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08](https://deepnote.com/project/38755e4e-23f6-4f5f-8c68-8a2e4114b1c1/notebook/b5ba27f2c5ba4ec0922208a7d5dc80a4)保存的同图重建与冻结读出。你的产出是一张对照：模型恢复了哪些像素，哪些下游能力另外测过，两者为什么不能互相代替。","sha256":"2864bf0006511ecd508b2de87e5c323f9679695de8967deee935355762a68f4c","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-64c15dd32072c87e96376ba1","nextBlock":"b-d89a6282013296dbf4db0f2a","prerequisites":[{"id":"b-6ac8c5c3a2366ba0eede7b88","url":"#b-6ac8c5c3a2366ba0eede7b88","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-d89a6282013296dbf4db0f2a","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-d89a6282013296dbf4db0f2a","content":"PCA、去噪与词典的推导保留在前半部，遇到瓶颈或误差问题再回读。需要一个无需神经网络的实际优化入口时，选[LAB 05像素词典](https://deepnote.com/project/38755e4e-23f6-4f5f-8c68-8a2e4114b1c1/notebook/de39b998077f44cca4b8407085547766)。可直接引用[隐藏loss分母](/course/blocks/mae-masked-loss-denominator.html)或[重建限制](/course/blocks/reconstruction-restrictions.html)。","sha256":"9e07fc295fe95877d90b697ad5fa48296f312a7ab8d0889774a5f486fd8d090d","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-fe0457a8f7b6e51047660171","nextBlock":"b-f1ee20f85060946d0c146c07","prerequisites":[{"id":"b-6ac8c5c3a2366ba0eede7b88","url":"#b-6ac8c5c3a2366ba0eede7b88","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-f1ee20f85060946d0c146c07","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-f1ee20f85060946d0c146c07","content":"如果不提供“猫”“杯子”这样的类别标签，图像还有什么可以当答案？最直接的是图像自身：把它压缩、破坏或遮住一部分，再让模型恢复。困难不在于是否有监督数值，而在于我们允许模型怎样取得它。把输入原样复制到输出可以获得零误差，却没有迫使模型形成有用的表示。重建方法之间真正改变的是限制条件：代码可以取什么值、信息从哪里通过、哪些位置可见，以及误差计算在哪里。","sha256":"a8be541fcaf19bfce2bf41f6779590be20eb214a5f4faf49be22dcf9a7d9bbc0","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-d89a6282013296dbf4db0f2a","nextBlock":"b-f0f2460339c541a50b93cc15","prerequisites":[{"id":"b-6ac8c5c3a2366ba0eede7b88","url":"#b-6ac8c5c3a2366ba0eede7b88","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-f0f2460339c541a50b93cc15","type":"h2","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-f0f2460339c541a50b93cc15","content":"## 同样是压缩，离散词典与线性子空间留下不同东西","sha256":"9a8088a16b7096a914d6ede16623634ce17a57528701ccdb6e9cb90e6d1c8e7f","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-f1ee20f85060946d0c146c07","nextBlock":"b-aaf8a4c6229911d55b2809e5","prerequisites":[{"id":"b-6ac8c5c3a2366ba0eede7b88","url":"#b-6ac8c5c3a2366ba0eede7b88","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-aaf8a4c6229911d55b2809e5","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-aaf8a4c6229911d55b2809e5","content":"把一个 $2\\times2$ 灰度 patch 展平为 $x\\in\\mathbb R^4$。K-means 学习 $K$ 个中心 $c_k$，用最近中心的索引编码：","sha256":"ab7a10e9728afbecdcec79d3f12505658449566b8277e7632a8f41cdc23677f7","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"2\\times2","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"x\\in\\mathbb 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x=c_{a(x)}.$","sha256":"aef13a0897067ed8c716ad7ba8fabf84019f69a5ea2017415804af017d79c7b6","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"a(x)=\\arg\\min_k\\|x-c_k\\|^2,\\qquad \\hat 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05](https://deepnote.com/workspace/ise3309jingao-5ba66b0a-4283-4f5a-ae06-3f68d3201e99/project/38755e4e-23f6-4f5f-8c68-8a2e4114b1c1/notebook/de39b998077f44cca4b8407085547766?utm_source=openai&utm_medium=mcp&utm_campaign=openaimcp&utm_content=de39b998077f44cca4b8407085547766&utm_term=get_notebook) 在真实 MNIST patch 上进行 Lloyd 拟合，可查看中心怎样随数据形成，而非只背“无监督”这个标签。","sha256":"b95322b58e86aab97b9239ea97f9fbcd76a279b78c510fe88e115594dd0cd9ea","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"c_1=(1,0,1,0)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"c_2=(0,1,0,1)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"(0.5,0.5,0.5,0.5)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\log_2 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个方向。","sha256":"314d138cb51e798d2754f0d3a4ea9e9f5db2ea2051da32427b62b50db0088186","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"x","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"U\\in\\mathbb R^{d\\times k}","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"z=U^Tx","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\hat 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的最优重建轴可能删去标签所需的差异。","sha256":"da8a5c96a3492ad378af6c0cbf696fd54ccd53c2c29bbc8e794d30c3a07e225e","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"(2,0),(-2,0),(0,1),(0,-1)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"1/4","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\mathrm{diag}(2,0.5)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"(2,0),(-2,0),(0,0),(0,0)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"0,0,1,1","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"}],"previousBlock":"b-7018bb2724db3fb6295b3e5c","nextBlock":"b-3a749a19b75d670318a9d610","prerequisites":[{"id":"b-f0f2460339c541a50b93cc15","url":"#b-f0f2460339c541a50b93cc15","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-3a749a19b75d670318a9d610","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-3a749a19b75d670318a9d610","content":"![PCA 的平方误差与 MAE 的位置恢复](/course/repair/assets/cls05-pca-mask.svg)","sha256":"5e93b74f70085ca5ec3b76b71d57b2a105f5a18720df2082b57f0ac8f1df1499","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-7f9f639f6ff8d85cb675e6be","nextBlock":"b-06134419dc5614ecb1bacddb","prerequisites":[{"id":"b-f0f2460339c541a50b93cc15","url":"#b-f0f2460339c541a50b93cc15","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-06134419dc5614ecb1bacddb","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-06134419dc5614ecb1bacddb","content":"左图按真实坐标画出四点和一维投影；右图展示 patch 的乱序、可见项与恢复位置。两者都在重建，但一个限制子空间，一个限制输入可见性。","sha256":"7ed392f7e5f8093e27fad86ce7c3c1dd94875138594c09c1db623b85a4946a51","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-3a749a19b75d670318a9d610","nextBlock":"b-227906796d16fea01bdd5e37","prerequisites":[{"id":"b-f0f2460339c541a50b93cc15","url":"#b-f0f2460339c541a50b93cc15","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-227906796d16fea01bdd5e37","type":"h2","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-227906796d16fea01bdd5e37","content":"## PCA 为什么可以看成受约束的线性自编码器","sha256":"859cfeee5fb3ba73e84cd3892eaf1ea08e08ffef5ebf530789c7027945093b85","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-06134419dc5614ecb1bacddb","nextBlock":"b-b27b8edbce328f22474c2ab1","prerequisites":[{"id":"b-f0f2460339c541a50b93cc15","url":"#b-f0f2460339c541a50b93cc15","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-b27b8edbce328f22474c2ab1","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-b27b8edbce328f22474c2ab1","content":"自编码器由 $z=f_\\theta(x)$ 与 $\\hat x=g_\\phi(z)$ 组成，以 $\\|x-\\hat x\\|^2$ 等重建损失更新参数。令 $f(x)=U^Tx,g(z)=Uz$ 并要求 $U^TU=I$，就得到上面的 PCA 形式。一般线性、低秩、平方误差自编码器也能学到 PCA 的主子空间，尽管编码坐标不一定等于标准主成分；平移、旋转等参数化差异要与重建子空间区分。","sha256":"1ec743964fe5a8b6a41542c2f230348bcf69b21f317b8a22ea2f1483ada27a4b","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"z=f_\\theta(x)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\hat x=g_\\phi(z)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\|x-\\hat 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本身没有执行一个另行承诺的非线性网络训练。引入非线性编码器和解码器以后，表示可以适应弯曲的数据结构；也可能凭较大容量记忆训练样本。瓶颈维数、正则化、跳连是否绕过瓶颈、训练与保留误差，都影响我们能否把低损失解释为学到了结构。","sha256":"e62fcdbdf8f5d621235c6d665a17152c1736f9eb0ff056daa0c2d302d56ee703","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[],"previousBlock":"b-b27b8edbce328f22474c2ab1","nextBlock":"b-04abe5ed92075079aadcaede","prerequisites":[{"id":"b-227906796d16fea01bdd5e37","url":"#b-227906796d16fea01bdd5e37","role":"section-title"},{"id":"b-4eb2ac231288a23b4634477c","url":"#b-4eb2ac231288a23b4634477c","role":"problem-context"}],"relatedBlocks":[]},{"id":"b-04abe5ed92075079aadcaede","type":"p","scope":"document","url":"https://codingai-lec03.pages.dev/course/notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559.html#b-04abe5ed92075079aadcaede","content":"去噪自编码器进一步改变训练对：先从破坏分布采样 $\\tilde x\\sim q(\\tilde x\\mid x)$，再最小化 $\\mathbb E\\|g(f(\\tilde x))-x\\|^2$。输入是带噪图片，目标仍是干净图片。训练时不给模型干净图，不能在可视化时偷偷把目标的边缘或 mask 喂回去。Gaussian 噪声、置零像素和随机遮块提出不同恢复任务；噪声太轻会接近复制，太重可能留下不足以判断的证据。在 [去噪自编码器原论文](https://www.jmlr.org/papers/v11/vincent10a.html) 中，破坏过程正是组织表示学习的一个关键约束。","sha256":"4d82f52f7518bc4d90a45a49b8188143486de499bd9a390f4e9be7d7b2985f15","chapter":{"id":"course","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"unit":{"id":"notebook-blocks/a06579c00c0d4cf0a190f71ae8b83559","title":"同样是压缩，离散词典与线性子空间留下不同东西"},"page":{"id":"reading-opening","title":"同样是压缩，离散词典与线性子空间留下不同东西","number":1},"sectionTitle":"同样是压缩，离散词典与线性子空间留下不同东西","currentQuestion":"同样是压缩，离散词典与线性子空间留下不同东西","sourceSlides":[],"knowledgeBlock":null,"formulas":[{"latex":"\\tilde x\\sim q(\\tilde x\\mid x)","definitionContext":"同样是压缩，离散词典与线性子空间留下不同东西"},{"latex":"\\mathbb E\\|g(f(\\tilde 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