QUEST: A robust attention formulation using query-modulated spherical attention

In standard attention, three separately learned projections decide how strongly a token attends to each neighbor ($W_Q$, $W_K$) and how the attended features are transformed before aggregation ($W_V$). We study Quat-Sphere-Vision (QSV), a sparse spherical vision model that replaces this projection triple with a single learned unit quaternion per token: the relative quaternion $r_{ij} = q_i^{*} \otimes q_j$ supplies both the attention logit $\operatorname{Re}(r_{ij})$ and a sandwich-product feature transport $x \mapsto r_{ij} \otimes x \otimes r_{ij}^{*}$, with messages passed over sparse kNN graphs on concentric Fibonacci spheres. Ablations that change only the targeted component show the two roles to be asymmetric. Removing the transport reduces test accuracy by about four percentage points on CIFAR-10 and CIFAR-100 (single runs per CIFAR-100 variant), while replacing the learned attention weights with uniform averaging leaves it essentially unchanged. Parameter-matched controls then remove the geometry itself: standard attention on the same graph exceeds QSV (mean $87.3\%$ vs. $85.9\%$), and the same model on a flat 2D lattice reaches $91.1\%$, within $2.1$ points of a ResNet-20 trained under the same pipeline (single run). In the coupled kernel, nearly all of the learned pairwise computation resides in the transport channel.

The projection of queries and keys are central to the attention mechanism in Transformer architectures. While they are mathematically symmetric, they play different roles in attention mechanisms. The question of whether there is an effect from their functional distinction on their geometric development in training remains unanswered. We investigate the problem through the training of small GPT-like Transformers on character-level WikiText-103 for three different depths (4, 6, and 8 layers), three types of initialization for queries and keys, and four random seeds, resulting in 36 runs and 54 trajectories of average layers across seeds. We track the effective dimensionality of those layers using participation ratios and discover that effective dimension of queries expand while keys shrink, and that $PR_Q - PR_K$ is positive in all trajectories studied. In connection to attention, the shrinking of keys leads to a narrower spectrum of $QK^\top$ and more peaked attention weights. In order to determine if this connection is causal or coincidental, we directly control the spectrum of keys during training across five seeds: restricting it to make it shrink sharpens the attention with high directional confidence, while keeping it constant to the level of initial dispersion makes attention softer. Additional token-level checkpoint analyses show that the monotonic paired-contrast trend is not universal across pretrained families, but survives as an early-training regime that later decays over a full pretraining run, and the link between interaction-rank geometry and attention entropy remains visible in several models.

Standard attention scales quadratically with sequence length. Efficient attention methods reduce this O(n^2) cost, but when retrofitted into pretrained models, they often degrade perplexity, downstream accuracy, or both. We introduce Focus, a method that learns which token pairs matter. Focus adds a small set of learnable centroids--as few as 148K parameters per layer--that act as gates: only token pairs belonging to the same centroid group attend to each other over long ranges. Focus is composable: it can be added to any pretrained model by training only the centroids while keeping all original weights frozen. Experiments show that composing Focus onto pretrained models yields zero degradation on downstream benchmarks across model sizes from 124M to 70B parameters and five attention architectures. Surprisingly, sparse Focus attention outperforms full attention at 124M scale (30.3 vs. 31.4 perplexity) and matches full attention when trained from scratch at 7B scale (13.82 vs. 13.89). Focus is also fast: top-k group membership gives a 2x speedup with better quality than the original pretrained model. Using our FlashAttention decomposition, Focus achieves an 8.6x speedup at 1M tokens without custom kernels.