矩阵值扩散模型的前向演化误差分析与自适应设计
Forward-Evolution Error Analysis and Adaptive Design for Matrix-Valued Diffusion Models
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中文总结 AI 辅助
该论文针对矩阵值扩散模型的时间离散化与噪声调度问题，分析两种数值格式的前向演化误差，提出自适应矩阵调度与网格，实验验证其改进效果。
中文摘要 AI 辅助
扩散模型学习逆转预定义的损坏过程，但采样仍需代价高昂的时间离散化，且依赖所选的噪声调度。我们针对具有矩阵值调度的方差保持扩散研究这两个问题。我们的分析将逆时间离散化误差转移至前向损坏规律，并在统一框架内处理两种数值格式：第一种冻结分数，通过对矩阵敏感的局部比较和前向信息耗散，对于KL精度ε²，得到环境维度步复杂度，其主导因子为d/ε²；第二种保持已知高斯漂移精确，冻结后验均值。对于度量熵维度为k的数据，利用前向马尔可夫恒等式、各向异性覆盖估计及斯蒂尔杰斯分部积分，给出对应因子k log k/ε²。两种情形下，证明均识别局部误差，通过前向演化累积该误差，并将结果纳入统一的KL分解。局部误差进一步为矩阵调度提供方向准则，并给出渐近最优的平方根自适应网格。高维高斯混合实验说明了所得调度与网格的改进。
英文摘要
Diffusion models learn to reverse a predefined corruption process, but sampling still requires a costly time discretization and depends on the chosen noise schedule. We study these two issues for variance-preserving diffusions with matrix-valued schedules. Our analysis transfers reverse-time discretization errors to the forward corruption law and treats two numerical schemes within a common framework. The first freezes the score and yields, through a matrix-sensitive local comparison and forward information dissipation, an ambient-dimensional step complexity with leading factor $d/\varepsilon^2$ for KL accuracy $\varepsilon^2$. The second keeps the known Gaussian drift exact and freezes the posterior mean. For data of metric-entropy dimension $k$, a forward Markov identity, an anisotropic covering estimate, and Stieltjes integration by parts give the corresponding factor $k\log k/\varepsilon^2$. In both cases, the proof identifies a local error, accumulates it through the forward evolution, and inserts the result into a common KL decomposition. The local errors further provide directional criteria for matrix schedules and an asymptotically optimal square-root adaptive grid. A high-dimensional Gaussian-mixture experiment illustrates the resulting schedule and grid improvements.
