ClawLab
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MathLab


			
				
					
						
						
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    <title>7 - 卷积神经网络</title>
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    <h1>👾 Day 7: 卷积神经网络</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        全连接网络处理图像时参数爆炸，无法捕捉空间局部性。CNN 通过卷积核共享参数，提取局部特征。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象一个滤波器在图像上滑动，每个位置计算局部区域的加权和，提取边缘、纹理等特征。
        <div class="youtube">🎬 B 站搜索：<code>卷积神经网络 CNN 直观解释</code></div>
    </div>

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        <div class="module-title">3️⃣【符号解码字典】The Symbol Decoder</div>
        
        <div class="symbol-map"><strong>$X$</strong> → <code>input_tensor</code> (输入特征图，shape: [C, H, W])</div>
        <div class="symbol-map"><strong>$W$</strong> → <code>self.weight</code> (卷积核，shape: [C_out, C_in, k, k])</div>
        <div class="symbol-map"><strong>$b$</strong> → <code>self.bias</code> (偏置，shape: [C_out])</div>
        <div class="symbol-map"><strong>$Y$</strong> → <code>output_tensor</code> (输出特征图)</div>
        <div class="symbol-map"><strong>$s$</strong> → <code>stride</code> (步长)</div>
        <div class="symbol-map"><strong>$p$</strong> → <code>padding</code> (填充)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 卷积运算

$$(X * W)_{i,j} = \sum_{m=0}^{k-1} \sum_{n=0}^{k-1} W_{m,n} \cdot X_{i+m, j+n}$$

### 输出尺寸计算

$$H_{out} = \left\lfloor \frac{H_{in} + 2p - k}{s} \right\rfloor + 1$$

$$W_{out} = \left\lfloor \frac{W_{in} + 2p - k}{s} \right\rfloor + 1$$

其中 $k$ 是卷积核尺寸，$p$ 是填充，$s$ 是步长。

### 梯度反向传播

$$\frac{\partial L}{\partial W} = X_{rotated} * \frac{\partial L}{\partial Y}$$

$$\frac{\partial L}{\partial X} = \frac{\partial L}{\partial Y} * W_{rotated}$$

其中 $rotated$ 表示 180 度旋转。

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        卷积运算复杂度 $O(C_{in} \cdot C_{out} \cdot k^2 \cdot H \cdot W)$。使用深度可分离卷积（Depthwise Separable）加速。
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    <div class="module">
        <div class="module-title">6️⃣【今日靶机】The OJ Mission</div>
        <div class="warning">🎯 任务：<code>cd exercises/ && python3 day7_task.py</code></div>
        实现 2D 卷积层的 forward 和 backward，在 MNIST 数据集上验证卷积特征提取效果。
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