Scaling language models to handle longer input sequences typically necessitates large key-value (KV) caches, resulting in substantial memory overhead during inference. In this paper, we propose Tensor Product Attention (TPA), a novel attention mechanism that uses tensor decompositions to represent queries, keys, and values compactly, substantially shrinking the KV cache size at inference time. By factorizing these representations into contextual low-rank components and seamlessly integrating with RoPE and any possible position encoding mechanisms, TPA achieves improved model quality alongside memory efficiency. Based on TPA, we introduce the Tensor ProducTATTenTion Transformer (T6), a new model architecture for sequence modeling. Through extensive empirical evaluation on language modeling tasks, we demonstrate that T6 surpasses or matches the performance of standard Transformer baselines, including Multi-Head Attention (MHA), Multi-Query Attention (MQA), Grouped-Query Attention (GQA), and Multi-Head Latent Attention (MLA) across various metrics, including perplexity and a range of established evaluation benchmarks. Notably, TPA's memory efficiency and computational efficiency at the decoding stage enable processing longer sequences under fixed resource constraints, addressing a critical scalability challenge in modern language models.

Associative memory models based on Hopfield networks and self-attention based on key-value mechanisms have been popular approaches in the study of memory mechanisms in deep learning. It has been pointed out that the state update rule of the modern Hopfield network (MHN) in the adiabatic approximation is in agreement with the self-attention layer of Transformer. In this paper, we go beyond this approximation and investigate the relationship between MHN and selfattention. Our results show that the correspondence between Hopfield networks and Transformers can be established in a more generalized form by adding a new variable, the hidden state derived from the MHN, to self-attention. This new attention mechanism, modern Hopfield attention (MHA), allows the inheritance of attention scores from the input layer of the Transformer to the output layer, which greatly improves the nature of attention weights. In particular, we show both theoretically and empirically that MHA hidden states significantly improve serious problem of deep Transformers known as rank collapse and token uniformity. We also confirm that MHA can systematically improve accuracy without adding training parameters to the Vision Transformer or GPT. Our results provide a new case in which Hopfield networks can be a useful perspective for improving the Transformer architecture.

Large language models (LLMs) frequently generate hallucinations--content that deviates from factual accuracy or provided context--posing challenges for diagnosis due to the complex interplay of underlying causes. This paper introduces a subsequence association framework to systematically trace and understand hallucinations. Our key insight is that hallucinations arise when dominant hallucinatory associations outweigh faithful ones. Through theoretical and empirical analyses, we demonstrate that decoder-only transformers effectively function as subsequence embedding models, with linear layers encoding input-output associations. We propose a tracing algorithm that identifies causal subsequences by analyzing hallucination probabilities across randomized input contexts. Experiments show our method outperforms standard attribution techniques in identifying hallucination causes and aligns with evidence from the model's training corpus. This work provides a unified perspective on hallucinations and a robust framework for their tracing and analysis.

Designing model architectures requires decisions such as selecting operators (e.g., attention, convolution) and configurations (e.g., depth, width). However, evaluating the impact of these decisions on model quality requires costly pretraining, limiting architectural investigation. Inspired by how new software is built on existing code, we ask: can new architecture designs be studied using pretrained models? To this end, we present grafting, a simple approach for editing pretrained diffusion transformers (DiTs) to materialize new architectures under small compute budgets. Informed by our analysis of activation behavior and attention locality, we construct a testbed based on the DiT-XL/2 design to study the impact of grafting on model quality. Using this testbed, we develop a family of hybrid designs via grafting: replacing softmax attention with gated convolution, local attention, and linear attention, and replacing MLPs with variable expansion ratio and convolutional variants. Notably, many hybrid designs achieve good quality (FID: 2.38-2.64 vs. 2.27 for DiT-XL/2) using < 2% pretraining compute.

Associative memory models based on Hopfield networks and self-attention based on key-value mechanisms have been popular approaches in the study of memory mechanisms in deep learning. It has been pointed out that the state update rule of the modern Hopfield network (MHN) in the adiabatic approximation is in agreement with the self-attention layer of Transformer. In this paper, we go beyond this approximation and investigate the relationship between MHN and self-attention. Our results show that the correspondence between Hopfield networks and Transformers can be established in a more generalized form by adding a new variable, the hidden state derived from the MHN, to self-attention. This new attention mechanism, modern Hopfield attention (MHA), allows the inheritance of attention scores from the input layer of the Transformer to the output layer, which greatly improves the nature of attention weights. In particular, we show both theoretically and empirically that MHA hidden states significantly improve serious problem of deep Transformers known as rank collapse and token uniformity. We also confirm that MHA can systematically improve accuracy without adding training parameters to the Vision Transformer or GPT. Our results provide a new case in which Hopfield networks can be a useful perspective for improving the Transformer architecture.

We develop a rigorous statistical theory of multi-head attention (MHA) as an ensemble of Nadaraya-Watson (NW) kernel regression estimators. Building on the algebraic identity between single-head softmax attention and the NW estimator, we prove that MHA is a structured ensemble of H NW estimators, each operating in a distinct learned projection subspace of the key space. We derive an explicit Bias-Variance-Covariance decomposition of the MHA mean squared error, showing that variance reduction depends not merely on the number of heads H but fundamentally on the decorrelation of head outputs. Decorrelation is governed by the principal angles between learned projection subspaces: orthogonal projections yield maximum variance reduction; aligned projections yield none. We introduce the Head Diversity Index (HDI), a computable spectral measure of inter-head decorrelation, and prove that MHA mean squared error is monotonically decreasing in HDI. This provides the first rigorous theoretical explanation for the empirically observed specialization of attention heads. Under a fixed total-dimension budget D = H * d_k, we solve the optimal head-dimension allocation problem, deriving the MSE-minimizing pair (H*, d_k*) from data distribution and regression smoothness. The solution yields a new architectural scaling law: the optimal per-head dimension grows logarithmically with training set size, while the optimal number of heads grows nearly linearly with the total budget D. Our framework unifies three strands of prior work: the NW theory of single-head attention, the general weighting theory for ensemble learning, and the decorrelation-variance-reduction isomorphism between biological and computational ensembles. Multi-head attention is the Transformer's instantiation of a universal principle: identical agents plus diversity-enforcing mechanisms yields emergent optimality.

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

The Hierarchical Kernel Transformer (HKT) is a multi-scale attention mechanism that processes sequences at L resolution levels via trainable causal downsampling, combining level-specific score matrices through learned convex weights. The total computational cost is bounded by 4/3 times that of standard attention, reaching 1.3125x for L = 3. Four theoretical results are established. (i) The hierarchical score matrix defines a positive semidefinite kernel under a sufficient condition on the symmetrised bilinear form (Proposition 3.1). (ii) The asymmetric score matrix decomposes uniquely into a symmetric part controlling reciprocal attention and an antisymmetric part controlling directional attention; HKT provides L independent such pairs across scales, one per resolution level (Propositions 3.5-3.6). (iii) The approximation error decomposes into three interpretable components with an explicit non-Gaussian correction and a geometric decay bound in L (Theorem 4.3, Proposition 4.4). (iv) HKT strictly subsumes single-head standard attention and causal convolution (Proposition 3.4). Experiments over 3 random seeds show consistent gains over retrained standard attention baselines: +4.77pp on synthetic ListOps (55.10+-0.29% vs 50.33+-0.12%, T = 512), +1.44pp on sequential CIFAR-10 (35.45+-0.09% vs 34.01+-0.19%, T = 1,024), and +7.47pp on IMDB character-level sentiment (70.19+-0.57% vs 62.72+-0.40%, T = 1,024), all at 1.31x overhead.

When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the