Recently, diffusion models have emerged as a powerful class of generative models. Despite their success, there is still limited understanding of their semantic spaces. This makes it challenging to achieve precise and disentangled image generation without additional training, especially in an unsupervised way. In this work, we improve the understanding of their semantic spaces from intriguing observations: among a certain range of noise levels, (1) the learned posterior mean predictor (PMP) in the diffusion model is locally linear, and (2) the singular vectors of its Jacobian lie in low-dimensional semantic subspaces. We provide a solid theoretical basis to justify the linearity and low-rankness in the PMP. These insights allow us to propose an unsupervised, single-step, training-free LOw-rank COntrollable image editing (LOCO Edit) method for precise local editing in diffusion models. LOCO Edit identified editing directions with nice properties: homogeneity, transferability, composability, and linearity. These properties of LOCO Edit benefit greatly from the low-dimensional semantic subspace. Our method can further be extended to unsupervised or text-supervised editing in various text-to-image diffusion models (T-LOCO Edit). Finally, extensive empirical experiments demonstrate the effectiveness and efficiency of LOCO Edit. The codes will be released at https://github.com/ChicyChen/LOCO-Edit.

本研究针对扩散模型语义空间理解不足的问题，提出了一种新的无监督、单步的低秩可控图像编辑方法（LOCO Edit）。通过发现学习的后验均值预测器在特定噪声水平范围内的局部线性特性和雅可比矩阵的奇异向量位于低维语义子空间，LOCO Edit实现在扩散模型中的精确本地编辑，展现了良好的可组合性和转移性。实验结果表明该方法在多个文本到图像扩散模型中的有效性和效率。
  

在扩散模型中探索低维子空间以实现可控图像编辑