Shih-Hsin Wang 王士欣
中文

Mathematical Foundations

A plain-language introduction to the mathematical tools that recur in machine learning, and to the mathematics the other courses on this site rely on.

7 units 31 notes Read 0 / 31 Sep 2026
  1. M0 Calculus 工具箱 0 / 4 Gradient and direction, divergence and flux, Taylor expansion and "assume a straight line", differentiating under the integral and the total derivative along a path.
    1. M0.0 霧中下山：Gradient 與方向
    2. M0.1 閘門的進出：Divergence 與通量
    3. M0.2 導航的三十秒：Taylor 展開與「假設直線」
    4. M0.3 漂流的溫度計：全導數、微分穿過積分與 JVP
  2. M1 機率與 Conditional Expectation 0 / 5 Multivariate Gaussians, conditional expectation, the minimizer of MSE, Bayes and the posterior, and denoising as the posterior mean (Tweedie).
    1. M1.0 兩台體重計：Gaussian 的線性組合
    2. M1.1 從鞋子猜身高：Conditional Expectation
    3. M1.2 天氣預報該報幾度：MSE 的最小值是 Conditional Expectation
    4. M1.3 快篩陽性：Bayes 與 Posterior
    5. M1.4 去噪就是取 Posterior Mean（Tweedie）
  3. M2 Deterministic Dynamics 0 / 5 Vector fields and ODEs, pushforward and flow maps, the continuity equation, Euler error and higher-order methods.
    1. M2.0 河面上的箭頭：Vector Field 與 ODE
    2. M2.1 從一片葉子到一群葉子：Flow Map 與 Pushforward
    3. M2.2 站內人數變化 = 進 − 出：Continuity Equation
    4. M2.3 導航每 30 秒更新一次：Euler 法與它的誤差
    5. M2.4 在 30 秒的中點多看一眼：高階方法
  4. M3 Stochastic Dynamics 0 / 5 From random walks to Brownian motion, SDEs as noisy ODEs, Fokker–Planck, Langevin and stationary distributions, weak vs. strong convergence.
    1. M3.0 醉漢一小時後在哪：Random Walk 到 Brownian Motion
    2. M3.1 水流加亂流：SDE 是加了噪聲的 ODE
    3. M3.2 一千片葉子在亂流裡：Fokker–Planck 方程式
    4. M3.3 山谷裡隨機走的人群：Langevin Dynamics 與 Stationary Distribution
    5. M3.4 每條路徑對，還是最後的分佈對：Weak 與 Strong Convergence
  5. M4 Markov Chains 0 / 4 Markov chains and transition matrices, absorbing states, continuous-time Markov chains and rate matrices, time reversal and ratios.
    1. M4.0 明天的天氣只看今天：Markov Chain 與 Transition Matrix
    2. M4.1 進去就出不來：Absorbing State 與吸收時間
    3. M4.2 電話隨時會響：Continuous-Time Markov Chain 與 Rate Matrix
    4. M4.3 影片倒著播看得出來嗎：Time Reversal 與比值
  6. M5 Optimization、Convexity 與距離 0 / 4 Jensen's inequality, KL divergence and cross-entropy, moving targets and EMA, differentiating through a generator (the KL gradient is a difference of scores).
    1. M5.0 保險為什麼存在：Convex Function 與 Jensen 不等式
    2. M5.1 用錯的機率表下注一年會多輸多少：KL Divergence 與 Cross-Entropy
    3. M5.2 追一隻自己也在跑的狗：移動目標、EMA 與 Stop-Gradient
    4. M5.3 只看得到樣本，怎麼調機器：對 Generator 微分
  7. M6 Optimal Transport 0 / 4 The warehouse problem (Monge / Kantorovich), the W₂ distance, one-dimensional monotone matching, and large-scale approximations (Sinkhorn, minibatch).
    1. M6.0 搬倉庫問題：Monge、Kantorovich 與 Coupling
    2. M6.1 兩個城市的人口分佈差多少：W₂ 距離
    3. M6.2 最省油的配法一定不交叉：一維的單調性與 Brenier
    4. M6.3 五萬個倉庫怎麼算：Sinkhorn 與 Minibatch 近似

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