user: In Transformer machine learning models: How is multi-head attention different from normal attention? First describe the intuition and then the mathematical perspective.

assistant: In a Transformer machine learning model, multi-head attention is a variation of the attention mechanism that allows the model to attend to different parts of the input sequence in parallel.

Intuition:
Normal attention works by computing a weighted sum of the values of the input sequence, where the weights are determined by the similarity between a query vector and each of the input vectors. Multi-head attention takes this idea one step further by allowing the model to compute multiple sets of attention weights, or "heads", in parallel, each focusing on different parts of the input sequence. The outputs of each head are then concatenated and transformed linearly to produce the final attention output. By attending to multiple parts of the input sequence simultaneously, multi-head attention can potentially capture more complex relationships between different parts of the input.

Mathematical perspective:
Formally, given a set of input vectors {v_1, v_2, ..., v_n} and a query vector q, the output of multi-head attention is computed as follows:

1. First, the input vectors are linearly transformed into a set of key vectors K = {k_1, k_2, ..., k_n} and a set of value vectors V = {v_1, v_2, ..., v_n} using learned weight matrices W_k and W_v:
K_i = W_k * v_i
V_i = W_v * v_i

2. The query vector q is also transformed using a learned weight matrix W_q:
Q = W_q * q

3. For each of h attention "heads", the query vector Q and key vectors K are used to compute a set of attention weights a^h = {a^h_1, a^h_2, ..., a^h_n}:
a^h_i = softmax((Q * K_i^T) / sqrt(d_k))
where d_k is the dimensionality of the key vectors, and the softmax function ensures that the attention weights sum to 1.

4. Each attention head produces a weighted sum of the value vectors V using the attention weights a^h:
h^h = sum(a^h_i * V_i)

5. The outputs of all the attention heads are concatenated into a single vector h^c, and linearly transformed using a learned weight matrix W_o to produce the final attention output:
h = W_o * h^c

By computing multiple sets of attention weights in parallel, multi-head attention allows the model to attend to different parts of the input sequence with different priorities, potentially capturing more complex relationships between different parts of the input.

user: How does the implementation of attention differ when used in vision transformer and in language transformers?

assistant: The implementation of attention in vision transformers and language transformers differs mainly in the input representations and the sequence lengths.

In language transformers, the input is typically a sequence of discrete tokens, such as words or subwords, represented as one-hot vectors or embeddings. The sequence length can be quite long, up to several thousand tokens in some cases. The attention mechanism is used to selectively attend to different parts of the sequence, allowing the model to learn complex relationships between the tokens and generate output sequences, such as in natural language generation tasks.

In contrast, in vision transformers, the input is usually a set of image patches, each of which is represented as a high-dimensional vector. The sequence length is much shorter compared to language transformers, typically only a few hundred patches. The attention mechanism is used to attend to different patches and capture spatial relationships between them, allowing the model to learn representations that are suitable for downstream tasks, such as object detection or image classification.

Although the input representations and sequence lengths differ, the basic idea of the attention mechanism remains the same in both types of transformers: attend to different parts of the input with varying levels of importance, and use the attended information to generate output representations.