Analyze Transformer Self-Attention Complexity

The self-attention mechanism is a critical component of the Transformer architecture. It processes a sequence of $N$ input vectors (tokens or patches), each with embedding dimension $D$.

The computation requires calculating pairwise affinities between every element (query $q$) and every other element (key $k$) in the sequence, typically followed by scaling and multiplying by the value matrix $V$.

Given:

• $N$: Sequence length
• $D$: Embedding dimension
• $K$: Convolution kernel size (for comparison)

Task: Determine the asymptotic time complexity for:

1. A single layer of self-attention: $O(N^2 \cdot D)$
2. A standard convolution layer: $O(N \cdot D^2 \cdot K^2)$

Return a tuple of strings representing both complexities.