为扩散模型的潜在空间建立信息几何框架，实现更自然的数据编辑路径。
The Spacetime of Diffusion Models: An Information Geometry Perspective
- 引入时空潜变量 $z=(x_t,t)$，构建非平凡几何结构。
- 基于指数族性质，无需模拟即可计算测地线长度与编辑距离。
- 适用于分子系统路径采样，支持低方差和避障约束编辑。
我们提出扩散模型潜在空间的新几何视角。标准确定性概率流反向微分方程解码器存在根本缺陷：它强制测地线在数据空间中表现为直线，忽略数据内在几何结构。而通过反向随机微分方程（SDE）解码，可采用费雪-罗氏度量进行信息几何分析。但若以 $x_T$ 作为潜在表示，该度量会因记忆缺失而退化。为此，我们引入潜时空 $z=(x_t,t)$，索引所有噪声尺度下的去噪分布 $p(x_0 | x_t)$，形成非平凡几何结构。我们证明这些分布构成指数族，并推导出无需模拟的曲线长度估计器，实现高效测地线计算。由此产生的原理性扩散编辑距离，使测地线可追踪数据间最小噪声-去噪编辑序列。此外，该方法在分子系统过渡路径采样中表现优异，包括低方差和区域回避等约束情形。代码已公开于 https://github.com/rafalkarczewski/spacetime-geometry。
原文摘要 · Abstract (English)
We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of $x_T$ as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime $z=(x_t,t)$ that indexes the family of denoising distributions $p(x_0 | x_t)$ across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.
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