Linear attention has attracted interest as a computationally efficient approximation to softmax attention, especially for long sequences. Recent studies have explored distilling softmax attention in pre-trained Transformers into linear attention. However, a critical challenge remains: how to choose the feature dimension that governs the approximation quality. Existing methods fix this dimension uniformly across all attention layers, overlooking the diverse roles and complexities of them. In this paper, we propose a principled method to automatically determine the feature dimension in linear attention using the concept of statistical degrees of freedom, which represent the effective dimensionality of the inputs. We provide a theoretical bound on the approximation error and show that the dimension chosen by our method achieves smaller errors under a fixed computational budget. Furthermore, we introduce an efficient layerwise training strategy to learn nonlinear features tailored to each layer. Experiments on multiple pre-trained transformers demonstrate that our method improves the performance of distilled models compared to baselines without increasing the inference cost. Our findings also provide insight into how the complexity of the attention mechanism evolves across layers.

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. The code implementation is available at this link.

Transformers have achieved state-of-the-art performance across various tasks, but suffer from a notable quadratic complexity in sequence length due to the attention mechanism. In this work, we propose MonarchAttention-a novel approach to sub-quadratic attention approximation via Monarch matrices, an expressive class of structured matrices. Based on the variational form of softmax, we describe an efficient optimization-based algorithm to compute an approximate projection of softmax attention onto the class of Monarch matrices with Θ(N Nd) computational complexity and Θ(Nd)memory/IO complexity.

Transformers and their attention mechanism have been revolutionary in the field of Machine Learning. While originally proposed for the language data, they quickly found their way to the image, video, graph, etc. data modalities with various signal geometries. Despite this versatility, generalizing the attention mechanism to scenarios where data is presented at different scales from potentially different modalities is not straightforward. The attempts to incorporate hierarchy and multimodality within transformers are largely based on ad hoc heuristics, which are not seamlessly generalizable to similar problems with potentially different structures. To address this problem, in this paper, we take a fundamentally different approach: we first propose a mathematical construct to represent multi-modal, multi-scale data. We then mathematically derive the neural attention mechanics for the proposed construct from the first principle of entropy minimization. We show that the derived formulation is optimal in the sense of being the closest to the standard Softmax attention while incorporating the inductive biases originating from the hierarchical/geometric information of the problem. We further propose an efficient algorithm based on dynamic programming to compute our derived attention mechanism. By incorporating it within transformers, we show that the proposed hierarchical attention mechanism not only can be employed to train transformer models in hierarchical/multi-modal settings from scratch, but it can also be used to inject hierarchical information into classical, pre-trained transformer models post training, resulting in more efficient models in zero-shot manner.

Gating mechanisms have been widely utilized, from early models like LSTMs and Highway Networks to recent state space models, linear attention, and also softmax attention. Yet, existing literature rarely examines the specific effects of gating. In this work, we conduct comprehensive experiments to systematically investigate gating-augmented softmax attention variants. Specifically, we perform a comprehensive comparison over 30 variants of 15B Mixture-of-Experts (MoE) models and 1.7B dense models trained on a 3.5 trillion token dataset. Our central finding is that a simple modification--applying a head-specific sigmoid gate after the Scaled Dot-Product Attention (SDPA)--consistently improves performance. This modification also enhances training stability, tolerates larger learning rates, and improves scaling properties. By comparing various gating positions and computational variants, we attribute this effectiveness to two key factors: (1) introducing non-linearity upon the low-rank mapping in the softmax attention, and (2) applying query-dependent sparse gating scores to modulate the SDPA output. Notably, we find this sparse gating mechanism mitigates, and enhances long-context extrapolation performance. We also release related codes (https://github.com/qiuzh20/gated

Transformers have achieved state-of-the-art performance across various tasks, but suffer from a notable quadratic complexity in sequence length due to the attention mechanism. In this work, we propose MonarchAttention -- a novel approach to sub-quadratic attention approximation via Monarch matrices, an expressive class of structured matrices. Based on the variational form of softmax, we describe an efficient optimization-based algorithm to compute an approximate projection of softmax attention onto the class of Monarch matrices with $\Theta(N\sqrt{N} d)$ computational complexity and $\Theta(Nd)$ memory/IO complexity. Unlike previous approaches, MonarchAttention is both (1) transferable, yielding minimal performance loss with no additional training, even when replacing every attention layer of the transformer, and (2) hardware-efficient, utilizing the highest-throughput tensor core units on modern GPUs. With optimized kernels, MonarchAttention achieves substantial speed-ups in wall-time over FlashAttention-2: $1.4\times$ for shorter sequences $(N=256)$, $4.5\times$ for medium-length sequences $(N=4K)$, and $8.2\times$ for longer sequences $(N=16K)$. We demonstrate the quality of MonarchAttention on diverse tasks and architectures in vision and language problems, showing that it flexibly and accurately approximates softmax attention in a variety of contexts.

Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation.

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.

Various linear complexity models, such as Linear Transformer (LinFormer), State Space Model (SSM), and Linear RNN (LinRNN), have been proposed to replace the conventional softmax attention in Transformer structures. However, the optimal design of these linear models is still an open question. In this work, we attempt to answer this question by finding the best linear approximation to softmax attention from a theoretical perspective. We start by unifying existing linear complexity models as the linear attention form and then identify three conditions for the optimal linear attention design: (1) Dynamic memory ability; (2) Static approximation ability; (3) Least parameter approximation. We find that none of the current linear models meet all three conditions, resulting in suboptimal performance. Instead, we propose Meta Linear Attention (MetaLA) as a solution that satisfies these conditions. Our experiments on Multi-Query Associative Recall (MQAR) task, language modeling, image classification, and Long-Range Arena (LRA) benchmark demonstrate that MetaLA is more effective than the existing linear models.