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GeoSteer: Faithful Chain-of-Thought Steering via Latent Manifold Gradients

Kentaro Kazama, Daiki Shirafuji, Tatsuhiko Saito

TL;DR

GeoSteer presents a geometry-aware, latent-manifold approach to improving intermediate reasoning in LLMs. By constructing a CoT dataset with stepwise quality scores, training a VAE to learn a latent representation of hidden states, and regressing a latent quality function, the method pulls back gradients to steer hidden states toward higher-quality reasoning. This latent-space steering preserves final-answer accuracy while enhancing coherence and structural faithfulness of the CoT, with model- and steering-strength-dependent effects. The work demonstrates that intermediate reasoning quality and final correctness are related but not identical, and it proposes a principled, non-Euclidean mechanism for steering internal representations during inference.

Abstract

Recent advances in Large Language Models (LLMs) have improved multi-step reasoning. Most approaches rely on Chain-of-Thought (CoT) rationales. Previous studies have shown that LLMs often generate logically inconsistent reasoning steps even when their final answers are correct. These inconsistencies reduce the reliability of step-level reasoning. We propose GeoSteer, a manifold-based framework that improves the quality of intermediate reasoning. The method consists of: (1) constructing a CoT dataset with segment-level scores, (2) training a Variational Autoencoder (VAE) model and a quality estimation model to learn a low-dimensional manifold of high-quality CoT trajectories, and (3) steering hidden states of target LLMs toward higher-quality regions in the latent space. This update in a latent space behaves like a natural-gradient adjustment in the original hidden-state space. It ensures geometrically coherent steering. We evaluate GeoSteer on the GSM8k dataset using the Qwen3 series. We measure via answer accuracy and overall reasoning performance. GeoSteer improved the exact match accuracy by up to 2.6 points. It also enhanced the pairwise win rate by 5.3 points. These results indicate that GeoSteer provides an effective and controllable mechanism for improving the quality of intermediate reasoning in LLMs.

GeoSteer: Faithful Chain-of-Thought Steering via Latent Manifold Gradients

TL;DR

GeoSteer presents a geometry-aware, latent-manifold approach to improving intermediate reasoning in LLMs. By constructing a CoT dataset with stepwise quality scores, training a VAE to learn a latent representation of hidden states, and regressing a latent quality function, the method pulls back gradients to steer hidden states toward higher-quality reasoning. This latent-space steering preserves final-answer accuracy while enhancing coherence and structural faithfulness of the CoT, with model- and steering-strength-dependent effects. The work demonstrates that intermediate reasoning quality and final correctness are related but not identical, and it proposes a principled, non-Euclidean mechanism for steering internal representations during inference.

Abstract

Recent advances in Large Language Models (LLMs) have improved multi-step reasoning. Most approaches rely on Chain-of-Thought (CoT) rationales. Previous studies have shown that LLMs often generate logically inconsistent reasoning steps even when their final answers are correct. These inconsistencies reduce the reliability of step-level reasoning. We propose GeoSteer, a manifold-based framework that improves the quality of intermediate reasoning. The method consists of: (1) constructing a CoT dataset with segment-level scores, (2) training a Variational Autoencoder (VAE) model and a quality estimation model to learn a low-dimensional manifold of high-quality CoT trajectories, and (3) steering hidden states of target LLMs toward higher-quality regions in the latent space. This update in a latent space behaves like a natural-gradient adjustment in the original hidden-state space. It ensures geometrically coherent steering. We evaluate GeoSteer on the GSM8k dataset using the Qwen3 series. We measure via answer accuracy and overall reasoning performance. GeoSteer improved the exact match accuracy by up to 2.6 points. It also enhanced the pairwise win rate by 5.3 points. These results indicate that GeoSteer provides an effective and controllable mechanism for improving the quality of intermediate reasoning in LLMs.
Paper Structure (35 sections, 22 equations, 3 figures, 3 tables)

This paper contains 35 sections, 22 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Activation Steering and Representation Control
  4. Modeling and Evaluating Reasoning Trajectories
  5. Latent Manifold Learning and Geometric Analysis
  6. Preliminaries
  7. Notation and Setup
  8. Latent Representation of Hidden States
  9. Quality of Intermediate Reasoning
  10. Proposed Method
  11. Construction of CoT Dataset
  12. Learning a Latent Manifold and Quality Function
  13. Latent Manifold of Hidden States
  14. Latent Quality Function
  15. Steering of Hidden States
  16. ...and 20 more sections

Figures (3)

  • Figure 1: Overview of GeoSteer. (1) A dataset of chain-of-thought (CoT) trajectories is constructed by generating multiple reasoning paths for each problem and assigning a quality score to each intermediate step. (2) A variational autoencoder (VAE) is trained on these trajectories to learn a latent manifold that captures the structural properties of high-quality reasoning. (3) During inference, the target model’s hidden states are steered by updating their latent representations toward regions associated with higher-quality reasoning.
  • Figure 2: Latent reasoning trajectories projected onto a 2D plane defined by the quality-gradient direction (x-axis) and its orthogonal direction (y-axis). Red line indicates the baseline trajectory, and blue line indicates the steered trajectory.
  • Figure 3: Step-wise reasoning-quality scores averaged over correct (EM=1) and incorrect (EM=0) solutions.