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Looped Transformers (LT) have emerged as a powerful architecture by iterating their layers multiple times before decoding the final token. However, pairing them with full attention retains quadratic complexity, making them computationally…

Machine Learning · Computer Science 2026-05-26 Chunyuan Deng , Yizhe Zhang , Rui-Jie Zhu , Yuanyuan Xu , Jiarui Liu , T. S. Eugene Ng , Hanjie Chen

Softmax is popular normalization method used in machine learning. Deep learning solutions like Transformer or BERT use the softmax function intensively, so it is worthwhile to optimize its performance. This article presents our methodology…

Mathematical Software · Computer Science 2019-05-28 Jacek Czaja , Michal Gallus , Tomasz Patejko , Jian Tang

Transformers have achieved significant success across various domains, relying on self-attention to capture dependencies. However, the standard first-order attention mechanism is often limited by a low-rank bottleneck, struggling to capture…

Computation and Language · Computer Science 2025-12-05 Hanting Chen , Chong Zhu , Kai Han , Yuchuan Tian , Yuchen Liang , Tianyu Guo , Xinghao Chen , Dacheng Tao , Yunhe Wang

The quadratic complexity of the attention mechanism represents one of the biggest hurdles for processing long sequences using Transformers. Current methods, relying on sparse representations or stateful recurrence, sacrifice token-to-token…

Machine Learning · Computer Science 2025-06-06 Tobias Christian Nauen , Sebastian Palacio , Andreas Dengel

Softmax attention is a central component of transformer architectures, yet its nonlinear structure poses significant challenges for theoretical analysis. We develop a unified, measure-based framework for studying single-layer softmax…

Machine Learning · Computer Science 2025-12-15 Etienne Boursier , Claire Boyer

Large language models have achieved remarkable success in recent years, primarily due to self-attention. However, traditional Softmax attention suffers from numerical instability and reduced performance as the number of inference tokens…

Computation and Language · Computer Science 2026-02-02 Bo Gao , Michael W. Spratling , Letizia Gionfrida

Vision transformers (ViTs) have pushed the state-of-the-art for visual perception tasks. The self-attention mechanism underpinning the strength of ViTs has a quadratic complexity in both computation and memory usage. This motivates the…

Computer Vision and Pattern Recognition · Computer Science 2024-03-18 Jiachen Lu , Junge Zhang , Xiatian Zhu , Jianfeng Feng , Tao Xiang , Li Zhang

Multimodal Transformers serve as the backbone for state-of-the-art vision-language models, yet their quadratic attention complexity remains a critical barrier to scalability. In this work, we investigate the viability of Linear Attention…

Computer Vision and Pattern Recognition · Computer Science 2026-04-14 Armin Gerami , Seyedehanita Madani , Ramani Duraiswami

Large language models (LLMs) have brought about significant transformations in human society. Among the crucial computations in LLMs, the softmax unit holds great importance. Its helps the model generating a probability distribution on…

Machine Learning · Computer Science 2023-07-18 Yichuan Deng , Zhihang Li , Sridhar Mahadevan , Zhao Song

Linear Transformers and State Space Models have emerged as efficient alternatives to softmax Transformers for causal sequence modeling, enabling parallel training via matrix multiplication and efficient RNN-style inference. However, despite…

Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as…

Computation and Language · Computer Science 2026-05-14 Albert Alcalde , Giovanni Fantuzzi , Enrique Zuazua

Transformer networks have seen great success in natural language processing and machine vision, where task objectives such as next word prediction and image classification benefit from nuanced context sensitivity across high-dimensional…

Machine Learning · Computer Science 2022-12-13 Yuxuan Li , James L. McClelland

We show the formal equivalence of linearised self-attention mechanisms and fast weight controllers from the early '90s, where a ``slow" neural net learns by gradient descent to program the ``fast weights" of another net through sequences of…

Machine Learning · Computer Science 2021-06-10 Imanol Schlag , Kazuki Irie , Jürgen Schmidhuber

We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the…

Machine Learning · Computer Science 2024-10-31 Mingze Wang , Weinan E

While the Transformer architecture dominates many fields, its quadratic self-attention complexity hinders its use in large-scale applications. Linear attention offers an efficient alternative, but its direct application often degrades…

Computer Vision and Pattern Recognition · Computer Science 2026-01-15 Kewei Zhang , Ye Huang , Yufan Deng , Jincheng Yu , Junsong Chen , Huan Ling , Enze Xie , Daquan Zhou

Vision Transformers have achieved impressive performance in video classification, while suffering from the quadratic complexity caused by the Softmax attention mechanism. Some studies alleviate the computational costs by reducing the number…

Computer Vision and Pattern Recognition · Computer Science 2022-10-18 Kaiyue Lu , Zexiang Liu , Jianyuan Wang , Weixuan Sun , Zhen Qin , Dong Li , Xuyang Shen , Hui Deng , Xiaodong Han , Yuchao Dai , Yiran Zhong

Position modeling plays a critical role in Transformers. In this paper, we focus on length extrapolation, i.e., training on short texts while evaluating longer sequences. We define attention resolution as an indicator of extrapolation. Then…

Computation and Language · Computer Science 2022-12-21 Yutao Sun , Li Dong , Barun Patra , Shuming Ma , Shaohan Huang , Alon Benhaim , Vishrav Chaudhary , Xia Song , Furu Wei

Transformers with linear attention (i.e., linear transformers) and state-space models have recently been suggested as a viable linear-time alternative to transformers with softmax attention. However, these models still underperform…

Machine Learning · Computer Science 2025-01-16 Songlin Yang , Bailin Wang , Yu Zhang , Yikang Shen , Yoon Kim

Transformers with linearised attention (''linear Transformers'') have demonstrated the practical scalability and effectiveness of outer product-based Fast Weight Programmers (FWPs) from the '90s. However, the original FWP formulation is…

Machine Learning · Computer Science 2021-10-28 Kazuki Irie , Imanol Schlag , Róbert Csordás , Jürgen Schmidhuber

Transformer attention is typically implemented using softmax normalization, which enforces attention weights with unit sum normalization. While effective in many settings, this constraint can limit flexibility in controlling attention…

Computation and Language · Computer Science 2026-02-27 Jeongin Bae , Baeseong Park , Gunho Park , Minsub Kim , Joonhyung Lee , Junhee Yoo , Sunghyeon Woo , Jiwon Ryu , Se Jung Kwon , Dongsoo Lee