0. 脚注/ 边注 (Sidenotes) 这是正文中的第一个脚注引用1 1 AutoFocus-IL 使用 VLM（具体为 GPT-4o）自动识别演示视频中的关键物体并生成时间显著性图，无需人工标注眼动数据。该方法在 CARLA 仿真和真机 WidowX 实验中均超越了需要人工监督的 SOTA 方法。2 2 在标准 Behavior Cloning 中，损失函数为 L BC = 1 N ∑ i ∥ a i − π θ ( o i ) ∥ 2 \mathcal{L}_{\text{BC}} = \frac{1}{N} \sum_i \| a_i - \pi_\theta(o_i) \|^2 L BC = N 1 ∑ i ∥ a i − π θ ( o i ) ∥ 2 L = 1 N ∑ i ∑ p ∈ Ω M i ( p ) ⋅ ∥ a i − π θ ( o i ) ∥ 2 \mathcal{L} = \frac{1}{N} \sum_i \sum_{p \in \Omega} M_i(p) \cdot \| a_i - \pi_\theta(o_i) \|^2 L = N 1 ∑ i ∑ p ∈ Ω M i ( p ) ⋅ ∥ a i − π θ ( o i ) ∥ 2 3 3 本文提出的架构在 ICRA 2026 接收。
0.5 文献引用 (Citations) The normal distribution, often called the Gaussian distribution or bell curve, is one of the most fundamental concepts in statistics and probability theory [1 . Its characteristic symmetric, bell-shaped curve appears throughout nature and human activity, from heights and test scores to measurement errors and biological variations [2 .
正文中用 [@id] 引用，文末自动生成编号参考文献（Bibliography）。也支持一处多引 [3 , 4 ，编号顺序与 frontmatter 中 references 的声明顺序一致。点击正文里的 [1 可跳到文末，点文末的 [1] 可跳回正文。
1. 标题层级H3 三级标题H4 四级标题正文段落。这是加粗 ，这是斜体 ，这是删除线，这是行内代码。这是超链接 。
2. 列表无序列表
机器人学习三大方向
模仿学习 (Imitation Learning)
强化学习 (Reinforcement Learning)
视觉-语言-动作模型 (VLA)
仿真平台
Isaac Sim / Isaac Lab
RoboCasa / LIBERO
MuJoCo / MotrixSim
有序列表
收集演示数据 D = { ( o i , a i ) } i = 1 N \mathcal{D} = \{(o_i, a_i)\}_{i=1}^N D = {( o i , a i ) } i = 1 N
训练策略 π θ \pi_\theta π θ
仿真评估 → 真机部署
任务列表
3. 引用
核心洞察 ：VLM 的视觉理解能力可以自动为模仿学习提供显著性监督，无需人工标注眼动数据。
— AutoFocus-IL 论文摘要, ICRA 2026
嵌套引用：
机器人学习的第一性原理
数据驱动 + 仿真迁移 是当前最优路径
但真机数据的价值不可替代
4. 表格论文发表汇总
论文 会议 年份 类型 AutoFocus-IL ICRA 2026 VLM + Imitation Learning ORIC CVPR 2026 VLM Benchmarking DynaMem CoRL 2026 Memory-Augmented VLA EvoMoE In Prep. 2026 Continual MoE for VLA
对齐方式
左对齐 居中 右对齐 imitationlearning0.85reinforcementfrom0.72humanfeedback0.91
5. 代码块Python import torch import torch.nn.functional as F def behavior_cloning_loss ( pred_actions: torch.Tensor, expert_actions: torch.Tensor, saliency_map: torch.Tensor | None = None ) -> torch.Tensor: """AutoFocus-IL: saliency-weighted BC loss.""" mse = F.mse_loss(pred_actions, expert_actions, reduction = 'none' ) if saliency_map is not None : mse = mse * saliency_map.unsqueeze( - 1 ) # Eq. (5) return mse.mean() Bash #!/bin/bash # 训练启动脚本 ACCELERATE_LOG_LEVEL = info accelerate launch \ --num_processes=4 \ --mixed_precision=bf16 \ train.py \ --model pi05_base \ --dataset robocasa-365 \ --batch_size 256 \ --lr 3e-4 YAML 配置# astro.config.mjs 等效配置 markdown : remarkPlugins : - remark-math rehypePlugins : - rehype-katex 6. 行内公式行内公式示例：策略梯度 ∇ θ J ( θ ) = E π [ ∇ θ log π θ ( a ∣ s ) Q π ( s , a ) ] \nabla_\theta J(\theta) = \mathbb{E}_\pi \left[ \nabla_\theta \log \pi_\theta(a|s) \, Q^\pi(s,a) \right] ∇ θ J ( θ ) = E π [ ∇ θ log π θ ( a ∣ s ) Q π ( s , a ) ] V π ( s ) = E π [ ∑ t = 0 ∞ γ t r t ∣ s 0 = s ] V^\pi(s) = \mathbb{E}_\pi \left[ \sum_{t=0}^\infty \gamma^t r_t \mid s_0 = s \right] V π ( s ) = E π [ ∑ t = 0 ∞ γ t r t ∣ s 0 = s ]
常见符号：μ , σ , θ , ϕ , π , α , β , γ , ϵ , λ , τ , ω , ∇ , ∂ , ∑ , ∏ , ∫ , ∞ , ≈ , ∝ , ∼ , N , L , E , R \mu, \sigma, \theta, \phi, \pi, \alpha, \beta, \gamma, \epsilon, \lambda, \tau, \omega, \nabla, \partial, \sum, \prod, \int, \infty, \approx, \propto, \sim, \mathcal{N}, \mathcal{L}, \mathbb{E}, \mathbb{R} μ , σ , θ , ϕ , π , α , β , γ , ϵ , λ , τ , ω , ∇ , ∂ , ∑ , ∏ , ∫ , ∞ , ≈ , ∝ , ∼ , N , L , E , R
7. 独立公式 + 编号策略梯度定理
∇ θ J ( θ ) = E τ ∼ π θ [ ∑ t = 0 T ∇ θ log π θ ( a t ∣ s t ) A ^ t ] (1) \nabla_\theta J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta} \left[ \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t \mid s_t) \, \hat{A}_t \right] \tag{1} ∇ θ J ( θ ) = E τ ∼ π θ [ t = 0 ∑ T ∇ θ log π θ ( a t ∣ s t ) A ^ t ] ( 1 ) 其中 A ^ t \hat{A}_t A ^ t
A ^ t = ∑ l = 0 ∞ ( γ λ ) l δ t + l , δ t = r t + γ V ( s t + 1 ) − V ( s t ) (2) \hat{A}_t = \sum_{l=0}^{\infty} (\gamma \lambda)^l \delta_{t+l}, \quad \delta_t = r_t + \gamma V(s_{t+1}) - V(s_t) \tag{2} A ^ t = l = 0 ∑ ∞ ( γ λ ) l δ t + l , δ t = r t + γ V ( s t + 1 ) − V ( s t ) ( 2 ) Diffusion Policy 损失L diff ( θ ) = E ϵ , t , a 0 [ ∥ ϵ − ϵ θ ( a t , t , s ) ∥ 2 ] (3) \mathcal{L}_{\text{diff}}(\theta) = \mathbb{E}_{\epsilon, t, \mathbf{a}_0} \left[ \| \epsilon - \epsilon_\theta(\mathbf{a}_t, t, \mathbf{s}) \|^2 \right] \tag{3} L diff ( θ ) = E ϵ , t , a 0 [ ∥ ϵ − ϵ θ ( a t , t , s ) ∥ 2 ] ( 3 ) VLA 记忆模块 (DynaMem) m t = GRU ϕ ( z t , m t − 1 ) , z t = f ψ ( v t , q t ) (4) \mathbf{m}_t = \text{GRU}_\phi(\mathbf{z}_t, \mathbf{m}_{t-1}), \quad \mathbf{z}_t = f_\psi(\mathbf{v}_t, \mathbf{q}_t) \tag{4} m t = GRU ϕ ( z t , m t − 1 ) , z t = f ψ ( v t , q t ) ( 4 ) a t = g ω ( m t , o t ) (5) a_t = g_\omega(\mathbf{m}_t, \mathbf{o}_t) \tag{5} a t = g ω ( m t , o t ) ( 5 ) KL 散度正则化D KL ( p ∥ q ) = ∑ x p ( x ) log p ( x ) q ( x ) (6) D_{\text{KL}}(p \parallel q) = \sum_x p(x) \log \frac{p(x)}{q(x)} \tag{6} D KL ( p ∥ q ) = x ∑ p ( x ) log q ( x ) p ( x ) ( 6 ) 8. 矩阵协方差矩阵
Σ = [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] (7) \Sigma = \begin{bmatrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix} \tag{7} Σ = σ xx σ y x σ z x σ x y σ y y σ z y σ x z σ y z σ z z ( 7 ) 旋转矩阵 (SO(3))
R x ( θ ) = [ 1 0 0 0 cos θ − sin θ 0 sin θ cos θ ] , R y ( θ ) = [ cos θ 0 sin θ 0 1 0 − sin θ 0 cos θ ] (8) R_x(\theta) = \begin{bmatrix}
1 & 0 & 0 \\
0 & \cos\theta & -\sin\theta \\
0 & \sin\theta & \cos\theta
\end{bmatrix}, \quad
R_y(\theta) = \begin{bmatrix}
\cos\theta & 0 & \sin\theta \\
0 & 1 & 0 \\
-\sin\theta & 0 & \cos\theta
\end{bmatrix} \tag{8} R x ( θ ) = 1 0 0 0 cos θ sin θ 0 − sin θ cos θ , R y ( θ ) = cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ ( 8 ) 大矩阵
[ K 11 K 12 ⋯ K 1 n K 21 K 22 ⋯ K 2 n ⋮ ⋮ ⋱ ⋮ K n 1 K n 2 ⋯ K n n ] [ w 1 w 2 ⋮ w n ] = [ y 1 y 2 ⋮ y n ] (9) \begin{bmatrix}
K_{11} & K_{12} & \cdots & K_{1n} \\
K_{21} & K_{22} & \cdots & K_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
K_{n1} & K_{n2} & \cdots & K_{nn}
\end{bmatrix}
\begin{bmatrix}
w_1 \\ w_2 \\ \vdots \\ w_n
\end{bmatrix}
=
\begin{bmatrix}
y_1 \\ y_2 \\ \vdots \\ y_n
\end{bmatrix} \tag{9} K 11 K 21 ⋮ K n 1 K 12 K 22 ⋮ K n 2 ⋯ ⋯ ⋱ ⋯ K 1 n K 2 n ⋮ K nn w 1 w 2 ⋮ w n = y 1 y 2 ⋮ y n ( 9 ) 9. 多行对齐公式L total = L BC + λ 1 L saliency + λ 2 L reg L BC = 1 N ∑ i = 1 N ∥ a i − π θ ( o i ) ∥ 2 L saliency = 1 N ∑ i = 1 N ∑ p ∈ Ω M i ( p ) ⋅ ∥ ∇ π θ ( o i ) p ∥ 2 (10) \begin{aligned}
\mathcal{L}_{\text{total}} &= \mathcal{L}_{\text{BC}} + \lambda_1 \mathcal{L}_{\text{saliency}} + \lambda_2 \mathcal{L}_{\text{reg}} \\[4pt]
\mathcal{L}_{\text{BC}} &= \frac{1}{N} \sum_{i=1}^N \| a_i - \pi_\theta(o_i) \|^2 \\[4pt]
\mathcal{L}_{\text{saliency}} &= \frac{1}{N} \sum_{i=1}^N \sum_{p \in \Omega} M_i(p) \cdot \| \nabla \pi_\theta(o_i)_p \|^2 \tag{10}
\end{aligned} L total L BC L saliency = L BC + λ 1 L saliency + λ 2 L reg = N 1 i = 1 ∑ N ∥ a i − π θ ( o i ) ∥ 2 = N 1 i = 1 ∑ N p ∈ Ω ∑ M i ( p ) ⋅ ∥∇ π θ ( o i ) p ∥ 2 ( 10 ) 10. 分段函数f ( x ) = { 0 if x < 0 x 2 if 0 ≤ x < 1 2 x − 1 if x ≥ 1 (11) f(x) = \begin{cases}
0 & \text{if } x < 0 \\
x^2 & \text{if } 0 \leq x < 1 \\
2x - 1 & \text{if } x \geq 1
\end{cases} \tag{11} f ( x ) = ⎩ ⎨ ⎧ 0 x 2 2 x − 1 if x < 0 if 0 ≤ x < 1 if x ≥ 1 ( 11 ) 11. 公式引用如公式 ( 1 ) (1) ( 1 ) ( 3 ) (3) ( 3 ) ( 10 ) (10) ( 10 )
对比 ( 6 ) (6) ( 6 ) ( 3 ) (3) ( 3 )
L ELBO = E q [ − log p θ ( x ∣ z ) ] + D KL ( q ϕ ( z ∣ x ) ∥ p ( z ) ) (12) \mathcal{L}_{\text{ELBO}} = \mathbb{E}_q \left[ -\log p_\theta(x|z) \right] + D_{\text{KL}}(q_\phi(z|x) \parallel p(z)) \tag{12} L ELBO = E q [ − log p θ ( x ∣ z ) ] + D KL ( q ϕ ( z ∣ x ) ∥ p ( z )) ( 12 ) 12. 综合示例：AutoFocus-IL 方法问题形式化
给定演示数据集 D = { ( o i , a i ) } i = 1 N \mathcal{D} = \{(o_i, a_i)\}_{i=1}^N D = {( o i , a i ) } i = 1 N o i ∈ R H × W × 3 o_i \in \mathbb{R}^{H \times W \times 3} o i ∈ R H × W × 3 a i ∈ R d a_i \in \mathbb{R}^d a i ∈ R d
VLM 生成显著性图：
M ( o ) = VLM ( o , prompt ) ∈ [ 0 , 1 ] H × W (13) M(o) = \text{VLM}(o, \text{prompt}) \in [0, 1]^{H \times W} \tag{13} M ( o ) = VLM ( o , prompt ) ∈ [ 0 , 1 ] H × W ( 13 ) 显著性加权的 BC 损失：
L ( θ ) = 1 N ∑ i = 1 N ∑ p ∈ Ω M i ( p ) ⋅ ∥ a i − π θ ( o i ) ∥ 2 + λ ∥ θ ∥ 2 2 (14) \mathcal{L}(\theta) = \frac{1}{N} \sum_{i=1}^N \sum_{p \in \Omega} M_i(p) \cdot \| a_i - \pi_\theta(o_i) \|^2 + \lambda \| \theta \|_2^2 \tag{14} L ( θ ) = N 1 i = 1 ∑ N p ∈ Ω ∑ M i ( p ) ⋅ ∥ a i − π θ ( o i ) ∥ 2 + λ ∥ θ ∥ 2 2 ( 14 ) 实验结果
方法 CARLA (Success %) WidowX (Success %) BC (baseline) 42.3 ± 3.1 42.3 \pm 3.1 42.3 ± 3.1 38.7 ± 4.2 38.7 \pm 4.2 38.7 ± 4.2 BC + Gaze 61.5 ± 2.8 61.5 \pm 2.8 61.5 ± 2.8 55.2 ± 3.6 55.2 \pm 3.6 55.2 ± 3.6 BC + GradCAM 54.8 ± 3.3 54.8 \pm 3.3 54.8 ± 3.3 47.1 ± 4.0 47.1 \pm 4.0 47.1 ± 4.0 AutoFocus-IL 67.2 ± 2.1 \mathbf{67.2 \pm 2.1} 67.2 ± 2.1 60.8 ± 3.0 \mathbf{60.8 \pm 3.0} 60.8 ± 3.0
AutoFocus-IL 无需人工标注 即超越了需要眼动数据监督的 SOTA 方法。
13. 分隔线以上就是 Astro + KaTeX 博客的全部 Markdown 渲染能力演示。
总结 ：支持标准 Markdown（标题/列表/表格/代码/引用），KaTeX 公式渲染 （行内 + 独立 + \tag{n} 编号 + 矩阵 + 多行对齐 + 分段函数 + \ref 交叉引用），以及 深色模式自动适配 ✨
如果你看到所有公式都正确渲染、表格对齐、代码高亮正常——恭喜，你的 Astro 学术博客已经完全就绪！
Bibliography
[1 ] Probability and Statistics , 4th ed. Pearson, 2012.[2 ] Mathematical Statistics and Data Analysis , 3rd ed. Duxbury Press, 2006.[3 ] Werke , vol. 7, pp. 1–280, 1809.[4 ] The American Statistician , vol. 36, no. 2, pp. 137–138, 1982.
