Transformers rely on position embedding mechanisms in long context modeling in most cases. Rotary Position Embedding (RoPE) embeds positional information with independent 2D rotations, forming relative position terms in self-attention. However, its pairwise, block-based, and decoupled structure limits deep mixing and robustness across channels. We propose HD-RoPE, which extends RoPE from independent 2D rotations to higher-dimensional rotations and introduces a Paley-I orthogonal basis to obtain balanced, isotropic, and dense phase mixing within each rotation subspace. This significantly enhances channel coupling and rotational degrees of freedom while maintaining orthogonal stability and the relative position closure property. Furthermore, HD-RoPE is easily optimized for engineering efficiency without introducing additional trainable parameters. We have conducted extensive evaluation results demonstrating that HD-RoPE achieves significant performance improvements over standard RoPE across various popular benchmarks and in both long and short contexts.
In a Transformer, each layer attends to past tokens only through KV produced at its own depth, despite the presence of deeper representations during autoregressive decoding. Feedback architectures allow shallow consumer layers to attend to KV produced by deeper past-token representations, but give all consumer layers the same fixed connection patterns to source layers. We propose WhiteMatter, which connects every attention layer to the representations from all layers of each past token, with connection weights that can vary across consumer layers and adapt to the source token. For each token, a router implements these connections by mixing its $L$ layer states into $k$ KV channels that are cached for subsequent tokens; each consumer layer attends to one of the channels. The number of channels $k$ controls the KV-cache size. Setting $k<L$ reduces the cache's memory footprint. In our pretraining experiments, WhiteMatter outperforms a vanilla Transformer with 50% more layers and retains most of this gain with a 50% KV-cache compression.
While large language models (LLMs) can solve advanced reasoning problems in seconds, we show that even frontier models fail to perform a much simpler operation: exactly copying an input string that lies well within their context windows. We attribute this failure to positional encodings in Transformer architectures, whose inductive bias favors copying through a shortcut based on matching local contexts rather than carefully locating the corresponding input positions. To address this issue, we introduce 2D-RoPE, which organizes text into a 2D grid rather than a 1D sequence and assigns each token a row ID and a column ID. Under this view, copying becomes simply retrieving input tokens at a fixed column offset, which makes the task easy to learn. In synthetic copy experiments, shallow Transformers with 2D-RoPE achieve perfect copying at input lengths hundreds of times longer than those seen during training, whereas standard positional encodings fall far behind. We further show that the advantage of 2D-RoPE language models on copy tasks consistently holds in large-scale pretraining on DCLM with model sizes up to 1.4B parameters. Overall, our results suggest that viewing text in 2D can benefit language modeling, and we hope this encourages future work to further explore the potential of 2D positional encodings.