# Training Tensor Attention Efficiently: From Cubic to Almost Linear Time

arXiv: https://arxiv.org/abs/2405.16411 · HTML: https://mlsanity.com/paper/2405.16411/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 60 of its 99 references that are in the map.
- First posted on arXiv 26 May 2024.

## Builds on (most cited first, 30 of 60)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [BLIP-2: Bootstrapping Language-Image Pre-training with Frozen Image Encoders and Large Language Models](https://mlsanity.com/paper/2301.12597/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Improved Baselines with Visual Instruction Tuning](https://mlsanity.com/paper/2310.03744/) (2023)
- [Flamingo: a Visual Language Model for Few-Shot Learning](https://mlsanity.com/paper/2204.14198/) (2022)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [MiniGPT-4: Enhancing Vision-Language Understanding with Advanced Large Language Models](https://mlsanity.com/paper/2304.10592/) (2023)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Qwen-VL: A Versatile Vision-Language Model for Understanding, Localization, Text Reading, and Beyond](https://mlsanity.com/paper/2308.12966/) (2023)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [A Survey on Evaluation of Large Language Models](https://mlsanity.com/paper/2307.03109/) (2023)
- [LLaMA-Adapter: Efficient Fine-tuning of Language Models with Zero-init Attention](https://mlsanity.com/paper/2303.16199/) (2023)
- [Large Language Models](https://mlsanity.com/paper/2307.05782/) (2023)
- [LLaMA-Adapter V2: Parameter-Efficient Visual Instruction Model](https://mlsanity.com/paper/2304.15010/) (2023)
- [YaRN: Efficient Context Window Extension of Large Language Models](https://mlsanity.com/paper/2309.00071/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [CamemBERT: a Tasty French Language Model](https://mlsanity.com/paper/1911.03894/) (2019)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [LongRoPE: Extending LLM Context Window Beyond 2 Million Tokens](https://mlsanity.com/paper/2402.13753/) (2024)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [LongLoRA: Efficient Fine-tuning of Long-Context Large Language Models](https://mlsanity.com/paper/2309.12307/) (2023)

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

- [Why Larger Language Models Do In-context Learning Differently?](https://mlsanity.com/paper/2405.19592/) (2024)
- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Do Large Language Models Have Compositional Ability? An Investigation into Limitations and Scalability](https://mlsanity.com/paper/2407.15720/) (2024)
- [Computational Limits of Low-Rank Adaptation (LoRA) Fine-Tuning for Transformer Models](https://mlsanity.com/paper/2406.03136/) (2024)
- [Circuit Complexity Bounds for RoPE-based Transformer Architecture](https://mlsanity.com/paper/2411.07602/) (2024)
- [Multi-Layer Transformers Gradient Can be Approximated in Almost Linear Time](https://mlsanity.com/paper/2408.13233/) (2024)
- [Bypassing the Exponential Dependency: Looped Transformers Efficiently Learn In-context by Multi-step Gradient Descent](https://mlsanity.com/paper/2410.11268/) (2024)
- [The Computational Limits of State-Space Models and Mamba via the Lens of Circuit Complexity](https://mlsanity.com/paper/2412.06148/) (2024)
- [A Tighter Complexity Analysis of SparseGPT](https://mlsanity.com/paper/2408.12151/) (2024)
- [Looped ReLU MLPs May Be All You Need as Practical Programmable Computers](https://mlsanity.com/paper/2410.09375/) (2024)
- [HSR-Enhanced Sparse Attention Acceleration](https://mlsanity.com/paper/2410.10165/) (2024)
- [Your Vision-Language Model Can't Even Count to 20: Exposing the Failures of VLMs in Compositional Counting](https://mlsanity.com/paper/2510.04401/) (2025)
- [Provable Differentially Private Computation of the Cross-Attention Mechanism](https://mlsanity.com/paper/2407.14717/) (2024)
- [Theoretical Constraints on the Expressive Power of mathsfRoPE-based Tensor Attention Transformers](https://mlsanity.com/paper/2412.18040/) (2024)
- [Video Latent Flow Matching: Optimal Polynomial Projections for Video Interpolation and Extrapolation](https://mlsanity.com/paper/2502.00500/) (2025)
- [Towards Infinite-Long Prefix in Transformer](https://mlsanity.com/paper/2406.14036/) (2024)
- [On Computational Limits of FlowAR Models: Expressivity and Efficiency](https://mlsanity.com/paper/2502.16490/) (2025)
- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [Modern Hopfield Networks Require Chain-of-Thought to Solve mathsfNC1-Hard Problems](https://mlsanity.com/paper/2412.05562/) (2024)
- [Fast RoPE Attention: Combining the Polynomial Method and Fast Fourier Transform](https://mlsanity.com/paper/2505.11892/) (2025)
- [Only Large Weights (And Not Skip Connections) Can Prevent the Perils of Rank Collapse](https://mlsanity.com/paper/2505.16284/) (2025)
- [Scaling Law Phenomena Across Regression Paradigms: Multiple and Kernel Approaches](https://mlsanity.com/paper/2503.01314/) (2025)
- [Differentially Private Kernel Density Estimation](https://mlsanity.com/paper/2409.01688/) (2024)
- [When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?](https://mlsanity.com/paper/2502.16912/) (2025)
- [Time and Memory Trade-off of KV-Cache Compression in Tensor Transformer Decoding](https://mlsanity.com/paper/2503.11108/) (2025)
- [Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiency](https://mlsanity.com/paper/2503.14076/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [Can Language Models Compose Skills In-Context?](https://mlsanity.com/paper/2510.22993/) (2025)
- [How Smoothing is N-simplicial Attention?](https://mlsanity.com/paper/2512.15600/) (2025)
- [When AI meets quantum information: A comprehensive review](https://mlsanity.com/paper/2607.00365/) (2026)

## Abstract (arXiv)

Tensor Attention, a multi-view attention that is able to capture high-order correlations among multiple modalities, can overcome the representational limitations of classical matrix attention. However, the O(n3) time complexity of tensor attention poses a significant obstacle to its utilization in transformers, where n is the input sequence length. In this work, we prove that the backward gradient of tensor attention training can be computed in almost linear time n1+o(1), the same complexity as its forward computation under the bounded entries assumption. We provide a closed-form solution for the gradient and propose a fast computation method utilizing polynomial approximation methods and tensor algebraic techniques. Furthermore, we prove the necessity and tightness of our assumption through hardness analysis, showing that slightly weakening it renders the gradient problem unsolvable in truly subcubic time. Our theoretical results establish the feasibility of efficient higher-order transformer training and may facilitate practical applications of tensor attention architectures.