风险敏感薛定谔桥：并非KL投影

中文摘要 AI 辅助

薛定谔桥的计算能力源于一个单一的结构性事实：根据Girsanov定理，受控问题是对固定参考测度的Kullback-Leibler（KL）投影，可通过交替投影求解。本文表明，这一事实在风险敏感情形下不再成立。当期望路径代价被熵风险度量取代，且两个端点边际保持为硬约束时，所得不动点桥值$J_θ$（在强制执行终端约束的乘子处的软问题值）无法表示为对任何具有正则端点律的固定路径空间参考测度的约束KL最小值（该参考测度类别严格大于一致椭圆扩散参考测度：不要求马尔可夫性质），即使允许依赖于初始边际的加性归一化也不行。此外，没有任何单一参考测度能生成风险参数中的单参数族。该障碍以闭式形式计算得出：高斯桥值恰好违反$θ/2$，违反了任何固定端点密度的高斯平滑必须满足的热方程。代替投影，该理论依赖于终端乘子不动点和惩罚后向因子得分能量的非对称分解。

英文摘要

The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value $J_θ$ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly $θ/2$, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.

发表机构

• Institute of Systems Medicine, Chinese Academy of Medical Sciences（中国医学科学院系统医学研究所）

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