Transformer-based large models have demonstrated remarkable generalization abilities across different tasks by leveraging a context-aware attention module for in-context learning. With richer context, transformers adapt more effectively to the current use case without any parameter updates. However, the quadratic computational and memory complexity with respect to context length significantly slows data processing in softmax transformers. Linear transformers were proposed to address this issue by reducing the complexity to linear dependence on context length, but the design and understanding of the feature mapping in linear attention, from a theoretical viewpoint, remain unclear. In this paper, we investigate the approximation and generalization abilities of linear transformers under a two-staged sampling process from domain generalization. We show that linear transformers perform in-context learning as learning a mapping from context distributions to response functions. A dimension-independent convergence rate is obtained for our generalization analysis, which also exhibits the tradeoff between the regularities of data distributions and latent features. Guided by our theoretical framework, we propose a new perspective on activation and loss design for linearizing pretrained softmax large language models.

Attention mechanisms have achieved significant empirical success in multiple fields, but their underlying optimization objectives remain unclear yet. Moreover, the quadratic complexity of self-attention has become increasingly prohibitive. Although interpretability and efficiency are two mutually reinforcing pursuits, prior work typically investigates them separately. In this paper, we propose a unified optimization objective that derives inherently interpretable and efficient attention mechanisms through algorithm unrolling. Precisely, we construct a gradient step of the proposed objective with a set of forward-pass operations of our Contractand-Broadcast Self-Attention (CBSA), which compresses input tokens towards low-dimensional structures by contracting a few representatives of them. This novel mechanism can not only scale linearly by fixing the number of representatives, but also covers the instantiations of varied attention mechanisms when using different sets of representatives. We conduct extensive experiments to demonstrate comparable performance and superior advantages over black-box attention mechanisms on visual tasks.

Previous research has explored the expressivity of Transformer models in simulating Boolean circuits or Turing machines. However, the efficient learnability of Transformers from data has remained an open question.

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 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 their success in causal tasks, no unified framework exists for applying Linear Transformers to bidirectional sequence modeling. We introduce LION, the first framework to systematically extend Linear Transformers to the bidirectional setting. LION generalizes three core representations commonly used in the causal case--full Linear Attention, bidirectional RNN, and chunkwise parallel form--to the bidirectional setting. These forms are theoretically equivalent and enable models to exploit the strengths of each during training and inference. We prove that a broad class of Linear Transformers can be extended using LION and validate our framework via three core examples based on the choice of decay type: LION-LIT, the bidirectional extension of [25]; LION-D, based on [44]; and LION-S, a variant using selective decay [34, 13]. Across standard bidirectional tasks, LION enables models to match or exceed the performance of softmax Transformers, while offering significantly faster training and more efficient inference than existing State Space Models.

Many real-world graphs exhibit diverse and complex topological structures that are not well captured by geometric manifolds with uniform global curvature, such as hyperbolic or spherical spaces. Recently, there has been growing interest in embedding graphs into pseudo-Riemannian manifolds, which generalize both hyperbolic and spherical geometries. However, existing approaches face three significant limitations, including the ineffective pseudo-Riemannain framework, the shallow architectures, and the absence of clear guideline for selecting suitable pseudo-Riemannian manifolds. To address these issues, we introduce a novel diffeomorphic framework for graph embedding that aligns with the nature of pseudo-Riemannian manifolds. Subsequently, we propose the pseudo-Riemannian Graph Transformer for learning representations of complex graph structures. Our diffeomorphic framework in pseudo-Riemannian geometry enables the principled definitions of core transformer components, including linear attention, residual connection, and layer normalization. Finally, we develop a lightweight space searching algorithm to automatically identify the most suitable pseudo-Riemannian manifold for an input graph. Extensive experiments on diverse real-world graphs demonstrate that our model consistently outperforms other baselines in both node classification and link prediction tasks.

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.

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 their success in causal tasks, no unified framework exists for applying Linear Transformers to bidirectional sequence modeling. We introduce LION, the first framework to systematically extend Linear Transformers to the bidirectional setting. LION generalizes three core representations commonly used in the causal case--full Linear Attention, bidirectional RNN, and chunkwise parallel form--to the bidirectional setting. These forms are theoretically equivalent and enable models to exploit the strengths of each during training and inference. We prove that a broad class of Linear Transformers can be extended using LION and validate our framework via three core examples based on the choice of decay type: LION-LIT, the bidirectional extension of [25]; LION-D, based on [44]; and LION-S, a variant using selective decay [34, 13]. Across standard bidirectional tasks, LION enables models to match or exceed the performance of softmax Transformers, while offering significantly faster training and more efficient inference than existing State Space Models.

Linear RNNs with gating recently demonstrated competitive performance compared to Transformers in language modeling. Although their linear compute scaling in sequence length offers theoretical runtime advantages over Transformers, realizing these benefits in practice requires optimized custom kernels, as Transformers rely on the highly efficient Flash Attention kernels (Dao, 2024). Leveraging the chunkwise-parallel formulation of linear RNNs, Flash Linear Attention (FLA) (Yang & Zhang, 2024) shows that linear RNN kernels are faster than Flash Attention, by parallelizing over chunks of the input sequence. However, since the chunk size of FLA is limited, many intermediate states must be materialized in GPU memory. This leads to low arithmetic intensity and causes high memory consumption and IO cost, especially for long-context pre-training. In this work, we present Tiled Flash Linear Attention (TFLA), a novel kernel algorithm for linear RNNs, that enables arbitrary large chunk sizes and high arithmetic intensity by introducing an additional level of sequence parallelization within each chunk. First, we apply TFLA to the xLSTM with matrix memory, the mLSTM (Beck et al., 2024). Second, we propose an mLSTM variant with sigmoid input gate and reduced computation for even faster kernel runtimes at equal language modeling performance. In our speed benchmarks, we show that our new mLSTM kernels based on TFLA outperform highly optimized Flash Attention, Linear Attention and Mamba kernels, setting a new state of the art for efficient long-context sequence modeling primitives.

Understanding architectural differences in language models is challenging, especially at academic-scale pretraining (e.g., 1.3B parameters, 100B tokens), where results are often dominated by noise and randomness. To overcome this, we introduce controlled synthetic pretraining tasks that isolate and evaluate core model capabilities. Within this framework, we discover \emph{Canon layers}: lightweight architectural components--named after the musical term ``canon''--that promote horizontal information flow across neighboring tokens. Canon layers compute weighted sums of nearby token representations and integrate seamlessly into Transformers, linear attention, state-space models, or any sequence architecture.