Matrix Mathematics: From Foundational Theory to Modern AI Architectures

Matrices are the lingua franca of ML. This path builds four core tools (decomposition, measurement, calculus, iteration), covers classical methods (SVD, PCA, NMF) and operator analysis (PageRank, spectral clustering), then converges on modern architectures (LoRA, Efficient Attention, SSM/Mamba). The "decompose → propagate → converge" arc reveals how one mathematical tool manifests across seemingly different domains.

1. 1

   矩阵数学全景图：ML 的通用语言

   Advanced Coming Soon

   #matrix-math#linear-algebra#overview
2. 2

   核心性质速查：概念关系图与公式速查表

   Advanced Coming Soon

   #matrix-math#linear-algebra#cheatsheet
3. 3

   数据矩阵分解概述：问题、工具与方法谱系

   Advanced Coming Soon

   #matrix-math#decomposition#overview
4. 4

   向量空间的几何：内积、投影、秩与子空间

   Advanced Coming Soon

   #matrix-math#inner-product#projection#rank#null-space#orthogonality
5. 5

   矩阵结构的几何：二次型、正定性与协方差

   Advanced Coming Soon

   #matrix-math#quadratic-form#positive-definite#covariance#gram-matrix#trace#determinant
6. 6

   特征分解与对角化：万物之基

   Advanced Coming Soon

   #matrix-math#eigendecomposition#diagonalization#spectral-theorem
7. 7

   奇异值分解：核心中的核心

   Advanced Coming Soon

   #matrix-math#svd#low-rank-approximation#pseudoinverse#eckart-young
8. 8

   矩阵范数、内积与条件数：度量的艺术

   Advanced Coming Soon

   #matrix-math#norms#condition-number#inner-product#frobenius#spectral#nuclear
9. 9

   矩阵微积分：从 Jacobian 到损失曲面

   Advanced Coming Soon

   #matrix-math#calculus#jacobian#hessian#backpropagation#loss-surface#taylor-expansion
10. 10

    优化算法：从梯度下降到牛顿法

    Advanced Coming Soon

    #matrix-math#optimization#gradient-descent#newton-method#sgd#convergence
11. 11

    PCA 与 Eigenfaces：从方差最大化到人脸识别

    Advanced Coming Soon

    #matrix-math#pca#eigenfaces#dimensionality-reduction#covariance-matrix#svd
12. 12

    随机化 SVD：当精确分解算不动的时候

    Advanced Coming Soon

    #matrix-math#randomized-svd#johnson-lindenstrauss#random-projection#low-rank-approximation
13. 13

    矩阵补全：从极少观测恢复低秩矩阵

    Advanced Coming Soon

    #matrix-math#matrix-completion#nuclear-norm#convex-relaxation#incoherence#low-rank
14. 14

    NMF：非负约束下的 Parts-Based 分解

    Advanced Coming Soon

    #matrix-math#nmf#non-negative-matrix-factorization#parts-based#topic-modeling
15. 15

    MF 与 FM：协同过滤的矩阵分解视角

    Advanced Coming Soon

    #matrix-math#matrix-factorization#factorization-machines#recommender-systems#collaborative-filtering
16. 16

    Word2Vec 与 GloVe：隐式 vs 显式矩阵分解

    Advanced Coming Soon

    #matrix-math#word2vec#glove#pmi#word-embeddings#implicit-factorization
17. 17

    Robust PCA：低秩 + 稀疏分解

    Advanced Coming Soon

    #matrix-math#robust-pca#low-rank#sparse#nuclear-norm#convex-optimization#principal-component-pursuit
18. 18

    张量分解与知识图谱嵌入：从二维到高阶

    Advanced Coming Soon

    #matrix-math#tensor-decomposition#knowledge-graph#CP-decomposition#Tucker-decomposition#DistMult#ComplEx
19. 19

    算子矩阵全景：当矩阵不再装数据

    Advanced Coming Soon

    #matrix-math#operator#markov#laplacian#kernel#overview
20. 20

    马尔可夫链与转移矩阵：当矩阵编码概率

    Advanced Coming Soon

    #matrix-math#markov-chains#transition-matrix#perron-frobenius#mixing-time
21. 21

    隐马尔可夫模型：当状态看不见

    Advanced Coming Soon

    #matrix-math#hmm#hidden-markov-model#forward-backward#viterbi#baum-welch
22. 22

    连续时间线性系统与 Kalman 滤波：从离散步进到平滑流动

    Advanced Coming Soon

    #matrix-math#linear-systems#kalman-filter#matrix-exponential#state-space#continuous-time#discretization
23. 23

    PageRank 与幂迭代：图上的马尔可夫链

    Advanced Coming Soon

    #matrix-math#pagerank#power-iteration#markov-chains#spectral-gap
24. 24

    随机游走与图嵌入：DeepWalk/Node2Vec

    Advanced Coming Soon

    #matrix-math#random-walk#graph-embedding#deepwalk#node2vec#transition-matrix
25. 25

    Kernel 矩阵与再生核：数据定义的给定算子

    Advanced Coming Soon

    #matrix-math#kernel#mercer-theorem#kernel-pca#gaussian-process#rkhs
26. 26

    图 Laplacian 与谱聚类：从图结构到最优分割

    Advanced Coming Soon

    #matrix-math#graph-laplacian#spectral-clustering#fiedler-vector#graph-partitioning
27. 27

    图扩散、热核与 GNN 消息传递：从热方程到图神经网络

    Advanced Coming Soon

    #matrix-math#graph-diffusion#heat-kernel#gnn#message-passing#graph-laplacian#gcn
28. 28

    学习算子中的低秩结构：为什么神经网络权重是低秩的？

    Advanced Coming Soon

    #matrix-math#learned-operator#low-rank#intrinsic-dimension#lora#overview
29. 29

    LoRA：低秩分解在 LLM 微调中的应用

    Advanced Coming Soon

    #matrix-math#lora#low-rank#fine-tuning#parameter-efficient#qlora
30. 30

    Attention 的低秩结构与 Efficient Attention

    Advanced Coming Soon

    #matrix-math#attention#low-rank#linformer#performer#efficient-attention#kernel
31. 31

    SSM / Mamba：矩阵对角化的胜利

    Advanced Coming Soon

    #matrix-math#ssm#mamba#hippo#diagonalization#state-space#s4#selective-ssm