Rotate2Think: Geometric Priming via Orthogonal Rotation to Improve Language Model Reasoning
Authors: Aditya Sharma, Christopher J. Pal, Amal Zouaq
Organizations: Polytechnique Montréal · Mila - Quebec AI Institute · LAMA-WeST Lab · Canada CIFAR AI Chair
Abstract
Reasoning models achieve strong performance on challenging tasks by generating explicit intermediate reasoning traces before producing a final answer. Yet the internal structure of representation space when reasoning remains poorly understood: how do a model's hidden representations differ during thinking versus the embeddings of the input prompt, and can this structure be exploited to elicit stronger reasoning at inference time? We show that both input embeddings and thinking embeddings (mean-pooled last-layer hidden states over the prompt and reasoning trace, respectively) exhibit extremely high conicity, with all vectors clustering tightly around a single mean direction. Crucially, these mean input and thinking directions are non-collinear, with thinking embeddings occupying a geometrically distinct region of embedding space across many different models and benchmark tasks. This observation motivates casting the input-to-thinking transition as a rotation problem admitting a closed-form solution via orthogonal Procrustes analysis. We propose Rotate2Think, a training-free method that estimates this rotation from a small set of correctly solved examples and injects the resulting synthetic thinking vector between thinking delimiters at inference time, providing a geometric primer at the onset of the reasoning trace. Evaluated across multiple benchmarks and model families, Rotate2Think improves accuracy in 30 of 32 model-benchmark configurations across mathematics, science, and code tasks, and generalizes zero-shot to multimodal reasoning on MATH-Vision.
Large language models often improve on difficult tasks by spending inference-time compute on a reasoning trace before producing the final answer. That extra computation can be useful, but it also raises latency, token cost, and deployment complexity. We introduce \textbf{ThinkSwitch}, a low-compute procedure for co-training paired instruct and thinking checkpoints. Starting from compatible Qwen3-4B instruct and thinking models, each iteration asks the thinking checkpoint to generate answers, removes the reasoning trace, distills the answer-only pairs into the instruct checkpoint with QLoRA, and reconstructs a thinking checkpoint with spherical weight interpolation. The only human-supplied inputs are task prompts; the labels are generated by the model itself. On a 30-question AIME 2026 evaluation, ThinkSwitch improves the instruct checkpoint from 10/30 to 20/30 and the thinking checkpoint from 14/30 to 22/30. On a 30-question PubMedQA subset, it improves the instruct checkpoint from 13/30 to 18/30 and the thinking checkpoint from 18/30 to 25/30. The complete experiment uses 15 training prompts per domain and costs $2.86 on a single cloud RTX 3070. The results are small-scale, but they indicate that targeted distillation loops can move part of the benefit of explicit reasoning into weights while preserving a separate thinking mode.
Large language models solve complex problems by generating lengthy chains of explicit reasoning tokens. While effective, this makes reasoning expensive, length-sensitive, and constrained to (discrete) natural language. While latent reasoning offers a continuous alternative, determining useful structures for intermediate latent states is an open challenge. In this paper, we formulate latent reasoning as a geometric path-approximation problem within the model's pretrained token-embedding space. We introduce Geometric Latent Reasoning (GLR), which uses a lightweight transition head to predict iterative direction updates in embedding space. Using textual chain-of-thought traces as anchors, GLR learns to approximate discrete reasoning trajectories while permitting continuous deviations from exact token embeddings. Evaluations on mathematical reasoning benchmarks using Qwen3 models reveal an emergent phenomenon: geometric latent reasoning induces substantially shorter generations without an explicit length objective. By replacing early explicit reasoning with continuous latent steps, models often reach correct answers using substantially fewer total generation steps. These findings suggest that continuous trajectories act as compact intermediate reasoning states, exposing a new tradeoff between latent computation budget, output length, and accuracy.
Large language models (LLMs) excel at generating long chains of thought, but long reasoning traces are often verbose and memory-inefficient. In this work, we introduce Structured Thoughts, a framework that organizes reasoning into alternating <try> and <outcome> blocks: <try> captures exploratory scratch work, while <outcome> contains the distilled conclusion of that step. We construct a dataset of structured thoughts by segmenting reasoning traces into <try> blocks and prompting an LLM to summarize each step into its corresponding <outcome>. Fine-tuning pretrained foundation models on this reformatted data produces models that adopt the structured reasoning style, leading to performance gains of up to 8.08% on reasoning benchmarks compared to standard SFT. The explicit structure also enables context pruning: after each <try>/<outcome> pair, the <try> can be pruned, allowing the model to retain conclusions without keeping the full scratch work in the context. A proof-of-concept pruning implementation achieves an average of 85% memory / context savings with an 8.67% performance drop across mathematical tasks.