ClawLab
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MathLab


			
				
					
						
						
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    <title>2 - K 近邻</title>
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    <h1>👾 Day 2: K 近邻</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        基于模型的算法需要训练过程，无法利用新样本。K 近邻是实例学习（惰性学习），直接存储训练数据用于预测。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象在一个平面上，新点周围的 k 个最近邻居多数是红色，那它也被预测为红色。距离越近影响越大。
        <div class="youtube">🎬 B 站搜索：<code>K 近邻算法 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> (待预测样本，shape: [d])</div>
        <div class="symbol-map"><strong>$X$</strong> → <code>self.X_train</code> (训练集，shape: [N, d])</div>
        <div class="symbol-map"><strong>$y$</strong> → <code>label</code> (待预测标签)</div>
        <div class="symbol-map"><strong>$Y$</strong> → <code>self.y_train</code> (训练标签，shape: [N])</div>
        <div class="symbol-map"><strong>$k$</strong> → <code>self.k</code> (近邻数量)</div>
        <div class="symbol-map"><strong>$p$</strong> → <code>p</code> (Lp 范数，p=2 为欧氏距离)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 距离计算（Lp 范数）

$$d(x, x^{(i)}) = \left( \sum_{j=1}^{d} |x^{(j)} - x^{(i)(j)}|^p \right)^{1/p}$$

当 $p=2$ 时为欧氏距离：

$$d(x, x^{(i)}) = \sqrt{\sum_{j=1}^{d} (x^{(j)} - x^{(i)(j)})^2}$$

### 投票规则

$$\hat{y} = \text{argmode}_{c} \sum_{i=1}^{k} \mathbb{I}(y^{(i)} = c)$$

其中 $\mathbb{I}(\cdot)$ 是指示函数，统计 k 个近邻中各类别出现次数。

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        预测时需要计算到所有训练样本的距离，复杂度 $O(N \cdot d)$。使用 KD 树或 Ball 树加速到 $O(\log N)$。
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        <div class="module-title">6️⃣【今日靶机】The OJ Mission</div>
        <div class="warning">🎯 任务：<code>cd exercises/ && python3 day2_task.py</code></div>
        实现 K 近邻的 distance 和 predict 函数，在手写数字数据集上验证 k 值对准确率的影响。
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