42 lines
1.6 KiB
Markdown
42 lines
1.6 KiB
Markdown
---
|
||
title: "语义 Extent (Semantic Extent)"
|
||
created: 2026-07-04
|
||
updated: 2026-07-04
|
||
type: concept
|
||
tags: [semantics, parameterization, vlm, geometry]
|
||
sources: ["arXiv:2606.18839"]
|
||
---
|
||
|
||
# 语义 Extent (Semantic Extent)
|
||
|
||
在 VLM 嵌入空间的语义平面 $P_{a,a'}$ 中，用角度参数 $\varphi \in (-\pi, \pi]$ 控制图像嵌入在 source semantic 和 target semantic 之间的语义强度变化。
|
||
|
||
## 定义
|
||
|
||
在语义平面 $P_{a,a'}$ 的标准正交基 $(e_1, e_2)$ 下：
|
||
|
||
- $e_1 = u_a$（source 文本嵌入）
|
||
- $e_2 = \frac{u_{a'} - \langle u_{a'}, u_a \rangle u_a}{\|u_{a'} - \langle u_{a'}, u_a \rangle u_a\|_2}$（正交化 target 方向）
|
||
|
||
source extent $\varphi_a = \text{atan2}(\langle z_\parallel, e_2 \rangle, \langle z_\parallel, e_1 \rangle)$
|
||
|
||
## 语义变换 $\gamma(\varphi)$
|
||
|
||
$$\gamma(\varphi) = r(\cos\varphi \cdot e_1 + \sin\varphi \cdot e_2) + z_\perp$$
|
||
|
||
- $r = \|z_\parallel\|_2$：平面内分量的模
|
||
- $\gamma(\varphi_a) = z$：起点对应原始图像嵌入
|
||
- $z_\perp$ 保持不变，保留与目标语义无关的剩余信息
|
||
|
||
## 两种 target 确定方式
|
||
|
||
- **Text-specified**：$\varphi_{a'} = \text{atan2}(\langle u_{a'}, e_2 \rangle, \langle u_{a'}, e_1 \rangle)$，纯文本驱动
|
||
- **Image-specified**：$\varphi_{a'} = \text{atan2}(\langle z'_\parallel, e_2 \rangle, \langle z'_\parallel, e_1 \rangle)$，参考图像驱动
|
||
|
||
## 参考
|
||
|
||
- [[semantic-robustness-certification|语义鲁棒性认证]]
|
||
- [[semantic-plane|语义平面]]
|
||
- [[prediction-invariant-intervals|预测不变区间]]
|
||
- [[semantic-robustness-certification-vlm-2026|论文原文]]
|