In the Neural Turing Machine (NTM) architecture, each write operation is decomposed into two parts: an **erase** phase and an **add** phase. Given an erase vector $$e_t$$ (whose $$M$$ elements lie in the range $$(0, 1)$$) and a weight vector $$w_t$$ emitted by the write head, the previous memory vectors $$M_{t-1}(i)$$ are modified as follows: $$M_{t}^{\sim}(i) \leftarrow M_{t-1}(i)[1 - w_t(i)e_t]$$ If both the weight and erase values at a location are equal to one, the memory is reset to 0; otherwise, the memory is partially erased or remains unchanged. After the erase step, an add vector $$a_t$$ is applied to update the memory: $$M_t(i) \leftarrow M_{t}^{\sim}(i) + w_t(i)a_t$$

San Diego State University

Google

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

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.

Differentiability of NTM Components

In the Neural Turing Machine (NTM) architecture, a read operation at time $$t$$ uses an $$N \times M$$ memory matrix, denoted as $$M_t$$, and a vector of weightings, $$w_t$$, over the $$N$$ memory locations emitted by a read head. The weightings are normalized such that:

$$\sum_i w_t(i) = 1, \quad 0 \le w_t(i) \le 1 \quad \forall i$$

The read vector, $$r_t$$, returned by the read operation is defined as the convex combination of the memory row vectors:

$$r_t \leftarrow \sum_i w_t(i)M_t(i)$$

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