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This paper introduces Bidirectional Manifold Consistency (BMC), a training-free metric that leverages the geometric stability of diffusion language models to verify reasoning traces without requiring ground truth answers. By treating valid generation paths as stable attractors on a learned manifold, the method enables self-verification, resource-efficient inference via rejection sampling, and fine-grained alignment rewards for model self-evolution. The work advances the theoretical understanding of how diffusion architectures can be harnessed for robust global planning and logical consistency in large-scale AI systems.
Reasoning on the Manifold: Bidirectional Consistency for Self-Verification in Diffusion Language Models
arXiv:2604.16565v4 Announce Type: replace Abstract: While Diffusion Large Language Models (dLLMs) offer structural advantages for global planning, efficiently verifying that they arrive at correct answers via valid reasoning traces remains a critical challenge. In this work, we propose a geometric perspective: Reasoning on the Manifold. We hypothesize that valid generation trajectories reside as stable attractors on the high-density manifold of the learned distribution, whereas invalid paths exhibit off-manifold drift. To operationalize this, we introduce Bidirectional Manifold Consistency (BMC), a training-free, unsupervised metric that quantifies the stability of the generated sequence through a forward-masking and backward-reconstruction cycle. Empirically, we demonstrate BMC's versatility across the full reasoning lifecycle: (1) in Diagnosis, it serves as a robust discriminator of solution validity without ground truth answer; (2) in Inference, it enables rejection resampling to effectively concentrate computational resources on complex reasoning tasks; and (3) in Alignment, it functions as a dense geometric reward that transforms sparse outcome supervision into fine-grained guidance, empowering models to self-evolve beyond standard baselines. Our results establish intrinsic geometric stability as a robust indicator of correctness for dLLMs.
Abstract:While Diffusion Large Language Models (dLLMs) offer structural advantages for global planning, efficiently verifying that they arrive at correct answers via valid reasoning traces remains a critical challenge. In this work, we propose a geometric perspective: Reasoning on the Manifold. We hypothesize that valid generation trajectories reside as stable attractors on the high-density manifold of the learned distribution, whereas invalid paths exhibit off-manifold drift. To operationalize this, we introduce Bidirectional Manifold Consistency (BMC), a training-free, unsupervised metric that quantifies the stability of the generated sequence through a forward-masking and backward-reconstruction cycle. Empirically, we demonstrate BMC's versatility across the full reasoning lifecycle: (1) in Diagnosis, it serves as a robust discriminator of solution validity without ground truth answer; (2) in Inference, it enables rejection resampling to effectively concentrate computational resources on complex reasoning tasks; and (3) in Alignment, it functions as a dense geometric reward that transforms sparse outcome supervision into fine-grained guidance, empowering models to self-evolve beyond standard baselines. Our results establish intrinsic geometric stability as a robust indicator of correctness for dLLMs.
31 pages, 7 figures. Accepted to the 43rd International Conference on Machine Learning (ICML 2026). Camera-ready version
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2604.16565 [cs.LG]
(or arXiv:2604.16565v4 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2604.16565 Focus to learn more arXiv-issued DOI via DataCite
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