gaojie
358 lines
20 KiB
HTML
358 lines
20 KiB
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<title>OU 过程与 Mehler 公式 · 为什么高阶成分被罚得更狠</title>
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<div class="hintkey">← → 翻页 · F 全屏</div>
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<!-- 1 封面 -->
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<section class="slide active cover">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>SELF-SUPERVISED · 谱分解 · EP.02</div>
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<h1 style="margin-top:34px;">OU 过程<br>与 <em>Mehler 公式</em></h1>
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<p class="lead" style="margin-top:38px;">正样本对怎么造？为什么高阶非线性成分被罚得更狠？</p>
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<div class="meta">配套播客 · 双主播对话 · 上接 Hermite 多项式，下启线性可识别性</div>
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</section>
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<!-- 2 核心问题 -->
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<section class="slide posBL">
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<div class="glow amber"></div>
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<div class="kicker"><span class="ln"></span>00 · 核心问题</div>
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<h2>正样本对从<em>哪里来</em>？</h2>
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<div class="content">
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<ul class="pts">
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<li>LeJEPA 训练需要「正样本对」——同一内容的两个视角 $(z, z')$。</li>
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<li>这对视角是<b>怎么生成</b>的？答案是 <b>OU 过程</b>。</li>
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<li>为什么这种生成方式，会让<b>高阶 Hermite 成分被更强地惩罚</b>？</li>
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<li>解开它的钥匙，是 <b>Mehler 公式</b>。</li>
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</ul>
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</div>
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</section>
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<!-- 3 OU 物理直觉 -->
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<section class="slide">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>01 · 物理直觉</div>
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<h2>弹簧上的<em>小球</em></h2>
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<div class="content" style="gap:60px;">
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<div class="card" style="flex:1;">
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<div class="ct">弹簧力 · 均值回归</div>
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<div class="cb">把小球持续拉回原点，不让它跑远。</div>
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</div>
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<div class="card" style="flex:1;">
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<div class="ct">随机扰动 · 布朗噪声</div>
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<div class="cb">周围的随机推搡，让小球抖动。</div>
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</div>
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<div class="card" style="flex:1;">
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<div class="ct">一拉一推 = OU 过程</div>
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<div class="cb">两股力量的拉扯，正是 Ornstein–Uhlenbeck 过程的图像。</div>
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</div>
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</div>
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</section>
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<!-- 4 离散 OU 定义 -->
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<section class="slide posCR">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>02 · 定义</div>
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<h2>LeJEPA 用的<em>离散一步转移</em></h2>
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<div class="content" style="gap:60px;">
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<div class="fbox" style="flex:1.1;font-size:42px;">
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$z' = \rho\,z + \sqrt{1-\rho^{2}}\;\eta$
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<div class="small">$\eta\sim\mathcal N(0,I),\quad \rho\in(0,1)$</div>
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</div>
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<div style="flex:1;">
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<p class="lead" style="font-size:30px;">$\rho$ 是<b style="color:var(--teal)">相关系数</b>——两个视角的相似度旋钮。</p>
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<p class="lead" style="font-size:28px;margin-top:18px;">$\rho\to1$：几乎相同视角；$\rho\to0$：相互独立。实践取 $\rho\in[0.8,0.95]$。</p>
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</div>
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</div>
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</section>
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<!-- 5 三性质 -->
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<section class="slide">
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<div class="glow amber"></div>
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<div class="kicker"><span class="ln"></span>03 · 三个性质</div>
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<h2>OU 过程的<em>三块基石</em></h2>
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<div class="content">
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<div class="cards">
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<div class="card"><div class="ct">① 平稳性</div><div class="cb">$z\sim\mathcal N(0,I)\Rightarrow z'\sim\mathcal N(0,I)$<br>均值 0、方差 $\rho^2+(1-\rho^2)=1$。两视角分布相同。</div></div>
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<div class="card"><div class="ct">② 相关可控</div><div class="cb">$\mathrm{Cov}(z',z)=\rho\,I$<br>$\rho$ 直接 = 相似度，一个旋钮说了算。</div></div>
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<div class="card"><div class="ct">③ 加性噪声</div><div class="cb">$z'=\rho z+\eta$：线性漂移 + 独立噪声。<br>满足论文的加性噪声假设。</div></div>
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</div>
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</div>
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</section>
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<!-- 6 Mehler 公式 -->
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<section class="slide posBL">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>04 · 谱定理</div>
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<h2>Mehler 公式：<em>每阶挂 ρᵈ</em></h2>
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<div class="content" style="gap:56px;">
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<div class="fbox" style="flex:1.15;font-size:30px;">
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$p(z'\mid z)=\varphi(z')\displaystyle\sum_{d=0}^{\infty}\rho^{d}\,\frac{He_d(z)He_d(z')}{d!}$
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<div class="small">OU 转移核在 Hermite 基下展开</div>
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</div>
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<div style="flex:1;">
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<div class="fbox" style="font-size:30px;">
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$\mathbb{E}[h_i(z)\,h_i(z')]=\displaystyle\sum_{d=0}^{\infty}\rho^{d}\,w_d$
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<div class="small">$w_d$＝第 $i$ 分量在 $d$ 阶上的谱权重</div>
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</div>
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<p class="lead" style="font-size:26px;margin-top:22px;">关键：<b style="color:var(--amber)">d 阶成分的系数 = ρ 的 d 次方</b>。</p>
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</div>
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</div>
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</section>
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<!-- 7 核心推论 -->
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<section class="slide posCR">
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<div class="glow amber"></div>
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<div class="kicker"><span class="ln"></span>05 · 核心推论</div>
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<h2>相关性<em>封顶在 ρ</em></h2>
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<div class="content" style="gap:56px;">
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<div class="fbox" style="flex:1.1;font-size:28px;line-height:2;">
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$\mathrm{corr}_i=\displaystyle\sum_{d\ge1}w_d\,\rho^{d}\le\sum_{d\ge1}w_d\,\rho=\rho$
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<div class="small">因为 $\rho^d\le\rho\ (d\ge1)$，且 $\sum w_d=1$</div>
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</div>
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<div style="flex:1;">
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<p class="lead" style="font-size:30px;">等号成立 <b style="color:var(--teal)">当且仅当 $w_1=1$</b>——编码器纯线性。</p>
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<p class="lead" style="font-size:27px;margin-top:18px;">只要有任何 $w_{d_0}>0\ (d_0\ge2)$，那一项就严格变小，和够不到 $\rho$。</p>
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</div>
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</div>
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</section>
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<!-- 8 数值例子 -->
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<section class="slide">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>06 · 数值例子</div>
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<h2>取 <em>ρ = 0.9</em> 算给你看</h2>
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<div class="content">
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<table class="cmp">
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<tr><th>编码器</th><th>谱权重</th><th>相关性 corrᵢ</th><th>与 0.9 的差距</th></tr>
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<tr><td class="k">纯线性 $h=z$</td><td><span class="hl">$w_1=1$</span></td><td><span class="hl">$0.9^1=0.900$</span></td><td>0 ｜ 最优</td></tr>
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<tr><td class="k">纯二次 $h=z^2-1$</td><td>$w_2=1$</td><td><span class="am">$0.9^2=0.810$</span></td><td>−0.090</td></tr>
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<tr><td class="k">纯三次 $h=z^3-3z$</td><td>$w_3=1$</td><td><span class="am">$0.9^3=0.729$</span></td><td>−0.171</td></tr>
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<tr><td class="k">混合 各半</td><td>$w_1=w_2=0.5$</td><td>$0.855$</td><td>−0.045</td></tr>
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</table>
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</div>
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</section>
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<!-- 9 与对齐损失 -->
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<section class="slide posBL">
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<div class="glow amber"></div>
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<div class="kicker"><span class="ln"></span>07 · 训练联系</div>
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<h2>对齐损失的<em>下界</em></h2>
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<div class="content" style="gap:56px;">
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<div class="fbox" style="flex:1.05;font-size:28px;line-height:1.9;">
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$\mathcal L_{\text{align}}=\mathbb{E}\|h(z')-h(z)\|^2$<br>
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$=2n-2\sum_i \mathrm{corr}_i\;\ge\;2(1-\rho)\,n$
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<div class="small">最小化损失 ⟺ 最大化相关性之和</div>
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</div>
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<div style="flex:1;">
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<p class="lead" style="font-size:31px;">等号成立 <b style="color:var(--teal)">当且仅当每个 $h_i$ 都是线性的</b>。</p>
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<p class="lead" style="font-size:28px;margin-top:18px;">这就是定理一的心脏：<b style="color:var(--amber)">最优编码器必须线性</b>。</p>
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</div>
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</div>
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</section>
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<!-- 10 衰减图示 -->
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<section class="slide posCR">
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<div class="glow"></div>
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<div class="kicker"><span class="ln"></span>08 · 直觉图示</div>
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<h2>ρ = 0.9 时<em>逐阶衰减</em></h2>
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<div class="content">
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<div style="width:100%;">
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<div class="barrow"><span class="chip">d=1 线性</span><div class="track"><i style="width:90%"></i></div><span class="val">ρ¹ = 0.900</span></div>
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<div class="barrow"><span class="chip">d=2 二次</span><div class="track"><i style="width:81%"></i></div><span class="val">ρ² = 0.810</span></div>
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<div class="barrow"><span class="chip">d=3 三次</span><div class="track"><i style="width:72.9%"></i></div><span class="val">ρ³ = 0.729</span></div>
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<div class="barrow"><span class="chip">d=4 四次</span><div class="track"><i style="width:65.6%"></i></div><span class="val">ρ⁴ = 0.656</span></div>
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<div class="barrow"><span class="chip">d=5 五次</span><div class="track"><i style="width:59%"></i></div><span class="val">ρ⁵ = 0.590</span></div>
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<p class="lead" style="font-size:27px;margin-top:20px;">非线性成分的相关性随阶数<b style="color:var(--amber)">指数衰减</b>——越高阶越吃亏。</p>
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</div>
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</div>
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</section>
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<!-- 11 小结 -->
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<section class="slide">
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<div class="glow amber"></div>
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<div class="kicker"><span class="ln"></span>09 · 小结</div>
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<h2>五句话<em>带走全场</em></h2>
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<div class="content">
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<div class="grid4">
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<div class="g"><div class="n">01</div><div class="t">OU 过程「一拉一推」造正样本对，相似度由 <b>ρ</b> 控制。</div></div>
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<div class="g"><div class="n">02</div><div class="t"><b>平稳性</b> + 协方差 = ρ + 线性漂移加噪声，三性质齐备。</div></div>
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<div class="g"><div class="n">03</div><div class="t"><b>Mehler</b>：d 阶 Hermite 成分相关性 = ρᵈ。</div></div>
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<div class="g"><div class="n">04</div><div class="t">核心不等式 corrᵢ = Σ wᵈ ρᵈ ≤ ρ，等号 ⟺ <b>纯线性</b>。</div></div>
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<div class="g" style="grid-column:1 / -1;"><div class="n">05</div><div class="t">最小化对齐损失 → 最大化相关性 → <b>编码器必须线性</b>（定理一）。</div></div>
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<div class="kicker"><span class="ln"></span>下一站</div>
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<h1 style="font-size:88px;margin-top:30px;">谱分解<br><em>与线性可识别性</em></h1>
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<p class="lead" style="margin-top:34px;font-size:30px;">下一讲把 Hermite 展开与 OU 衰减正式拼起来，组装成定理一的完整证明。</p>
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<div class="meta">谢谢收听 · 配套音频 ou_mehler-podcast.mp3 · 弹簧拉小球，ρ 拧大小 · 我们下次见</div>
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