Python與矩陣論——特征值與特征向量
The Unknown Word
| The First Column | The Second Column |
|---|---|
| receptive field | [ri'septiv] [fild]感受野 |
| filter | 濾波器['filte] |
| toggle movement | ['ta:gl]轉換 ['muvment]動作 |
| recurrent | 循環的[ri'ke:rent] |
| ReLU | The Rectified Linear Unit |
| rectified | ['rekte faie]整流器 |
| leaky | 有漏洞的['liki] |
| tuning | ['tju:ning]調諧 |
| SGD | Stochastic gradient descent |
| stochastic | [ste'kae stik]隨機的 |
| hypothesis | [hai'pa:thesis]假設 |
| SVD | Singular Value Decomposition 萬能矩陣分解 |
| singular | ['singjule(r)]奇特的 |
| decomposition | [dikamlpe'zition]分解 |
| format | n.版式 vt.格式化 |
PCA 降維舉例
1.X=\(\begin{bmatrix} -1 & -1 & 0 & 2 & 0 \\-2 & 0 & 0 & 1 & 1 \\ \end{bmatrix}\)，\(C_x\)=\(\begin{bmatrix} 6/5 & 4/5 \\ 4/5 & 6/5 \\ \end{bmatrix}\)
2.計算\(C_x\)特征值為：\(\lambda\)=2,\(\lambda_2\)=2/5,特征值特征向量為$\begin{bmatrix} \sqrt{2}\ \ \sqrt{2}\ \ \end{bmatrix} $ ，可驗證\(\Lambda\)=\(U^T\)\(C_x\)U
3.降維：\(\begin{bmatrix} 1/\sqrt{2}\ & -1/\sqrt{2}\ \end{bmatrix}\)X=\(\begin{bmatrix} -3/\sqrt{2}\ & -1/\sqrt{2}\ & 0 & 3/\sqrt{2}\ & -1/\sqrt{2}\ \end{bmatrix}\)
Gradient vector and Hessian matrix
- Function : f(x)=2\(x_1^3+3x_2^2+3x_1^2x_2-24x_2\)
- calculate(Gradient vector and Hessian matrix): \(\nabla\)f(x)=\(\begin{bmatrix} 6x_1^2+6x_1x_2 \\ 6x_2+3x_1^2-24 \\ \end{bmatrix}\)，\(\nabla^2\)f(x)=6\(\begin{bmatrix} 2x_1+x_2 & x_1 \\ x_1 & 1 \\ \end{bmatrix}\)
The Unknown Word
| The First Column | The Second Column |
|---|---|
| convex optimization | 凸規划 |
| optimization | [optemi'zetion]最優化 |