# Block-Sparse Attention with Semantic-Geometric Decoupled Routing

arXiv: https://arxiv.org/abs/2609.22884 · HTML: https://mlsanity.com/paper/2609.22884/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 46 references that are in the map.
- First posted on arXiv 19 Sep 2026.

## Builds on (most cited first, 30 of 36)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [LongBench: A Bilingual, Multitask Benchmark for Long Context Understanding](https://mlsanity.com/paper/2308.14508/) (2023)
- [Video-MME: The First-Ever Comprehensive Evaluation Benchmark of Multi-modal LLMs in Video Analysis](https://mlsanity.com/paper/2405.21075/) (2024)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [SnapKV: LLM Knows What You are Looking for Before Generation](https://mlsanity.com/paper/2404.14469/) (2024)
- [Compressive Transformers for Long-Range Sequence Modelling](https://mlsanity.com/paper/1911.05507/) (2019)
- [LongVideoBench: A Benchmark for Long-context Interleaved Video-Language Understanding](https://mlsanity.com/paper/2407.15754/) (2024)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Quest: Query-Aware Sparsity for Efficient Long-Context LLM Inference](https://mlsanity.com/paper/2406.10774/) (2024)
- [Ego-Exo4D: Understanding Skilled Human Activity from First- and Third-Person Perspectives](https://mlsanity.com/paper/2311.18259/) (2023)
- [MInference 1.0: Accelerating Pre-filling for Long-Context LLMs via Dynamic Sparse Attention](https://mlsanity.com/paper/2407.02490/) (2024)
- [LongBench v2: Towards Deeper Understanding and Reasoning on Realistic Long-context Multitasks](https://mlsanity.com/paper/2412.15204/) (2024)
- [MoBA: Mixture of Block Attention for Long-Context LLMs](https://mlsanity.com/paper/2502.13189/) (2025)
- [XAttention: Block Sparse Attention with Antidiagonal Scoring](https://mlsanity.com/paper/2503.16428/) (2025)
- [FlexPrefill: A Context-Aware Sparse Attention Mechanism for Efficient Long-Sequence Inference](https://mlsanity.com/paper/2502.20766/) (2025)
- [SpargeAttention: Accurate and Training-free Sparse Attention Accelerating Any Model Inference](https://mlsanity.com/paper/2502.18137/) (2025)
- [mPLUG-DocOwl2: High-resolution Compressing for OCR-free Multi-page Document Understanding](https://mlsanity.com/paper/2409.03420/) (2024)
- [LazyLLM: Dynamic Token Pruning for Efficient Long Context LLM Inference](https://mlsanity.com/paper/2407.14057/) (2024)
- [GLM-5V-Turbo: Toward a Native Foundation Model for Multimodal Agents](https://mlsanity.com/paper/2604.26752/) (2026)
- [Various Lengths, Constant Speed: Efficient Language Modeling with Lightning Attention](https://mlsanity.com/paper/2405.17381/) (2024)
- [IndexCache: Accelerating Sparse Attention via Cross-Layer Index Reuse](https://mlsanity.com/paper/2603.12201/) (2026)
- [The Sparse Frontier: Sparse Attention Trade-offs in Transformer LLMs](https://mlsanity.com/paper/2504.17768/) (2025)
- [Docopilot: Improving Multimodal Models for Document-Level Understanding](https://mlsanity.com/paper/2507.14675/) (2025)
- [HISA: Efficient Hierarchical Indexing for Fine-Grained Sparse Attention](https://mlsanity.com/paper/2603.28458/) (2026)
- [ResidualKV: Residual-Based KV Cache Compression for Efficient Long-Context Inference](https://mlsanity.com/paper/2602.08005/) (2026)
- [MISA: Mixture of Indexer Sparse Attention for Long-Context LLM Inference](https://mlsanity.com/paper/2605.07363/) (2026)

## Abstract (arXiv)

Long-context inference has become a defining capability of large language models, but exact dense attention remains costly due to its quadratic scaling with sequence length. Block-sparse attention offers a hardware-friendly alternative by routing each query block to a small set of relevant key blocks, yet accurate training-free block routing remains difficult. Existing routers often pool post-RoPE token representations, which entangles semantic aggregation with RoPE-induced geometry and attenuates local positional cues through high-frequency phase cancellation. To resolve this mismatch, we propose Semantic-Geometric Decoupled Routing, a training-free block routing framework that shifts semantic aggregation to the pre-RoPE space and reconstructs geometric bias with an offline structural prior and relative block distances. This decomposition yields an explicit closed-form block routing score without token-level search or post-hoc calibration. Experiments on long-context text and video tasks show that our method approaches full-attention accuracy across 4K--128K contexts, keeps routing overhead below 3.4 ms, and achieves a 5.03× speedup over FlashAttn at a 128K context length.