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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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量子机器学习
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<h1 id="_1">向量性质<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>向量性质描述了定义向量行为的几何和代数特征。本文涵盖模长、方向、单位向量、相等性、平行性、正交性和线性无关性，它们是每个 ML 特征空间的基石。</em></p>
|
||
<ul>
|
||
<li>向量的<strong>模长</strong>（或长度）告诉你它延伸了<em>多远</em>。把它想象成箭头的长度。对于向量 <span class="arithmatex">\(\mathbf{a} = (a_1, a_2, a_3)\)</span>，其模长为：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这只是勾股定理推广到更高维度，测量从原点到该点的直线距离。</p>
|
||
</li>
|
||
<li>
|
||
<p>向量的<strong>方向</strong>告诉你它指向<em>哪里</em>；只需想象从原点到坐标点的一条直线即可。</p>
|
||
</li>
|
||
<li>
|
||
<p>当没有明确指定原点时，我们通常隐含地使用 <span class="arithmatex">\((0,0,\ldots,0)\)</span> 即中心点，至少为了可视化目的如此。</p>
|
||
</li>
|
||
<li>
|
||
<p>位置并不重要，它总是关于位移：从原点画出的向量 <span class="arithmatex">\((3, 2)\)</span> 和从另一个点画出的同样的 <span class="arithmatex">\((3, 2)\)</span> 仍然是相等的。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="向量相等：从两个不同起点画出的相同 (3,2) 向量" src="../../images/vector_equality.svg" /></p>
|
||
<ul>
|
||
<li>两个向量可以有相同的长度但指向完全不同的方向，或者指向相同方向但长度不同。</li>
|
||
</ul>
|
||
<p><img alt="相同方向、不同模长（v 和 2v）对比相同模长、不同方向" src="../../images/magnitude_direction.svg" /></p>
|
||
<ul>
|
||
<li>两个向量<strong>相等</strong>当且仅当它们所有对应的分量都匹配；相同的长度，相同的方向，完全相同的箭头。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} = \mathbf{b} \iff a_i = b_i \text{ 对所有 } i\]</div>
|
||
<ul>
|
||
<li>两个向量<strong>平行</strong>如果一个是另一个的标量倍数。它们沿着同一条直线，要么同向，要么完全反向。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = k\mathbf{b} \text{ 对于某个标量 } k \neq 0\]</div>
|
||
<p><img alt="平行向量：a 和 b 指向相同方向，a 和 -b 指向相反方向" src="../../images/parallel_vectors.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>如果 <span class="arithmatex">\(k > 0\)</span>，它们指向相同方向。如果 <span class="arithmatex">\(k < 0\)</span>，它们指向相反方向。无论哪种情况，它们都位于经过原点的同一条直线上。</p>
|
||
</li>
|
||
<li>
|
||
<p>直观地说，平行向量不携带任何"新的"方向信息。一个只是另一个的拉伸或翻转版本。</p>
|
||
</li>
|
||
<li>
|
||
<p>两个向量<strong>正交</strong>（垂直）如果它们指向完全独立的方向。沿一个方向移动不会让你在另一个方向上有任何进展。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="正交向量：u 和 v 成直角相交" src="../../images/orthogonal_vectors.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>想象向北走然后向东走，这些是正交方向，无论向北走多远都不会使你向东移动。我们经常会遇到正交性。</p>
|
||
</li>
|
||
<li>
|
||
<p>正交性对 ML 至关重要：正交的特征携带完全独立的信息，这对于表示是最理想的。</p>
|
||
</li>
|
||
<li>
|
||
<p>更一般地，任意两个向量之间都有一个<strong>夹角</strong> <span class="arithmatex">\(\theta\)</span>，范围从 <span class="arithmatex">\(0°\)</span> 到 <span class="arithmatex">\(180°\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>这个角度捕捉了两个方向之间的全部关系：<span class="arithmatex">\(0°\)</span> 表示平行（相同方向），<span class="arithmatex">\(180°\)</span> 表示平行（相反方向），<span class="arithmatex">\(90°\)</span> 表示正交。介于之间的都是混合情况。</p>
|
||
</li>
|
||
<li>
|
||
<p>ML 中的大多数向量关系都处在这个范围的某处。稍后，我们将看到精确的工具（点积、余弦相似度）来计算这个角度。</p>
|
||
</li>
|
||
<li>
|
||
<p>一组向量是<strong>线性相关</strong>的，如果其中至少一个可以通过缩放和相加从其他向量构造出来。它没有为该集合带来新的信息。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，如果 <span class="arithmatex">\(\mathbf{c} = 2\mathbf{a} + 3\mathbf{b}\)</span>，那么 <span class="arithmatex">\(\mathbf{c}\)</span> 是冗余的，你已经通过 <span class="arithmatex">\(\mathbf{a}\)</span> 和 <span class="arithmatex">\(\mathbf{b}\)</span> 拥有了 <span class="arithmatex">\(\mathbf{c}\)</span> 所提供的全部信息。</p>
|
||
</li>
|
||
<li>
|
||
<p>平行向量总是线性相关的，因为一个只是另一个的缩放副本。任何包含零向量的集合也是线性相关的。</p>
|
||
</li>
|
||
<li>
|
||
<p>向量是<strong>线性无关</strong>的，如果其中没有一个能从其他向量构造出来。每个向量都贡献了一个真正的新方向。正交向量总是线性无关的。</p>
|
||
</li>
|
||
<li>
|
||
<p>在二维中，两个线性无关的向量可以到达平面上的任何点。在三维中，你需要三个。"需要多少个独立的向量"这个想法直接与维度相关。</p>
|
||
</li>
|
||
<li>
|
||
<p>当向量的大多数分量为零时，该向量是<strong>稀疏</strong>的。相反，大多数分量非零称为<strong>稠密</strong>。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathbf{s} = [0, 0, 3, 0, 0, 0, 1, 0, 0, 0]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>稀疏性很重要，因为它影响存储和计算。稀疏向量可以通过只跟踪非零条目来更高效地存储和处理。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>单位向量</strong>是模长正好为 1 的向量。它纯粹表示方向，不包含长度信息。你可以通过除以模长将任何向量变成单位向量：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\hat{\mathbf{a}} = \frac{\mathbf{a}}{\|\mathbf{a}\|}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这个过程称为<strong>归一化</strong>。它剥离了"多远"，只保留"往哪走。"</p>
|
||
</li>
|
||
<li>
|
||
<p>标准单位向量指向每个轴：<span class="arithmatex">\(\hat{\mathbf{i}} = (1, 0, 0)\)</span>，<span class="arithmatex">\(\hat{\mathbf{j}} = (0, 1, 0)\)</span>，<span class="arithmatex">\(\hat{\mathbf{k}} = (0, 0, 1)\)</span>。任何向量都可以写成这些向量的组合，例如 <span class="arithmatex">\((3, 2, 4) = 3\hat{\mathbf{i}} + 2\hat{\mathbf{j}} + 4\hat{\mathbf{k}}\)</span>。</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>计算向量的模长并验证它符合勾股定理，然后修改代码计算单位向量。
|
||
<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">3.0</span><span class="p">,</span> <span class="mf">4.0</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">magnitude</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">a</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Magnitude of a: </span><span class="si">{</span><span class="n">magnitude</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>通过测试一个向量是否是另一个的标量倍数来检查两个向量是否平行。
|
||
<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="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</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="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>
|
||
<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">ratios</span> <span class="o">=</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Ratios: </span><span class="si">{</span><span class="n">ratios</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<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">"Parallel: </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">ratios</span><span class="p">,</span><span class="w"> </span><span class="n">ratios</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
</ol>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
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||
|
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|
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