Each position \(t\) is encoded as a $d$-dimensional vector \(p\). To encode the \(i\) th dimension of the vector for the \(t\) position, the following formula is used: \[p_t(i) = \begin{cases} \sin(\omega_k \cdot t) & \text{if } i = 2k\\ \cos(\omega_k \cdot t) & \text{if } i = 2k + 1\\ \end{cases}\] where \[\omega_k = \frac{1}{1000^{2k/d}}\]

Each encoder module uses multi-head attention. What is multi-head attention? In single head attention, we have projection matrices \(W^Q\), \(W^K\), and \(W^V\) for the query, key, and value embeddings respectively. In multi-head attention, we have \(W^Q_i\), \(W^K_i\), and \(W^V_i\) for each head \(i\). Let \(n\) be the source length, \(m\) be the target length, and \(d\) the embedding dimension. Each attention head outputs \(Z_i\), which has dimension \(m \times d\). These are concatenated and multiplied by a weight matrix \(W^o\).

The input to the first module are the embedded tokens of the input sequence. Added to these embeddings is a postional embedding that encodes the relative position of the token in the sequence.

The output \(Z\) passes through a feed forward network. There are residual connections and layer normalization operations around the self-attention layer as well as around the feed forward network.

Each decoder module takes as input: the final \(K\) and \(V\) matrices of the top-most encoder module, and the output of the previous decoder module. Each decoder module attends to the encoder output with an Encoder-Decoder attention layer. Tokens are predicted one at a time. At time \(i\), each decoder module attends to every output produced up until \(i\) with a self-attention layer.

A linear and softmax layer sit at the top.