LaplacianFormer:Rethinking Linear Attention with Laplacian Kernel

This paper introduces Exact Linear Attention (ELA), a mechanism that achieves linear computational complexity for Transformer attention by exploiting the exact decomposition property of kernel functions, thereby eliminating approximation error. We identify and address two key limitations of prior linear attention -- gradient explosion and token attention dilution -- by imposing kernel constraints that ensure non-negativity, discriminability, and geometric interpretability. Several kernel functions are proposed, including the Hadamard Exp Kernel, Summation Squared Euclidean Distance Kernel, and Subtraction Squared Euclidean Distance Kernel, each tailored for specific attention behaviors. Beyond the core attention formulation, the paper presents three engineering innovations: (1) a Hyper-Link structure that replaces traditional residual connections to mitigate gradient degradation; (2) a Memory Lobe module based on bidirectional linear attention, which captures "transformation flow" across layers to implement qualitative memory and an implicit reinforcement learning paradigm; and (3) a routing-score-based bias mechanism for Mixture-of-Experts (MoE) to improve interpretability and semantic alignment. Experimental results demonstrate that ELA achieves up to 6x faster decoding speed and 75% reduction in KV cache memory usage compared to full attention, while maintaining comparable or superior training performance. The proposed memory module accelerates convergence and enhances generalization. Furthermore, we extend the linear attention principle to vision models, yielding YOLO-LAT, which attains up to 4.3x GPU inference speedup and 7.9x parameter reduction with competitive detection accuracy. These results underline the broad applicability of exact linear attention for scaling Transformer models to ultra-long sequences and efficient visual tasks.

Transformer models rely on attention mechanism to capture long-range dependencies but suffer from quadratic complexity, limiting their scalability to long sequences. Kernel-based linear attention reduces this complexity but typically relies on fixed or weakly learnable kernels, restricting expressiveness and performance. In this work, we propose Flexformer, a flexible linear Transformer that learns attention kernels in a fully data-driven manner. Flexformer builds on random Fourier feature-based linear attention and treats spectral frequencies as trainable parameters, enabling the model to learn a broad family of attention kernels. We develop both stationary and nonstationary variants, with the latter offering strictly greater expressiveness. Extensive experiments on language modeling and sequence classification demonstrate that Flexformer consistently outperforms baselines. Moreover, Flexformer can be effectively distilled from pretrained Transformers to recover softmax attention and exhibits strong kernel transferability across domains, achieving both high efficiency and competitive performance on long-sequence tasks.

Existing attention accelerators often trade exact softmax semantics, depend on fused Tensor Core kernels, or incur sequential depth that limits FP32 throughput on long sequences. We present \textbf{ELSA}, an algorithmic reformulation of online softmax attention that (i)~preserves exact softmax semantics in real arithmetic with a \emph{provable} $\mathcal{O}(u\log n)$ FP32 relative error bound; (ii)~casts the online softmax update as a prefix scan over an associative monoid $(m,S,W)$, yielding $O(n)$ extra memory and $O(\log n)$ parallel depth; and (iii)~is Tensor-Core independent, implemented in Triton and CUDA C++, and deployable as a \emph{drop-in replacement} requiring no retraining or weight modification. Unlike FlashAttention-2/3, which rely on HMMA/GMMA Tensor Core instructions and provide no compatible FP32 path, ELSA operates identically on A100s and resource-constrained edge devices such as Jetson TX2 -- making it the only hardware-agnostic exact-attention kernel that reduces parallel depth to $O(\log n)$ at full precision. On A100 FP32 benchmarks (1K--16K tokens), ELSA delivers $1.3$--$3.5\times$ speedup over memory-efficient SDPA and $1.97$--$2.27\times$ on BERT; on Jetson TX2, ELSA achieves $1.5$--$1.6\times$ over Math (64--900 tokens), with $17.8$--$20.2\%$ throughput gains under LLaMA-13B offloading at $\ge$32K. In FP16, ELSA approaches hardware-fused baselines at long sequences while retaining full FP32 capability, offering a unified kernel for high-precision inference across platforms. Our code and implementation are available at https://github.com/ming053l/ELSA.