本系列为MIT Gilbert Strang教授的"数据分析、信号处理和机器学习中的矩阵方法"的学习笔记。

  • Gilbert Strang & Sarah Hansen | Sprint 2018
  • 18.065: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
  • 视频网址: https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/
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Lecture 0: Course Introduction

Lecture 1 The Column Space of AAA Contains All Vectors AxAxAx

Lecture 2 Multiplying and Factoring Matrices

Lecture 3 Orthonormal Columns in QQQ Give Q′Q=IQ'Q=IQ′Q=I

Lecture 4 Eigenvalues and Eigenvectors

Lecture 5 Positive Definite and Semidefinite Matrices

Lecture 6 Singular Value Decomposition (SVD)

Lecture 7 Eckart-Young: The Closest Rank kkk Matrix to AAA

Lecture 8 Norms of Vectors and Matrices

Lecture 9 Four Ways to Solve Least Squares Problems

Lecture 10 Survey of Difficulties with Ax=bAx=bAx=b

Lecture 11 Minimizing ||x|| Subject to Ax=bAx=bAx=b

Lecture 12 Computing Eigenvalues and Singular Values

Lecture 13 Randomized Matrix Multiplication

Lecture 14 Low Rank Changes in AAA and Its Inverse

Lecture 15 Matrices A(t)A(t)A(t) Depending on ttt, Derivative = dA/dtdA/dtdA/dt

Lecture 16 Derivatives of Inverse and Singular Values

Lecture 17 Rapidly Decreasing Singular Values

Lecture 18 Counting Parameters in SVD, LU, QR, Saddle Points

Lecture 19 Saddle Points Continued, Maxmin Principle

Lecture 20 Definitions and Inequalities

Lecture 21 Minimizing a Function Step by Step

Lecture 22 Gradient Descent: Downhill to a Minimum

Lecture 23 Accelerating Gradient Descent (Use Momentum)

Lecture 24 Linear Programming and Two-Person Games

Lecture 25 Stochastic Gradient Descent

Lecture 26 Structure of Neural Nets for Deep Learning

Lecture 27 Backpropagation: Find Partial Derivatives

Lecture 28 Computing in Class [No video available]

Lecture 29 Computing in Class (cont.) [No video available]

Lecture 30 Completing a Rank-One Matrix, Circulants!

Lecture 31 Eigenvectors of Circulant Matrices: Fourier Matrix

Lecture 32 ImageNet is a Convolutional Neural Network (CNN), The Convolution Rule

Lecture 33 Neural Nets and the Learning Function

Lecture 34 Distance Matrices, Procrustes Problem

Lecture 35 Finding Clusters in Graphs

Lecture 36 Alan Edelman and Julia Language

文章目录

  • Lecure 8: Norms of Vectors and Matrices
  • 8.1 Vectors norms ∥v∥p\|v\|_p∥v∥p​
  • 8.2 Optimal Problem
  • 8.3 Matrix Norms
  • 本节小结

Lecure 8: Norms of Vectors and Matrices

  • 一个与本课程无关的问题(现象)：Probability Matching

    • Biased Coin: 但是参与者不知道 (无先验)

      ▪ 75% likely to produce heads

      ▪ 25% likely to produce tails

    • payoff 1 guess right or 1 guess wrong

    • Optimal strategy: 刚开始会根据之前的经验猜测都是 50%, 但尝试过程中会慢慢优化 ⇒\Rightarrow⇒ 最终，to guess heads all the time

    • 但人类实际上会：guess heads three quarters of the time and tails one quarter of the time

    • 该问题说明了: there is good math questions everywhere, 与本节课无关

  • 下面开始本节课的内容: Norms

8.1 Vectors norms ∥v∥p\|v\|_p∥v∥p​