What is positional encoding in a Transformer, and why is it necessary?

Positional encoding in a Transformer is a technique used to provide information about the position of words in a sequence, which is necessary because the self-attention mechanism in Transformers is permutation-invariant. Permutation-invariant means that the self-attention mechanism processes the words in the sequence regardless of their order. Unlike recurrent neural networks (RNNs) that inherently process sequences word by word, Transformers process the entire sequence at once. Therefore, without positional encoding, the Transformer would be unable to distinguish between sentences with the same words but in different orders. Positional encodings are added to the input embeddings (the numerical representations of the words) to inject information about the position of each word in the sequence. These encodings are typically represented as vectors of the same dimension as the word embeddings and can be learned or fixed. Common fixed positional encodings use sine and cosine functions of different frequencies to represent different positions. For example, if the sentence is 'The cat sat on the mat', the positional encoding would provide information about which word is the first, second, third, etc. This allows the Transformer to understand that 'The' is the first word and 'mat' is the last word, which is crucial for tasks like machine translation or text generation. Therefore, positional encoding is essential for Transformers to capture the order of words in a sequence and understand the meaning of the sequence correctly.