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concepts/kl-order.md

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+---
+title: "KL 阶 (KL Order)"
+created: 2026-06-10
+updated: 2026-06-10
+type: concept
+tags: ["singular-learning-theory", "information-geometry", "bridge-invariant"]
+sources: ["[[dead-directions-geometric-singular-learning]]"]
+---
+
+# KL 阶 (KL Order)
+
+**KL 阶** k 是沿 [[dead-direction|Dead Direction]] 接近奇异集时 KL 散度趋于零的速率：
+
+```
+K(theta(t)) = c · t^{2k} + O(t^{2k+1})
+```
+
+KL 散度在 t 处有 2k 阶零点——两倍于 KL 阶。
+
+## 桥接不变量
+
+KL 阶是[[singular-learning-theory|SLT]]和[[information-geometry|信息几何]]都可计算的少数不变量之一：
+
+- **SLT 解读**：法交形式中的指数——与 [[real-log-canonical-threshold|RLCT]] 直接相关 (lambda = 1/(2k))
+- **信息几何解读**：[[fisher-information-metric|Fisher 度量]]退化速率的驱动力 (u^T F u ~ t^{2(k-1)})
+
+## 与 Deep Direction Fisher 衰减的关系
+
+```
+k = 1: Fisher decay rate 0 (正则方向)
+k = 2: Fisher decay rate 2
+k = 3: Fisher decay rate 4
+```
+
+## 从 Checkpoint 计算
+
+Shirodkar (2026) 证明可通过一次前向+反向传播计算 KL 阶——无需后验采样，无需消解。这使 SLT 分析在大规模网络上首次进入实践领域。
+
+## 参考
+- [[dead-directions-geometric-singular-learning|Dead Directions]]
+- [[dead-direction|Dead Direction]]
+- [[real-log-canonical-threshold|RLCT]]
+- [[watanabe-triple|Watanabe's Triple]]