Transformers with linear attention (i.e., linear transfor mers) and state-space models have recently been suggested as a viable linear-time alt ernative to transformers with softmax attention. However, these models still underp erform transformers especially on tasks that require in-context retrieval. Whil e more expressive variants of linear transformers which replace the additive upda te in linear transformers with the delta rule [DeltaNet; 101 ] have been found to be more effective at associative recall, existing algorithms for training such mode ls do not parallelize over sequence length and are thus inefficient to train on modern ha rdware. This work describes a hardware-efficient algorithm for training line ar transformers with the delta rule, which exploits a memory-efficient representati on for computing products of Householder matrices [ 11 ]. This algorithm allows us to scale up DeltaNet to standard language modeling settings. We train a 1.3B mode l for 100B tokens and find that it outperforms recent linear-time baselines su ch as Mamba [ 31 ] and GLA [ 124 ] in terms of perplexity and zero-shot performance on downst ream tasks. We also experiment with two hybrid models which combine Delt aNet layers with (1) sliding-window attention layers every other layer or (2) two global attention layers, and find that these hybrids outperform strong transf ormer baselines.

In this paper, we propose QT -ViT models that improve the previous linear self-attention using quadratic Taylor expansion.

Standard attention stores keys/values losslessly but reads them via a per-head convex average, blocking channel-wise selection. We propose the Free Energy Mixer (FEM): a free-energy (log-sum-exp) read that applies a value-driven, per-channel log-linear tilt to a fast prior (e.g., from queries/keys in standard attention) over indices. Unlike methods that attempt to improve and enrich the $(q,k)$ scoring distribution, FEM treats it as a prior and yields a value-aware posterior read at unchanged complexity, smoothly moving from averaging to per-channel selection as the learnable inverse temperature increases, while still preserving parallelism and the original asymptotic complexity ($O(T^2)$ for softmax; $O(T)$ for linearizable variants). We instantiate a two-level gated FEM that is plug-and-play with standard and linear attention, linear RNNs and SSMs. It consistently outperforms strong baselines on NLP, vision, and time-series at matched parameter budgets.

Linear attention methods offer Transformers $O(N)$ complexity but typically underperform standard softmax attention. We identify two fundamental limitations affecting these approaches: the restriction to convex combinations that only permits additive information blending, and uniform accumulated weight bias that dilutes attention in long contexts. We propose Zero-Sum Linear Attention (ZeroS), which addresses these limitations by removing the constant zero-order term $1/t$ and reweighting the remaining zero-sum softmax residuals. This modification creates mathematically stable weights, enabling both positive and negative values and allowing a single attention layer to perform contrastive operations. While maintaining $O(N)$ complexity, ZeroS theoretically expands the set of representable functions compared to convex combinations. Empirically, it matches or exceeds standard softmax attention across various sequence modeling benchmarks.

Recent research has shown that Transformers with linear attention are capable of in-context learning (ICL) by implementing a linear estimator through gradient descent steps. However, the existing results on the optimization landscape apply under stylized settings where task and feature vectors are assumed to be IID and the attention weights are fully parameterized. In this work, we develop a stronger characterization of the optimization and generalization landscape of ICL through contributions on architectures, low-rank parameterization, and correlated designs: (1) We study the landscape of 1-layer linear attention and 1-layer H3, a state-space model. Under a suitable correlated design assumption, we prove that both implement 1-step preconditioned gradient descent. We show that thanks to its native convolution filters, H3 also has the advantage of implementing sample weighting and outperforming linear attention in suitable settings.

Vision transformer model (ViT) is widely used and performs well in vision tasks due to its ability to capture long-range dependencies. However, the time complexity and memory consumption increase quadratically with the number of input patches which limits the usage of ViT in real-world applications. Previous methods have employed linear attention to mitigate the complexity of the original self-attention mechanism at the expense of effectiveness. In this paper, we propose QT-ViT models that improve the previous linear self-attention using quadratic Taylor expansion. Specifically, we substitute the softmax-based attention with second-order Taylor expansion, and then accelerate the quadratic expansion by reducing the time complexity with a fast approximation algorithm. The proposed method capitalizes on the property of quadratic expansion to achieve superior performance while employing linear approximation for fast inference. Compared to previous studies of linear attention, our approach does not necessitate knowledge distillation or high-order attention residuals to facilitate the training process. Extensive experiments demonstrate the efficiency and effectiveness of the proposed QT-ViTs, showcasing the state-of-the-art results. Particularly, the proposed QT-ViTs consistently surpass the previous SOTA EfficientViTs under different model sizes, and achieve a new Pareto-front in terms of accuracy and speed.

Widely adopted in modern Vision Transformer designs, Softmax attention can effectively capture long-range visual information; however, it incurs excessive computational cost when dealing with high-resolution inputs. In contrast, linear attention naturally enjoys linear complexity and has great potential to scale up to higher-resolution images. Nonetheless, the unsatisfactory performance of linear attention greatly limits its practical application in various scenarios. In this paper, we take a step forward to close the gap between the linear and Softmax attention with novel theoretical analyses, which demystify the core factors behind the performance deviations. Specifically, we present two key perspectives to understand and alleviate the limitations of linear attention: the injective property and the local modeling ability. Firstly, we prove that linear attention is not injective, which is prone to assign identical attention weights to different query vectors, thus adding to severe semantic confusion since different queries correspond to the same outputs. Secondly, we confirm that effective local modeling is essential for the success of Softmax attention, in which linear attention falls short. The aforementioned two fundamental differences significantly contribute to the disparities between these two attention paradigms, which is demonstrated by our substantial empirical validation in the paper. In addition, more experiment results indicate that linear attention, as long as endowed with these two properties, can outperform Softmax attention across various tasks while maintaining lower computation complexity.