Detailed Explanation of Q, K, V Matrices in the Self-Attention Mechanism

1. Knowledge Point Description
The self-attention mechanism is the core component of the Transformer model. It enables each position in a sequence (e.g., each word in a sentence) to attend to all other positions in the sequence, thereby computing a weighted sum representation. The Q (Query), K (Key), and V (Value) matrices are key to implementing this mechanism. Simply put, the self-attention mechanism can be viewed as an information retrieval process: for a specific "query" (Q), we compute an attention distribution (i.e., weights) by comparing it with a set of "keys" (K), and then use this distribution to perform a weighted sum on a set of "values" (V), ultimately obtaining an enhanced representation for that query.

2. Step-by-Step Explanation

Step 1: From Input Sequence to Initial Vector Representation
Suppose we have an input sequence, for example, a sentence containing 3 words. First, each word is converted into a vector representation (e.g., via an embedding layer). Thus, we obtain an input matrix \(X\) with dimensions \(3 \times d_{model}\), where 3 is the sequence length and \(d_{model}\) is the model's hidden dimension (e.g., 512). Each row represents a word vector.

Step 2: Generating the Q, K, V Matrices
The input matrix \(X\) itself cannot be directly used for attention calculation. We need to linearly transform it (i.e., multiply by learnable weight matrices) into three different representations: Query, Key, and Value.

• Linear Transformation: We have three independent sets of learnable weight matrices: \(W^Q\), \(W^K\), and \(W^V\). Their dimensions are typically \(d_{model} \times d_k\) (for Q and K) and \(d_{model} \times d_v\) (for V). In practice, it's common to set \(d_k = d_v = d_{model}\) for simplicity.
• Calculation Process:
  • \(Q = X W^Q\) (Query matrix)
  • \(K = X W^K\) (Key matrix)
  • \(V = X W^V\) (Value matrix)

Now, the input sequence \(X\) is projected into three new matrices \(Q\), \(K\), and \(V\). Their number of rows (sequence length) is the same as \(X\), but the number of columns (feature dimension) becomes \(d_k\) or \(d_v\). Key Point: These three matrices originate from the same input \(X\), but through different linear transformations, they are assigned different roles and meanings.

Step 3: Understanding the Roles of Q, K, V
This is the most crucial step. We can use the analogy of an information retrieval system or dictionary lookup to understand:

• Query (Q): Can be thought of as "What information does the position I am currently looking at (e.g., the first word in the sentence) want to find?" It represents an active "questioner."
• Key (K): Can be thought of as the "label" or "index" of the information that each position in the sequence (including itself) can provide. It represents the identifier of the "entry" being queried.
• Value (V): Can be thought of as the "substantive information" actually contained in each position of the sequence that I want to obtain. It represents the actual content of the "entry" being queried.

A vivid example: Imagine a search engine.

• The keywords you enter are the Query (Q).
• The collection of keyword tags for all web pages on the internet are the Keys (K).
• The actual content of the web pages are the Values (V).
• The search engine matches your Query with all the web pages' Keys (calculating similarity) to obtain a relevance score (attention weight).
• Finally, based on this weight, the system performs a weighted sum on all the web pages' Values (actual text content) and returns you the most relevant answer, integrating information from multiple web pages.

In self-attention, each word simultaneously plays all three roles: it uses its own Query to query other words, and also uses its own Key and Value to respond to queries from other words.

Step 4: Calculating Attention Weights and Output
Now, we proceed with the formal calculation:

1. Calculate Attention Scores: We want to know to what extent a given Query (e.g., \(q_1\) corresponding to the first word) should attend to each word in the sequence (including itself). The method is to take the dot product of \(q_1\) with each word's Key (\(k_1, k_2, k_3\)). A larger dot product indicates greater similarity between the two vectors, and thus stronger attention should be paid.

   • Score matrix: \(\text{Scores} = Q K^T\) (dimension: \(3 \times 3\))
   • This \(3 \times 3\) matrix is often called the attention score matrix or attention map, where each element \(score_{ij}\) represents the attention score of the i-th word for the j-th word.

2. Scaling and Normalization: Using raw dot product scores directly may lead to gradient instability issues (especially when \(d_k\) is large). Therefore, we scale the scores by dividing by \(\sqrt{d_k}\), and then apply the Softmax function to convert the scores in each row into a probability distribution (all weights are positive and sum to 1).

   • \(\text{Attention Weights} = \text{Softmax}(\frac{Q K^T}{\sqrt{d_k}})\) (dimension: \(3 \times 3\))

3. Weighted Sum: Now we have the weights. For the Query of the first word, we use its computed weight distribution (the first row after Softmax) to perform a weighted sum on the Value vectors of all words (each row of the \(V\) matrix). This sum result is the new, enhanced representation of the first word after the self-attention mechanism.

   • \(\text{Output} = \text{Attention Weights} \cdot V\) (dimension: \(3 \times d_v\))

Finally, we obtain an output sequence of the same length as the input sequence. The new vector at each position in the sequence contains information from all other positions in the sequence, weighted according to relevance.

Summary:
The Q, K, V matrices are tools that project the original input sequence into three different functional subspaces. Through the interaction of Q and K to compute the attention distribution, and then using this distribution to aggregate V, the self-attention mechanism achieves dynamic fusion of information within the sequence. This allows the representation of each position to incorporate global contextual information, forming the foundation of the Transformer model's powerful capabilities.