Step 1: Dot Product of Query and Key Matrices in Self-Attention

The initial step in the self-attention mechanism, fundamental to Transformer models, involves computing the dot product between the Query (Q) matrix and the Key (K) matrix. This operation is the bedrock for determining the degree of focus or "attention" each word in a sequence should give to every other word within the same sequence.

Purpose of the Dot Product in Self-Attention

The dot product between the query and key matrices serves to generate similarity scores between words. For each word in a given sentence, the self-attention mechanism performs the following:

• Query-Key Comparison: It utilizes the query vector associated with a specific word to compare against all key vectors present in the sentence.
• Similarity Matrix Generation: This pairwise comparison results in a matrix where each element represents the similarity score between a query word and a key word. The rows of this matrix indicate how much attention a particular word should direct towards every other word in the sentence.

Mathematically, if we have a Query matrix $Q$ and a Key matrix $K$, the similarity scores are calculated as:

$$ \text{Scores} = QK^T $$

where $K^T$ is the transpose of the Key matrix.

Example: Dot Product Computation

Let's illustrate this with a simple sentence: "I am good."

Assume each word is represented by a vector, and we are considering the dot product computation for each word's query vector against all words' key vectors.

Row 1: Comparing the Word "I"

In the first row of the resulting similarity matrix, we are examining the attention paid by the word "I" to all other words. This involves computing the dot product of the query vector for "I" (let's denote it as $q_1$) with the key vectors of:

• "I" ($k_1$)
• "am" ($k_2$)
• "good" ($k_3$)

This yields three scores:

• Score($q_1, k_1$)
• Score($q_1, k_2$)
• Score($q_1, k_3$)

These scores quantify how relevant "I" is to itself, to "am," and to "good." Typically, a word is most similar to itself, so the highest score in this row is expected for the pair ($q_1, k_1$).

Row 2: Comparing the Word "Am"

For the second row, we compute the dot product of the query vector for "am" ($q_2$) with the key vectors of:

• "I" ($k_1$)
• "am" ($k_2$)
• "good" ($k_3$)

This generates scores:

• Score($q_2, k_1$)
• Score($q_2, k_2$)
• Score($q_2, k_3$)

Again, the dot product of $q_2$ with $k_2$ (i.e., "am" with "am") is anticipated to yield the highest score, indicating that "am" is most relevant to its own context.

Row 3: Comparing the Word "Good"

Similarly, the third row involves computing the dot product of the query vector for "good" ($q_3$) with the key vectors of:

• "I" ($k_1$)
• "am" ($k_2$)
• "good" ($k_3$)

The scores will be:

• Score($q_3, k_1$)
• Score($q_3, k_2$)
• Score($q_3, k_3$)

The strongest similarity score in this row will likely be for the pair ($q_3, k_3$), representing "good" attending to itself.

Conclusion: Understanding Word Relationships Through Similarity Scores

The dot product between the Query (Q) and Key (K) matrices generates a matrix of similarity scores. These scores are crucial for the subsequent steps in the attention mechanism, as they inform how much attention each word should distribute to every other word in the sentence. This process is vital for creating context-aware word representations, which are fundamental to the success of Transformer models in various Natural Language Processing (NLP) tasks such as machine translation, text summarization, and sentiment analysis.

---

SEO Keywords:

• Dot product in self-attention
• Query and key vectors in Transformer
• Similarity scores in NLP
• Attention matrix explained
• Self-attention similarity matrix
• QK^T in Transformer
• Contextual relationships in Transformers
• Sentence-level attention computation

---

Interview Questions:

1. What is the primary role of the dot product in the self-attention mechanism?
2. Why is it necessary to compare the query vector of a word with all key vectors in the sequence?
3. How does the resulting similarity matrix contribute to the overall attention calculation?
4. What does a high dot product score between two word vectors signify in this context?
5. In the sentence "I am good," what would the first row of the similarity matrix represent conceptually?
6. Explain why a word typically exhibits the highest similarity score with itself during the dot product step.
7. What are the typical formations or dimensions of the Query (Q) and Key (K) matrices if a sentence has N words and an embedding size of d_model?
8. How does the dot product operation directly influence the downstream computation of attention weights?
9. What potential issues might arise if the dot products are not scaled appropriately before the softmax function is applied?