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统计学
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抽样
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推断
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概率论
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计数
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</a>
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概率概念
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分布
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</span>
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</a>
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贝叶斯
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</span>
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</a>
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信息论
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</span>
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机器学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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机器学习
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经典机器学习
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梯度机器学习
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</span>
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</a>
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深度学习
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</span>
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</a>
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强化学习
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</span>
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</a>
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分布式深度学习
|
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</span>
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|
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|
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|
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</a>
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计算语言学
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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语言学基础
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</a>
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嵌入与序列模型
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</span>
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</a>
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Transformer 与语言模型
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</span>
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|
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</a>
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<span class="md-ellipsis">
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高级文本生成
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
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计算机视觉
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计算机视觉
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图像基础
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</span>
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</a>
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多模态学习
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多模态学习
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多模态表征
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视觉语言模型
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</span>
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</span>
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</a>
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统一多模态架构
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</span>
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|
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|
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自主系统
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感知
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视觉-语言-动作模型
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</span>
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</a>
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自动驾驶
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</a>
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太空与极端机器人
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</span>
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图神经网络
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几何深度学习
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几何深度学习
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对称性与群
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<li class="md-nav__item">
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<a href="#_3" class="md-nav__link">
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不变性与等变性
|
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</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#_4" class="md-nav__link">
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<span class="md-ellipsis">
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五个几何域
|
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</a>
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|
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|
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|
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<a href="#_5" class="md-nav__link">
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<span class="md-ellipsis">
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尺度分离与粗化
|
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||
</a>
|
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|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="#colabnotebook" class="md-nav__link">
|
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<span class="md-ellipsis">
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编程任务（使用CoLab或notebook）
|
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</a>
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</li>
|
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</ul>
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</nav>
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<span class="md-ellipsis">
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图论
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|
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|
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</span>
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|
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|
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|
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</a>
|
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</li>
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<span class="md-ellipsis">
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图神经网络
|
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|
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</span>
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|
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|
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|
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</a>
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<span class="md-ellipsis">
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图注意力网络
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../05.%203d%20graph%20networks/" class="md-nav__link">
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<span class="md-ellipsis">
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3D 图网络
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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计算机与操作系统
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计算机与操作系统
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离散数学
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|
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</span>
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|
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</a>
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计算机体系结构
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</span>
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|
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|
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</a>
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操作系统
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</span>
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</a>
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并发与并行
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
|
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</li>
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编程语言
|
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|
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|
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|
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</span>
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|
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|
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|
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数据结构与算法
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基础
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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|
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数组与哈希
|
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|
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|
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|
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</span>
|
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|
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|
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|
||
</a>
|
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</li>
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|
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|
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|
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<span class="md-ellipsis">
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链表、栈与队列
|
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|
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|
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|
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</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
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|
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树
|
||
|
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|
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|
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</span>
|
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|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
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|
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<span class="md-ellipsis">
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图
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|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
|
||
</li>
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2014%3A%20data%20structures%20and%20algorithms/05.%20sorting%20and%20search/" class="md-nav__link">
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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排序与搜索
|
||
|
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|
||
|
||
</span>
|
||
|
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|
||
|
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</a>
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|
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</ul>
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</nav>
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|
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生产级软件工程
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生产级软件工程
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<span class="md-ellipsis">
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Linux 与命令行
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</span>
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|
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</a>
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<span class="md-ellipsis">
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Git 与仓库管理
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</span>
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|
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</a>
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|
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</a>
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测试与质量保障
|
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</span>
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|
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|
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</a>
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|
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|
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<span class="md-ellipsis">
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|
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部署与 DevOps
|
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|
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|
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|
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</span>
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|
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|
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|
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SIMD 与 GPU 编程
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硬件基础
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ARM 与 NEON
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x86 与 AVX
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Triton、TPU 与 Pallas
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RISC-V 与嵌入式系统
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Vulkan Compute 与跨平台 GPU
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AI 推理
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量化
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边缘推理
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</a>
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<span class="md-ellipsis">
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扩缩与部署
|
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</span>
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|
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ML 系统设计
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应用 AI
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药物发现
|
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|
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</a>
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|
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医疗健康
|
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|
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</span>
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|
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|
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前沿 AI
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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前沿 AI
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/01.%20quantum%20machine%20learning/" class="md-nav__link">
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<span class="md-ellipsis">
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量子机器学习
|
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|
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</span>
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|
||
|
||
|
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</a>
|
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<span class="md-ellipsis">
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神经形态计算
|
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|
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</span>
|
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|
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|
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|
||
</a>
|
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太空数据中心
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</span>
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</a>
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<span class="md-ellipsis">
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去中心化 AI
|
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</span>
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</a>
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脑机接口
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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<div class="md-sidebar__inner">
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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目录
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|
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|
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<a href="#_2" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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对称性与群
|
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|
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</span>
|
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</a>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="#_3" class="md-nav__link">
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<span class="md-ellipsis">
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不变性与等变性
|
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</span>
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</a>
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</li>
|
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<li class="md-nav__item">
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<a href="#_4" class="md-nav__link">
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<span class="md-ellipsis">
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五个几何域
|
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|
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</span>
|
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</a>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="#_5" class="md-nav__link">
|
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<span class="md-ellipsis">
|
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|
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尺度分离与粗化
|
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|
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|
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</a>
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</li>
|
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<a href="#colabnotebook" class="md-nav__link">
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<span class="md-ellipsis">
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编程任务（使用CoLab或notebook）
|
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<div class="md-content" data-md-component="content">
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<article class="md-content__inner md-typeset">
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|
||
|
||
|
||
|
||
<h1 id="_1">几何深度学习<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>几何深度学习是揭示CNN、Transformer和GNN皆遵循同一原理——利用对称性——的统一框架。本章涵盖对称群、群作用、不变性、等变性、五个几何域以及尺度分离</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在本书中，我们已经学习了多种架构：图像的CNN（第8章）、语言的Transformer（第7章）以及序列决策的RL策略（第6章）。它们看上去像是为完全不同的问题设计的完全不同的模型。但背后存在一个更深层的模式。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>几何深度学习</strong>揭示出所有这些架构都是同一个思想的实例：构建尊重数据<strong>对称性</strong>的网络。CNN利用图像中的平移对称性。Transformer利用序列中的置换对称性（注意力不依赖于绝对位置）。GNN利用图中的置换对称性。一旦看清这一点，众多架构就变成了一个统一的连贯框架。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_2">对称性与群<a class="headerlink" href="#_2" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>一个对象的<strong>对称性</strong>是使其保持不变的变换。正方形有8种对称性：4种旋转（0°、90°、180°、270°）和4种反射。圆有无限多种：任何绕其中心的旋转。关键洞察在于，对称性告诉你什么是不重要的，而知道什么不重要的对于学习来说极为强大。</p>
|
||
</li>
|
||
<li>
|
||
<p>用机器学习的术语来说：如果一个任务具有对称性，那么无论看到输入的哪种"版本"，模型都应给出相同的答案。猫检测器无论猫在图像的左上角还是右下角都应能工作。这就是平移对称性。</p>
|
||
</li>
|
||
<li>
|
||
<p>对称性通过<strong>群</strong>来形式化。一个群 <span class="arithmatex">\(G\)</span> 是一个具有四个性质的变换集合：</p>
|
||
<ul>
|
||
<li><strong>封闭性</strong>：两个变换的组合产生集合中的另一个变换。先旋转90°再旋转90°得到180°，也属于该集合。</li>
|
||
<li><strong>结合律</strong>：<span class="arithmatex">\((g_1 \circ g_2) \circ g_3 = g_1 \circ (g_2 \circ g_3)\)</span>。分组的顺序无关紧要（回顾第2章中矩阵乘法的结合律）。</li>
|
||
<li><strong>单位元</strong>：存在一个"什么也不做"的变换 <span class="arithmatex">\(e\)</span>，使得 <span class="arithmatex">\(e \circ g = g \circ e = g\)</span>。</li>
|
||
<li><strong>逆元</strong>：每个变换都有撤销操作：<span class="arithmatex">\(g \circ g^{-1} = e\)</span>。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>这些公理与向量空间（第1章）的公理相同，但应用于变换而非向量。其联系十分深刻：群作用于向量空间，而神经网络必须尊重这种作用。</p>
|
||
</li>
|
||
<li>
|
||
<p>深度学习中出现的关键群：</p>
|
||
<ul>
|
||
<li><strong>平移群</strong> <span class="arithmatex">\((\mathbb{R}^n, +)\)</span>：平移图像或信号。这是CNN利用的对称性。</li>
|
||
<li><strong>对称群</strong> <span class="arithmatex">\(S_n\)</span>：<span class="arithmatex">\(n\)</span> 个元素的所有置换。这是GNN和Transformer利用的对称性（重新排序节点或标记不应改变结果）。</li>
|
||
<li><strong>旋转群</strong> <span class="arithmatex">\(SO(n)\)</span>：<span class="arithmatex">\(n\)</span> 维空间中的所有旋转。<span class="arithmatex">\(SO(2)\)</span> 是平面旋转，<span class="arithmatex">\(SO(3)\)</span> 是三维旋转（对分子和3D视觉任务至关重要）。</li>
|
||
<li><strong>欧几里得群</strong> <span class="arithmatex">\(E(n)\)</span>：所有旋转、反射和平移。物理空间的对称性。</li>
|
||
<li><strong>特殊欧几里得群</strong> <span class="arithmatex">\(SE(n)\)</span>：旋转和平移（不含反射）。刚体运动的对称性。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>群作用</strong>描述了群如何变换数据。如果 <span class="arithmatex">\(G\)</span> 是一个群，<span class="arithmatex">\(X\)</span> 是数据空间，则作用 <span class="arithmatex">\(\rho: G \times X \to X\)</span> 将每个群元素 <span class="arithmatex">\(g\)</span> 和数据点 <span class="arithmatex">\(x\)</span> 映射到一个变换后的点 <span class="arithmatex">\(\rho(g, x)\)</span>。对于图像，平移群通过平移像素坐标来作用。对于图，对称群通过重新标记节点来作用。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_3">不变性与等变性<a class="headerlink" href="#_3" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>给定一个对称群，函数可以通过两种重要方式与之关联：</p>
|
||
</li>
|
||
<li>
|
||
<p>函数 <span class="arithmatex">\(f\)</span> 对群 <span class="arithmatex">\(G\)</span> 是<strong>不变</strong>的，如果输入变换后输出不变：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(\rho(g, x)) = f(x) \quad \text{对于所有 } g \in G\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>示例：图像的总体亮度不因平移而改变。图像分类应是平移不变的："猫"的类别无论猫在何处都是一样的。</p>
|
||
</li>
|
||
<li>
|
||
<p>函数 <span class="arithmatex">\(f\)</span> 对群 <span class="arithmatex">\(G\)</span> 是<strong>等变</strong>的，如果变换输入会对等地变换输出：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(\rho_{\text{in}}(g, x)) = \rho_{\text{out}}(g, f(x)) \quad \text{对于所有 } g \in G\]</div>
|
||
<ul>
|
||
<li>示例：如果将图像向右平移5个像素，CNN中的特征图也会向右平移5个像素。卷积操作是平移等变的：它保留了空间关系。目标检测应该是等变的：如果猫移动了，边界框也应随之移动。</li>
|
||
</ul>
|
||
<p><img alt="不变性：输出不随变换而改变。等变性：输出随之对应变换" src="../../images/invariance_vs_equivariance.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>区分两者的重要性在于：<strong>中间层</strong>通常应是等变的（为下游层保留结构），而<strong>最终输出</strong>应是不变的（答案不应依赖于变换）。CNN通过堆叠等变卷积层，然后在末尾应用全局池化（它是不变的）来实现这一点。</p>
|
||
</li>
|
||
<li>
|
||
<p>将等变性构建到架构中比从数据中学习它要高效得多。一个具有权重共享的平移等变CNN所需的参数远少于一个必须独立学习"位置(10,10)处的猫"和"位置(200,150)处的猫"的全连接网络。对称性约束指数级地缩小了假设空间。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_4">五个几何域<a class="headerlink" href="#_4" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>几何深度学习识别出数据的<strong>五个基本域</strong>，每个域都有其自己的对称群。每一个神经网络架构都可以被理解为利用其中某个域的对称性。</li>
|
||
</ul>
|
||
<p><img alt="五个几何域：网格、集合、序列、图和流形，各有其对称性和架构" src="../../images/five_geometric_domains.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>1. 网格（欧几里得数据）</strong>：图像、音频频谱图、体数据。底层结构是具有平移对称性的规则网格。群是平移群（可能再加上旋转和反射）。利用这种对称性的架构是<strong>CNN</strong>：卷积正是平移等变的操作。空间位置上的权重共享就是平移等变性的具体实现。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>2. 集合（无序集合）</strong>：点云、粒子系统。对称性是置换不变性：元素的顺序无关紧要。架构是<strong>DeepSets</strong>（以及第8章的PointNet）：对每个元素应用共享函数，然后用置换不变操作（求和、均值或取最大值）进行聚合。形式上，<span class="arithmatex">\(f(\{x_1, \ldots, x_n\}) = \phi\left(\sum_i \psi(x_i)\right)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>3. 序列（有序数据）</strong>：文本、时间序列。序列是一维网格，但有一个微妙之处：对称性更加细致。绝对位置可能重要也可能不重要。RNN以自回归方式处理序列。带位置编码的Transformer可以关注任何位置，其自注意力在加入位置编码之前是置换等变的。这就是Transformer泛化能力如此之强的原因：它们从置换等变开始，然后仅添加必要的位置结构。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>4. 图（关系数据）</strong>：社交网络、分子、知识图谱。对称性是节点的置换：重新标记节点不应改变图的性质。架构是<strong>GNN</strong>：连接节点之间传递消息，使用不依赖于节点顺序的共享函数。这是本章剩余部分的重点。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>5. 流形和网格</strong>：曲面、3D形状。对称性包括微分同胚（光滑变形）。架构使用内在算子（例如拉普拉斯-贝尔特拉米算子），这些算子由曲面几何本身定义，与曲面在空间中的嵌入方式无关。这联系到微分几何，并适用于形状分析、球面上的气候建模和蛋白质表面分析。</p>
|
||
</li>
|
||
<li>
|
||
<p>这个框架的强大之处在于其统一性。CNN是网格图上的GNN。Transformer是完全连接图上的GNN。DeepSets是没有边的GNN。将这些视为同一原理的实例，指导着新架构的设计：识别数据的对称性，然后构建一个尊重它的网络。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_5">尺度分离与粗化<a class="headerlink" href="#_5" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>真实世界的数据具有多尺度结构。一幅图像有细粒度纹理（像素级）、局部模式（边缘、角点）、物体部件（车轮、窗户）和全局结构（整个场景）。一个分子有原子级特征、官能团和整体分子形状。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>尺度分离</strong>是这样一个原理：这些细节层次可以分层处理——先捕获局部结构，然后逐步聚合成更粗粒度的表示。这就是<strong>粗化</strong>或<strong>池化</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p>在CNN中，池化层（最大池化、平均池化）对空间分辨率进行下采样，迫使高层捕获更大尺度的模式。在感受野视角（第8章）中，更深层能"看到"更多的图像。这就是尺度分离的实际应用。</p>
|
||
</li>
|
||
<li>
|
||
<p>在图（graph）中，粗化意味着将节点群聚为"超节点"，生成一个保留基本结构的更小图。这就是图池化，我们将在文件3中详细讨论。它与图像池化直接类似：降低分辨率的同时保留重要特征。</p>
|
||
</li>
|
||
<li>
|
||
<p>在序列中，分层处理（例如句子→段落→文档）在不同时间或语义尺度捕获结构。Swin Transformer（第8章）通过其移位窗口层次结构将这一思想应用于图像。</p>
|
||
</li>
|
||
<li>
|
||
<p>数学上，粗化定义了一个<strong>逐渐抽象的表示层次</strong>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[x \xrightarrow{\text{局部特征}} h^{(1)} \xrightarrow{\text{粗化}} h^{(2)} \xrightarrow{\text{粗化}} \cdots \xrightarrow{\text{全局}} y\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>在每个层次，表示相对于该层次的对称群是等变的。最后的全局表示是不变的，捕获了输入的本质而不受无关变换的影响。</p>
|
||
</li>
|
||
<li>
|
||
<p>这就是为什么对于结构化数据，深层网络比浅层网络效果更好：每一层增加一个抽象层次，多个等变层的组合从简单的局部特征构建出复杂的不变特征。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colabnotebook">编程任务（使用CoLab或notebook）<a class="headerlink" href="#colabnotebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>验证卷积的平移等变性。对图像应用卷积，然后平移图像再次卷积。检查输出是否互为平移版本。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 一维信号和一个简单滤波器</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">signal</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">kernel</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># 先卷积再平移</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">conv_result</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">convolve</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="n">kernel</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s2">"same"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">shifted_signal</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">conv_shifted</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">convolve</span><span class="p">(</span><span class="n">shifted_signal</span><span class="p">,</span> <span class="n">kernel</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s2">"same"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">shifted_conv</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">conv_result</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"先卷积再平移: </span><span class="si">{</span><span class="n">shifted_conv</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"先平移再卷积: </span><span class="si">{</span><span class="n">conv_shifted</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"等变性: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">shifted_conv</span><span class="p">,</span><span class="w"> </span><span class="n">conv_shifted</span><span class="p">,</span><span class="w"> </span><span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>验证DeepSets风格聚合的置换不变性。对集合中的每个元素应用共享函数，求和结果，并检查输出是否不依赖于元素顺序。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 4个向量的"集合"（顺序应无关紧要）</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">6.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">]])</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="c1"># 简单的共享函数：逐元素平方</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">psi</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">v</span><span class="p">:</span> <span class="n">v</span> <span class="o">**</span> <span class="mi">2</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="c1"># 通过求和聚合</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="k">def</span><span class="w"> </span><span class="nf">deepsets</span><span class="p">(</span><span class="n">points</span><span class="p">):</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">vmap</span><span class="p">(</span><span class="n">psi</span><span class="p">)(</span><span class="n">points</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="c1"># 原始顺序</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">result1</span> <span class="o">=</span> <span class="n">deepsets</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="c1"># 置换后的顺序</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">perm</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">result2</span> <span class="o">=</span> <span class="n">deepsets</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">perm</span><span class="p">])</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"原始顺序: </span><span class="si">{</span><span class="n">result1</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"置换顺序: </span><span class="si">{</span><span class="n">result2</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"不变性: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">result1</span><span class="p">,</span><span class="w"> </span><span class="n">result2</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>探索群结构。通过检查封闭性、结合律、单位元和逆元，验证二维旋转矩阵构成群。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">rot2d</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)],</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="p">[</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)]])</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">R1</span> <span class="o">=</span> <span class="n">rot2d</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">6</span><span class="p">)</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">R2</span> <span class="o">=</span> <span class="n">rot2d</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">4</span><span class="p">)</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">R3</span> <span class="o">=</span> <span class="n">rot2d</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="c1"># 封闭性：两个旋转的乘积还是一个旋转</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">R12</span> <span class="o">=</span> <span class="n">R1</span> <span class="o">@</span> <span class="n">R2</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"封闭性 (行列式=1, 正交): det=</span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">R12</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, "</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="sa">f</span><span class="s2">"R^T R = I: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R12</span><span class="o">.</span><span class="n">T</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">R12</span><span class="p">,</span><span class="w"> </span><span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="c1"># 结合律</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"结合律: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">((</span><span class="n">R1</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">R2</span><span class="p">)</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">R3</span><span class="p">,</span><span class="w"> </span><span class="n">R1</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="p">(</span><span class="n">R2</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">R3</span><span class="p">),</span><span class="w"> </span><span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="c1"># 单位元</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">I</span> <span class="o">=</span> <span class="n">rot2d</span><span class="p">(</span><span class="mf">0.0</span><span class="p">)</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"单位元: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R1</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">I</span><span class="p">,</span><span class="w"> </span><span class="n">R1</span><span class="p">,</span><span class="w"> </span><span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="c1"># 逆元</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="n">R1_inv</span> <span class="o">=</span> <span class="n">rot2d</span><span class="p">(</span><span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">6</span><span class="p">)</span>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"逆元: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R1</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">R1_inv</span><span class="p">,</span><span class="w"> </span><span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="n">atol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
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