# PaTH Attention: Position Encoding via Accumulating Householder Transformations

arXiv: https://arxiv.org/abs/2505.16381 · HTML: https://mlsanity.com/paper/2505.16381/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- Builds on 69 of its 91 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 30 of 69)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [LongBench: A Bilingual, Multitask Benchmark for Long Context Understanding](https://mlsanity.com/paper/2308.14508/) (2023)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [Extending Context Window of Large Language Models via Positional Interpolation](https://mlsanity.com/paper/2306.15595/) (2023)
- [Compressive Transformers for Long-Range Sequence Modelling](https://mlsanity.com/paper/1911.05507/) (2019)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [The NarrativeQA Reading Comprehension Challenge](https://mlsanity.com/paper/1712.07040/) (2017)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [FlashAttention-3: Fast and Accurate Attention with Asynchrony and Low-precision](https://mlsanity.com/paper/2407.08608/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Ring Attention with Blockwise Transformers for Near-Infinite Context](https://mlsanity.com/paper/2310.01889/) (2023)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Repeat After Me: Transformers are Better than State Space Models at Copying](https://mlsanity.com/paper/2402.01032/) (2024)
- [BABILong: Testing the Limits of LLMs with Long Context Reasoning-in-a-Haystack](https://mlsanity.com/paper/2406.10149/) (2024)

## Built on by (most cited first, 30 of 34)

- [Kimi Linear: An Expressive, Efficient Attention Architecture](https://mlsanity.com/paper/2510.26692/) (2025)
- [Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models](https://mlsanity.com/paper/2601.07372/) (2026)
- [Comba: Improving Bilinear RNNs with Closed-loop Control](https://mlsanity.com/paper/2506.02475/) (2025)
- [Selective Rotary Position Embedding](https://mlsanity.com/paper/2511.17388/) (2025)
- [Deep Delta Learning](https://mlsanity.com/paper/2601.00417/) (2026)
- [Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria](https://mlsanity.com/paper/2508.07102/) (2025)
- [Group Representational Position Encoding](https://mlsanity.com/paper/2512.07805/) (2025)
- [RePo: Language Models with Context Re-Positioning](https://mlsanity.com/paper/2512.14391/) (2025)
- [HyperMLP: An Integrated Perspective for Sequence Modeling](https://mlsanity.com/paper/2602.12601/) (2026)
- [The Topological Trouble With Transformers](https://mlsanity.com/paper/2604.17121/) (2026)
- [Norm×Direction: Restoring the Missing Query Norm in Vision Linear Attention](https://mlsanity.com/paper/2506.21137/) (2025)
- [Beyond Real: Imaginary Extension of Rotary Position Embeddings for Long-Context LLMs](https://mlsanity.com/paper/2512.07525/) (2025)
- [Design Principles for Sequence Models via Coefficient Dynamics](https://mlsanity.com/paper/2510.09389/) (2025)
- [Rational Transductors](https://mlsanity.com/paper/2602.07599/) (2026)
- [LUCID: Attention with Preconditioned Representations](https://mlsanity.com/paper/2602.10410/) (2026)
- [Why Depth Matters in Parallelizable Sequence Models: A Lie Algebraic View](https://mlsanity.com/paper/2603.05573/) (2026)
- [When Good Enough Is Optimal: Multiplication-Only Matrix Inversion Approximation for Quantized Gated DeltaNet](https://mlsanity.com/paper/2606.06034/) (2026)
- [Comparing Transformers and Hybrid Models at the Token Level](https://mlsanity.com/paper/2606.20936/) (2026)
- [LeRoPE: Learnable RoPE Frequencies Improve Language Modeling](https://mlsanity.com/paper/2607.10134/) (2026)
- [Power law graph attention: exact generalization of scaled dot-product attention, empirical collapse at inference](https://mlsanity.com/paper/2608.10288/) (2026)
- [Kalman Delta Networks: Uncertainty-aware Associative Memory](https://mlsanity.com/paper/2609.07816/) (2026)
- [Causal Attention with Lookahead Keys](https://mlsanity.com/paper/2509.07301/) (2025)
- [An empirical study on the limitation of Transformers in program trace generation](https://mlsanity.com/paper/2509.25073/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [MapFormer: Self-Supervised Learning of Cognitive Maps with Input-Dependent Positional Embeddings](https://mlsanity.com/paper/2511.19279/) (2025)
- [The Transformer as a Polar State Estimator](https://mlsanity.com/paper/2605.11007/) (2026)
- [Long-Context Demonstration Selection Using State Space Models](https://mlsanity.com/paper/2609.17888/) (2026)
- [Length-Independent State Tracking Under a Parallel Scan](https://mlsanity.com/paper/2609.32447/) (2026)
- [RoPE is Dead, Long Live RoPE: Towards Scalable Data-aware Positional Encodings](https://mlsanity.com/paper/2609.34556/) (2026)
- [LLMs are General Asynchronous Agents](https://mlsanity.com/paper/2609.35427/) (2026)

## Abstract (arXiv)

The attention mechanism is a core primitive in modern large language models (LLMs) and AI more broadly. Since attention by itself is permutation-invariant, position encoding is essential for modeling structured domains such as language. Rotary position encoding (RoPE) has emerged as the de facto standard approach for position encoding and is part of many modern LLMs. However, in RoPE the key/query transformation between two elements in a sequence is only a function of their relative position and otherwise independent of the actual input. This limits the expressivity of RoPE-based transformers. This paper describes PaTH, a flexible data-dependent position encoding scheme based on accumulated products of Householder(like) transformations, where each transformation is data-dependent, i.e., a function of the input. We derive an efficient parallel algorithm for training through exploiting a compact representation of products of Householder matrices, and implement a FlashAttention-style blockwise algorithm. Across both targeted synthetic benchmarks and moderate-scale real-world language modeling experiments, we find that PaTH improves upon RoPE and other recent baselines. Finally, we show that we can convert pretrained RoPE transformers into PaTH with continued pretraining.