  The computational intensity of Large Language Models (LLMs) is a critical
bottleneck, primarily due to the $O(n^2)$ complexity of the attention mechanism
in transformer architectures. Addressing this, sparse attention emerges as a
key innovation, aiming to reduce computational load while maintaining model
performance. This study presents a rigorous theoretical analysis of the
sparsity in attention scores within LLMs, particularly under the framework of
Gaussian inputs. By establishing a set of foundational assumptions and
employing a methodical theoretical approach, we unravel the intrinsic
characteristics of attention score sparsity and its implications on
computational efficiency. Our main contribution lies in providing a detailed
theoretical examination of how sparsity manifests in attention mechanisms,
offering insights into the potential trade-offs between computational savings
and model effectiveness. This work not only advances our understanding of
sparse attention but also provides a scaffold for future research in optimizing
the computational frameworks of LLMs, paving the way for more scalable and
efficient AI systems.
