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|
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|
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|
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函数逼近
|
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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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优化
|
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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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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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</a>
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贝叶斯
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</a>
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信息论
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机器学习
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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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计算语言学
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语言学基础
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嵌入与序列模型
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Transformer 与语言模型
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</a>
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高级文本生成
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</span>
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计算机视觉
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图像基础
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目标检测与分割
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ViT 与生成模型
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视频与 3D 视觉
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音频与语音
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数字信号处理
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</a>
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<span class="md-ellipsis">
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自动语音识别
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
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<span class="md-ellipsis">
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语音合成
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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说话人与音频分析
|
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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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源分离与降噪
|
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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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</ul>
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</nav>
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<span class="md-ellipsis">
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多模态学习
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</span>
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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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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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<span class="md-ellipsis">
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图像与视频 Token 化
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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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跨模态生成
|
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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="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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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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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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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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自主系统
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感知
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</span>
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</a>
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机器人学习
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</span>
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</a>
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<span class="md-ellipsis">
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视觉-语言-动作模型
|
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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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<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
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自动驾驶
|
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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="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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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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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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</label>
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几何深度学习
|
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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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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AI 推理
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量化
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边缘推理
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扩缩与部署
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ML 系统设计
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应用 AI
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前沿 AI
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<h1 id="_1">度量与范数<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>范数衡量单个向量的大小；度量衡量两个向量之间的距离。本文涵盖 L1、L2 和 L-无穷范数、欧几里得距离和余弦距离，以及为什么为 kNN、聚类和 ML 中的检索选择合适的距离函数至关重要。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>我们知道向量有模长和方向。但我们如何实际衡量单个向量"有多大"，或者两个向量"有多远"？这就是<strong>范数</strong>和<strong>度量</strong>发挥作用的地方。</p>
|
||
</li>
|
||
<li>
|
||
<p>对标量而言，我们知道 10 > 5，因为它们的值对它们进行了量化，但是我们如何量化一个向量？它的<strong>范数</strong>衡量单个向量的大小。</p>
|
||
</li>
|
||
<li>
|
||
<p>最熟悉的范数是<strong>欧几里得范数</strong>（L2），它就是我们已知的模长公式：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{v}\|_2 = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}\]</div>
|
||
<ul>
|
||
<li>但还有其他衡量大小的方法。想象你在一个街道呈网格状的城市中。你不能斜穿建筑物，所以你旅程的"长度"是沿着每条街道行走的总街区数。这就是<strong>曼哈顿范数</strong>（L1）：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{v}\|_1 = |v_1| + |v_2| + \cdots + |v_n|\]</div>
|
||
<ul>
|
||
<li>或者你可能只关心单个最大的分量，忽略其余部分。这就是<strong>最大范数</strong>（L-无穷）：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{v}\|_\infty = \max(|v_1|, |v_2|, \ldots, |v_n|)\]</div>
|
||
<ul>
|
||
<li>这三个都是<strong>一般 Lp 范数</strong>的特例：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{v}\|_p = (|v_1|^p + |v_2|^p + \cdots + |v_n|^p)^{1/p}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>设置 <span class="arithmatex">\(p = 2\)</span> 得到欧几里得，<span class="arithmatex">\(p = 1\)</span> 得到曼哈顿，而当 <span class="arithmatex">\(p \to \infty\)</span> 时得到最大范数。随着 <span class="arithmatex">\(p\)</span> 增大，最大分量贡献越来越大，直到最终只有它重要。</p>
|
||
</li>
|
||
<li>
|
||
<p>每个范数必须遵守三条规则：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>非负性</strong>：<span class="arithmatex">\(\|\mathbf{v}\| \geq 0\)</span>，且 <span class="arithmatex">\(\|\mathbf{v}\| = 0\)</span> 仅当 <span class="arithmatex">\(\mathbf{v} = \mathbf{0}\)</span>。大小从不为负，只有零向量的大小为零。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>缩放性</strong>：<span class="arithmatex">\(\|c\mathbf{v}\| = |c| \cdot \|\mathbf{v}\|\)</span>。将向量加倍，其大小也加倍。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>三角不等式</strong>：<span class="arithmatex">\(\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|\)</span>。捷径永远不会比绕远路更长。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>现在，<strong>度量</strong>衡量<em>两个</em>向量之间的距离。把它想象成问："这两个点相距多远？"</p>
|
||
</li>
|
||
<li>
|
||
<p>获得度量的最简单方法是使用差值的范数：<span class="arithmatex">\(d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|\)</span>。减去两个向量，然后测量剩余部分的大小。</p>
|
||
</li>
|
||
<li>
|
||
<p>使用欧几里得范数，我们得到熟悉的<strong>欧几里得距离</strong>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[d(\mathbf{u}, \mathbf{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + \cdots + (u_n - v_n)^2}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>使用曼哈顿范数得到<strong>曼哈顿距离</strong>，沿着每个轴的总差异，就像计算两个位置之间的城市街区数。</p>
|
||
</li>
|
||
<li>
|
||
<p>每个度量必须遵守四条规则：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>非负性</strong>：<span class="arithmatex">\(d(\mathbf{u}, \mathbf{v}) \geq 0\)</span>。距离从不为负。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>同一性</strong>：<span class="arithmatex">\(d(\mathbf{u}, \mathbf{v}) = 0\)</span> 当且仅当 <span class="arithmatex">\(\mathbf{u} = \mathbf{v}\)</span>。零距离意味着同一点。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>对称性</strong>：<span class="arithmatex">\(d(\mathbf{u}, \mathbf{v}) = d(\mathbf{v}, \mathbf{u})\)</span>。从 A 到 B 的距离与从 B 到 A 的距离相同。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>三角不等式</strong>：<span class="arithmatex">\(d(\mathbf{u}, \mathbf{w}) \leq d(\mathbf{u}, \mathbf{v}) + d(\mathbf{v}, \mathbf{w})\)</span>。直接走永远不会比绕路更长。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>那么两者之间的关系是什么？范数衡量一个向量，度量衡量两个向量之间的差距。每个范数自然地创建一个度量（通过测量差值），但并非每个度量都来自范数。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，<strong>汉明距离</strong>计算两个向量不同的位置数量。它是一个有效的度量，但并非来自任何范数。</p>
|
||
</li>
|
||
<li>
|
||
<p>在 ML 中，选择合适的范数或度量很重要。</p>
|
||
</li>
|
||
<li>
|
||
<p>L2 距离在求和前对每个差值平方，因此单个大的差值会主导结果。</p>
|
||
</li>
|
||
<li>
|
||
<p>L1 距离对绝对差值求和，平等对待每个差值。与 L2 相比，单个大的差值影响较小。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>计算同一向量的 L1 和 L2 范数。尝试更改值，注意哪个范数对大的分量最敏感，哪个对许多小分量最敏感。然后尝试计算 p 值递增（例如 1、2、5、10、50、100）时的 Lp 范数，观察它如何收敛到 L-无穷值。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">v</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">3.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">l1</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">l2</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">v</span> <span class="o">**</span> <span class="mi">2</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="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"L1: </span><span class="si">{</span><span class="n">l1</span><span class="si">}</span><span class="s2">, L2: </span><span class="si">{</span><span class="n">l2</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>计算两个向量之间的欧几里得距离和曼哈顿距离。尝试让向量彼此靠近或远离，观察每种距离如何不同地响应。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">u</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="mf">3.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">v</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">4.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">euclidean</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">u</span> <span class="o">-</span> <span class="n">v</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">manhattan</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">v</span><span class="p">))</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Euclidean: </span><span class="si">{</span><span class="n">euclidean</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">, Manhattan: </span><span class="si">{</span><span class="n">manhattan</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
</ol>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
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