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<span class="md-ellipsis">
|
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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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</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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经典机器学习
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梯度机器学习
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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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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">向量积<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>向量积是衡量相似性和计算投影的基本运算。本文涵盖内积、点积、余弦相似度、叉积和外积，这些运算支撑了 AI 中的注意力机制、嵌入和几何推理。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>我们已经看到如何相加和缩放向量。但是我们可以<em>相乘</em>两个向量吗？事实证明不止一种方法，每种方法回答不同的问题。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>内积</strong>是一个广义概念：一个接受两个向量并产生一个标量的函数。它是"相乘"向量的抽象蓝图。</p>
|
||
</li>
|
||
<li>
|
||
<p>任何内积必须满足三条规则：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>正定性</strong>：<span class="arithmatex">\(\langle \mathbf{v}, \mathbf{v} \rangle \geq 0\)</span>，且仅对零向量等于零。向量与自身相乘总是给出非负结果。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>对称性</strong>：<span class="arithmatex">\(\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle\)</span>。顺序无关紧要。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>线性性</strong>：<span class="arithmatex">\(\langle a\mathbf{u} + b\mathbf{v}, \mathbf{w} \rangle = a\langle \mathbf{u}, \mathbf{w} \rangle + b\langle \mathbf{v}, \mathbf{w} \rangle\)</span>。它对加法和缩放具有分配性。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>点积</strong>是最常见的内积。它是你几乎到处都会用到的具体版本。对于两个向量 <span class="arithmatex">\(\mathbf{a} = (a_1, a_2, \ldots, a_n)\)</span> 和 <span class="arithmatex">\(\mathbf{b} = (b_1, b_2, \ldots, b_n)\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>将匹配的分量相乘，然后全部加起来。这就是全部。</p>
|
||
</li>
|
||
<li>
|
||
<p>但这个数字<em>意味着</em>什么？点积有一个优美的几何解释：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \, \|\mathbf{b}\| \cos(\theta)\]</div>
|
||
<p><img alt="点积：向量 a 投影到 b 上，显示角度 θ 和投影" src="../../images/dot_product.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>这将点积直接与两个向量之间的角度 <span class="arithmatex">\(\theta\)</span> 联系起来。结果告诉你两个向量在方向上"一致"的程度。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果它们指向相同方向（<span class="arithmatex">\(\theta = 0°\)</span>），<span class="arithmatex">\(\cos(\theta) = 1\)</span> 且点积最大。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果它们正交（<span class="arithmatex">\(\theta = 90°\)</span>），<span class="arithmatex">\(\cos(\theta) = 0\)</span> 且点积恰好为零。这给出了正交性的精确检验。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果它们指向相反方向（<span class="arithmatex">\(\theta = 180°\)</span>），<span class="arithmatex">\(\cos(\theta) = -1\)</span> 且点积为负。</p>
|
||
</li>
|
||
<li>
|
||
<p>向量与自身的点积给出其模长的平方：<span class="arithmatex">\(\mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>点积还给出了<strong>投影</strong>，即一个向量在另一个向量上投下的影子。<span class="arithmatex">\(\mathbf{a}\)</span> 在 <span class="arithmatex">\(\mathbf{b}\)</span> 上的投影为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \, \mathbf{b}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>想象一束光线直射到 <span class="arithmatex">\(\mathbf{b}\)</span> 上。<span class="arithmatex">\(\mathbf{a}\)</span> 在那条线上的影子就是投影。它告诉你 <span class="arithmatex">\(\mathbf{a}\)</span> 有多少位于 <span class="arithmatex">\(\mathbf{b}\)</span> 的方向上。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>余弦相似度</strong>通过除以两个模长来归一化点积：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这会给出一个介于 <span class="arithmatex">\(-1\)</span> 和 <span class="arithmatex">\(1\)</span> 之间的值，衡量方向对齐程度，忽略向量的长度。它广泛应用于 ML 中来比较文档、嵌入和用户偏好等事物。</p>
|
||
</li>
|
||
<li>
|
||
<p>现在，点积接受两个向量并返回标量。<strong>叉积</strong>则相反，它接受两个向量并返回一个<em>新向量</em>。</p>
|
||
</li>
|
||
<li>
|
||
<p>叉积 <span class="arithmatex">\(\mathbf{a} \times \mathbf{b}\)</span> 产生一个同时垂直于 <span class="arithmatex">\(\mathbf{a}\)</span> 和 <span class="arithmatex">\(\mathbf{b}\)</span> 的向量：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} \times \mathbf{b} = (a_2 b_3 - a_3 b_2, \; a_3 b_1 - a_1 b_3, \; a_1 b_2 - a_2 b_1)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>叉积只适用于三维。点积适用于任意维度，而叉积是三维空间特有的。</p>
|
||
</li>
|
||
<li>
|
||
<p>其模长等于由这两个向量形成的平行四边形的面积：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \, \|\mathbf{b}\| \sin(\theta)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>注意模式：点积使用 <span class="arithmatex">\(\cos(\theta)\)</span>，叉积使用 <span class="arithmatex">\(\sin(\theta)\)</span>。点积衡量两个向量对齐的程度，叉积衡量它们在方向上<em>差异</em>的程度。</p>
|
||
</li>
|
||
<li>
|
||
<p>结果的方向遵循<strong>右手定则</strong>：将右手的手指从 <span class="arithmatex">\(\mathbf{a}\)</span> 弯向 <span class="arithmatex">\(\mathbf{b}\)</span>，拇指指向 <span class="arithmatex">\(\mathbf{a} \times \mathbf{b}\)</span> 的方向。</p>
|
||
</li>
|
||
<li>
|
||
<p>与点积不同，叉积<strong>不可交换</strong>：<span class="arithmatex">\(\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\)</span>。交换顺序会翻转方向。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果两个向量平行，它们的叉积是零向量（因为 <span class="arithmatex">\(\sin(0°) = 0\)</span>）。没有面积，没有垂直方向。</p>
|
||
</li>
|
||
<li>
|
||
<p>当你使用两个乘积结合三个向量会发生什么？这就得到了<strong>三重积</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p>xxxxxxxxxx9 1import jax.numpy as jnp23u = jnp.array([1.0, 2.0, 3.0])4v = jnp.array([4.0, 0.0, 1.0])56euclidean = jnp.sqrt(jnp.sum((u - v) ** 2))7manhattan = jnp.sum(jnp.abs(u - v))89print(f"Euclidean: {euclidean:.2f}, Manhattan: {manhattan}")python</p>
|
||
</li>
|
||
<li>
|
||
<p>如果标量三重积为零，则这三个向量<strong>共面</strong>，它们都位于同一个平坦平面上，不形成体积。</p>
|
||
</li>
|
||
<li>
|
||
<p>顺序可以循环而不改变结果：<span class="arithmatex">\(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>向量三重积</strong> <span class="arithmatex">\(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\)</span> 应用两次叉积并返回一个向量。它可以使用恒等式简洁展开：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\]</div>
|
||
<ul>
|
||
<li>结果总是位于由 <span class="arithmatex">\(\mathbf{b}\)</span> 和 <span class="arithmatex">\(\mathbf{c}\)</span> 张成的平面内。注意叉积<strong>不满足结合律</strong>：<span class="arithmatex">\(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\)</span>。</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>计算两个向量的点积并用它求出它们之间的角度。尝试让它们正交、平行或反向，观察角度如何变化。
|
||
<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">a</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-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">b</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="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">dot</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">angle</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arccos</span><span class="p">(</span><span class="n">dot</span> <span class="o">/</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">)))</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Dot product: </span><span class="si">{</span><span class="n">dot</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Angle: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">degrees</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">°"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>计算两个三维向量的叉积，并通过检查结果与每个原始向量的点积为零来验证结果垂直于两者。
|
||
<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">a</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">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">b</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">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</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">cross</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"a x b = </span><span class="si">{</span><span class="n">cross</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<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">"Perpendicular to a: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">cross</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Perpendicular to b: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">cross</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
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