Architecture)

The weights are produced using two addressing mechanisms that are:
1. Content-based 
2. Location-based 

Addressing Mechanisms (NTM Architecture)

The most important architectural choice of the NTMs is the type of the neural network that would be used as a controller. More precisely, we have to decide whether we want to use feedforward or recurrent network. 
"If one compares the controller to the central processing unit in a digital computer (albeit with adaptive rather than predefined instructions) and the memory matrix to RAM, then the hidden activations of the recurrent controller are akin to the registers in the processor"
It is interesting that if we choose feedforward networks as a controller we can mimic RNNs by reading and writing at the same location at every time step.

Controller Network

In Neural Turing Machines (NTMs), components are differentiable because the read and write operations are 'blurry.' Instead of addressing a single memory element like a standard Turing machine or digital computer, NTM operations interact to varying degrees with all elements in the memory. An attentional focus mechanism determines this degree of blurriness, limiting input and output interactions to relatively small parts of the memory while neglecting the rest.

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