桥接图模型：用于生成建模的耦合、投影与保电流动力学
Bridge Graphical Models: Coupling, Projection, and Current-Preserving Dynamics for Generative Modeling
- Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出桥接图模型（BGMs），定义马尔可化间隙作为生成模型桥到解码器压缩的不可约损失，通过实验验证其可作为桥和耦合设计的训练前诊断工具。
AI中文摘要：
连续时间生成模型通常基于端点条件桥构建，但生成所需的是仅观测当前状态和时间的非预见性马尔可夫解码器。我们将这种桥到解码器的压缩确定为扩散模型、流匹配、修正流、薛定谔桥以及基于场的生成模型共有的结构瓶颈。我们引入马尔可夫化间隙，即给定马尔可夫状态时桥速度的时间积分条件方差，它是从采样器可用信息预测端点条件运动的最小均方误差（MMSE），用于衡量神经网络训练前产生的不可约损失。为使该瓶颈在不同模型族间具有可比性，我们定义桥接图模型（Bridge Graphical Models，BGMs），将端点耦合、桥律、马尔可夫投影与保电流动力学表示为独立设计选择。该形式体系还将泊松模型和静电模型表示为具有对应场线马尔可夫化间隙的场线桥核。在CIFAR-10和Fashion-MNIST上的合成、潜在及像素空间先导实验中，训练前几分钟估计的特征空间代理间隙，在固定架构、桥、采样器和计算资源的情况下，对设计选择的排序与下游训练损失和FID指标的排序一致。这些结果支持马尔可化间隙作为桥和耦合设计的训练前诊断工具。
英文摘要：
Continuous-time generative models are often built from endpoint-conditioned bridges, but generation requires a different object: a non-anticipative Markov decoder that only observes the current state and time. We identify this bridge-to-decoder compression as a structural bottleneck shared by diffusion models, flow matching, rectified flow, Schrödinger bridges, and field-based generative models. We introduce the \emph{Markovization gap}, the time-integrated conditional variance of the bridge velocity given the Markov state. It is the MMSE of predicting endpoint-conditioned motion from the information available to a sampler, and it measures an irreducible loss incurred before any neural network is trained. To make this bottleneck comparable across model families, we define \emph{Bridge Graphical Models} (BGMs), which separate endpoint coupling, bridge law, Markovian projection, and current-preserving dynamics representation as independent design choices. The same formalism also represents Poisson and electrostatic models as field-line bridge kernels with a corresponding field-line Markovization gap. Across synthetic, latent, and pixel-space pilots on CIFAR-10 and Fashion-MNIST, a feature-space proxy gap estimated in minutes before training ranks design choices in the same direction as downstream training loss and FID under fixed architecture, bridge, sampler, and compute. These results support the Markovization gap as a pre-training diagnostic for bridge and coupling design.
