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APS1070
Foundations of Data Analytics and Machine Learning
Fall 2021
Week 5:
• LinearAlgebra
• Analytical Geometry
• Data Augmentation
Prof.

Mid-term assessment logistics
➢Platform: Crowdmark
➢ is available now (emailed to you) to get familiar with the
logistics. It will not be graded. Due date 8 Oct at 21:00.
➢Some practice problems and solutions from past semesters are on Quercus
• Material in midterm:
• Week 1 to 5 of Lectures, Tutorials 1 and 2, Project 1, Reading assignments 1-4.
2

Mid-term assessment logistics
• Midterm Assessment Distribution: Oct 18th at 9:00 (Toronto time)
• Deadline for Submitting the Assessment: Oct 19th at 21:00 (Toronto time)
• A countdown starts as soon as you access the assessment (2 hours as per course schedule)
• 30 minutes are already added to the countdown time for contingency. No excuses.
• Time needed for writing answers: about 90 minutes
• Late submission or no submission: 0 mark (as per syllabus).
• Use course material + online resources – NO HELP FROM OTHERS!! No Piazza.
• Quercus announcement for common issues and typos,
• If needed, make assumptions and answer the questions. Do not contact us to ask.
• In case of a logistic problem, you can email Samin and Ali during your assessment. There
is no guarantee that we can respond to you before your time runs out.
3

Slide Attribution
These slides contain materials from various sources. Special thanks to the following authors:
• •
•

Last Time
➢Looked into assessing model performance ➢Probability Theory
➢ Gaussians ➢Confusion Matrix ➢ ROCs
➢Data Processing to the Rescue ➢Data Augmentation (today) ➢Dimensionality Reduction (Week 7)
5

What do these datasets have in common?
➢ How can we improve these datasets?
6

ML Performance Benchmarks
➢C10+ and C100+ highlight the error rates after data augmentation
➢Data augmentation found to consistently lower the error rates!
7

Agenda
➢Linear Algebra
➢Scalars, Vectors, Matrices
➢Solving Systems of Linear Equations ➢Linear Independence
➢Linear Mappings
➢Analytic Geometry
➢Norms, Inner Products, Lengths, etc. ➢Angles and Orthonormal Basis
➢Data Augmentation
Today’s Theme:
Data Processing
8

Part 1 Linear Algebra
Readings:
• Chapter 2.1-5 MML Textbook

Systems of Linear Equations
➢ The solution space of a system of two linear equations with two variables can be geometrically interpreted as the intersection of two lines
➢ intersection of planes in three variables x2
4×1+ 4×2=5
2×1- 4×2=1
x1
System in three variables – solution is at intersection
System with 2 equations and three variables – solution is typically a line
10

Matrix Representation
➢Used to solve systems of linear equations more systematically ➢Compact notation collects coefficients into vectors, and vectors
into matrices:
𝑎11 𝑎12 𝑎1𝑛 𝑏 ⋮⋮⋮1
𝑥1 +𝑥2 +…+𝑥𝑛 𝑎𝑚1 𝑎𝑚2
= ⋮ 𝑎𝑚𝑛 𝑏𝑚
𝑎11 ⋯ 𝑎1𝑛 𝑥1 𝑏
⋮⋮⋮1 ⟺=⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛 𝑥𝑛
𝑏𝑚
11

Matrix Notation
➢A matrix has m x n elements (with 𝑚,𝑛 ∈ N, and aij, i=1,…,m; j=1,…,n) which are ordered according to a rectangular scheme consisting of m rows and n columns:
12n
𝑎11 𝑎12 … 𝑎1𝑛 1 𝐴=𝑎21 𝑎22 …𝑎2𝑛2𝑎𝑖𝑗∈R
⋮⋮⋮
𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛 m
➢By convention (1 by n)-matrices are called rows and (m by 1)-matrices are called columns. These special matrices are also called row/column vectors.
➢A (1 by 1)-matrices is referred to as scalars
12

Addition and Scalar Multiplication
➢Vector addition:
𝑎+𝑏= 𝑎1 + 𝑏1 = 𝑎1+𝑏1 𝑎2 𝑏2 𝑎2 + 𝑏2
➢Scalar multiplication:
⍺𝑏=⍺𝑏1 =⍺𝑏1 𝑏2 ⍺𝑏2
13

Addition and Scalar Multiplication
➢Matrix addition: The sum of two matrices 𝐴 ∈ R𝑚×𝑛, 𝐵 ∈ R𝑚×𝑛 is defined as the element-wise sum:
𝑎+𝑏 …𝑎+𝑏 11 11 1𝑛 1𝑛
𝐴+𝐵=⋮⋮ 𝑎𝑚1+𝑏𝑚1 … 𝑎𝑚𝑛+𝑏𝑚𝑛
➢Scalar multiplication of a matrix 𝐴 ∈ R𝑚×𝑛is defined as:
𝛼∗𝑎11 … 𝛼∗𝑎1𝑛 ⋮⋮ 𝛼∗𝑎𝑚1 … 𝛼∗𝑎𝑚𝑛
𝛼∗𝐴=
14

Matrix Multiplication
➢ We can multiply a matrix by a column vector:
𝑎11 𝑎12 𝑎13 𝑥1 𝑎11𝑥1 + 𝑎12𝑥2 + 𝑎13𝑥3 𝐴𝑥 = 𝑎21 𝑎22 𝑎23 𝑥2 = 𝑎21𝑥1 +𝑎22𝑥2 +𝑎23𝑥3 𝑎31 𝑎32 𝑎33 𝑥3 𝑎31𝑥1 + 𝑎32𝑥2 + 𝑎33𝑥3
➢ We can multiply a matrix by a row vector:
𝑎11 𝑎12 𝑎13
𝑥𝑇𝐴 = 𝑥1 𝑥2 𝑥3 𝑎21 𝑎22
𝑎31 𝑎32 𝑎33
𝑎23 = 𝑎11𝑥1 + 𝑎21𝑥2 + 𝑎31𝑥3 𝑎12𝑥1 + 𝑎22𝑥2 + 𝑎32𝑥3 𝑎13𝑥1 + 𝑎23𝑥2 + 𝑎33𝑥3
➢ In general, we can multiply matrices A and B when the number of columns in A matches the number of rows in B:
𝑎11 𝑎12 𝑎13
𝑏11 𝑏12 𝑏13
𝑎11𝑏11 + 𝑎12𝑏21 + 𝑎13𝑏31 𝑎21𝑏11 + 𝑎22𝑏21 + 𝑎23𝑏31 𝑎31𝑏11 + 𝑎32𝑏21 + 𝑎33𝑏31
𝑎11𝑏12 + 𝑎12𝑏22 + 𝑎13𝑏32 𝑎21𝑏12 + 𝑎22𝑏22 + 𝑎23𝑏32 𝑎31𝑏12 + 𝑎32𝑏22 + 𝑎33𝑏32
𝑎11𝑏13 + 𝑎12𝑏23 + 𝑎13𝑏33 𝑎21𝑏13 + 𝑎22𝑏23 + 𝑎23𝑏33 𝑎31𝑏13 + 𝑎32𝑏23 + 𝑎33𝑏33
𝐴𝐵 = 𝑎21 𝑎22
𝑎31 𝑎32 𝑎33
𝑎23
𝑏21
𝑏31 𝑏32
=
𝑏22 𝑏23 𝑏33
15

Example: Matrix Multiplication
1232×3 023×2 ➢ For two matrices: 𝐴 = 3 2 1 ∈ R , 𝐵 = 1 −1 ∈ R ,
➢ we obtain:
01
12302 23 2×2 𝐴𝐵=3211−1=25∈R ,
01
02123 642 3×3 𝐵𝐴= 1 −1 3 2 1 = −2 0 2 ∈R
01 321 Not commutative! 𝐴𝐵 ≠ 𝐵𝐴
16

Basic Properties
➢ A few properties: ➢ Associativity:
∀𝐴 ∈ R𝑚×𝑛,𝐵 ∈ R𝑛×𝑝,𝐶 ∈ R𝑝×𝑞: 𝐴𝐵 𝐶 = 𝐴(𝐵𝐶)
➢ Distributivity:
∀𝐴,𝐵∈R𝑚×𝑛,𝐶,𝐷∈R𝑛×𝑝: 𝐴+𝐵 𝐶=𝐴𝐶+𝐵𝐶
𝐴𝐶+𝐷 =𝐴𝐶+𝐴𝐷
17

Inner Product and Outer Product
➢The inner product between vectors of the same length is: 𝑛
𝑎𝑇𝑏=෍𝑎𝑖𝑏𝑖 =𝑎1𝑏1+𝑎2𝑏2+ …+𝑎𝑛𝑏𝑛=𝛾 𝑖=1
➢The outer product between vectors of the same length is: 𝑎𝑏 𝑎𝑏 …𝑎𝑏
Theinner product is a scalar
11121𝑛
𝑎𝑏 𝑎𝑏 …𝑎𝑏 𝑎𝑏𝑇=2122 2𝑛
⋮⋮⋮
𝑎𝑏 𝑎𝑏 …𝑎𝑏 𝑛1𝑛2𝑛𝑛
The outer product is a matrix
18

Identity Matrix
➢ We define the identity matrix as shown:
10⋯0⋯0
01⋯0⋯0 𝐼:=⋮ ⋮ ⋱ ⋮ ⋱ ⋮
∈R𝑛×𝑛
𝑛
00⋯0⋯1
00⋯1⋯0 ⋮⋮⋱⋮⋱⋮
➢ Any matrix multiplied by the identity will not change the matrix: 1 2 3 1 0 0 1+0+0 0+2+0 0+0+3
4 5 6 X 0 1 0 = 4+0+0 0+5+0 0+0+6 7 8 9 0 0 1 7+0+0 0+8+0 0+0+9
19

Inverse
➢If square matrices 𝐴 ∈ R𝑛×𝑛 and 𝐵 ∈ R𝑛×𝑛 have the property that 𝐴𝐵 = 𝐼 = 𝐵𝐴. Then B is called the inverse of A and denoted by A-1.
➢Example, these matrices are inverse to each other:
1 2 1 −7 −7 6 𝐴=445, 𝐵=2 1−1 6 7 7 4 5 −4
➢ We’ll look at how to calculate the inverse later
𝑛
20

Transpose
➢Transpose definition: For 𝐴 ∈ R𝑚×𝑛 the matrix 𝐵 ∈ R𝑛×𝑚 with 𝑏𝑖𝑗 = 𝑎 is called transpose of A. We write 𝐵 = 𝐴𝑇.
➢Some useful identities:
𝑗𝑖
➢Symmetric Matrix: A matrix 𝐴 ∈ R𝑛×𝑛 is symmetric if 𝐴 = 𝐴𝑇.
𝐴𝐴−1 = 𝐼 = 𝐴−1𝐴 (𝐴𝐵)−1= 𝐵−1𝐴−1 (𝐴 + 𝐵)−1≠ 𝐴−1 + 𝐵−1 (𝐴𝑇)𝑇= 𝐴
(𝐴 + 𝐵)𝑇= 𝐴𝑇 + 𝐵𝑇 (𝐴𝐵)𝑇= 𝐵𝑇𝐴𝑇
21

Solving Systems of Linear Equations
➢Given A and b, we want to solve for x:
211𝑥1 5 𝐴𝑥=𝑏 4 −60𝑥2=−2
−272𝑥3 9
➢ Key to solving a system of linear equations are elementary transformations that keep the solution set the same, but that transform the equation system into a simpler form.
1. Exchange of two equations (rows in the matrix)
2. Multiplication of an equation (row) with a constant
3. Addition of two equations (rows)
➢This is known as Gaussian Elimination (aka row reduction)
22

Triangular Linear Systems
➢ Consider a square linear system with an upper triangular matrix (non- zero diagonals):
𝑎11 𝑎12 𝑎13 𝑥1
0 𝑎22 𝑎23 𝑥2 𝑏2 0 0 𝑎33 𝑥3 𝑏3
➢ We can solve this system bottom to top using substitution:
𝑎33𝑥3 = 𝑏3
𝑎22𝑥2 + 𝑎23𝑥3 = 𝑏2
𝑎 𝑥 +𝑎 𝑥 +𝑎 𝑥 =𝑏 11 1 12 2 13 3 1
𝑥3 = 𝑏3 𝑎33
𝑥2 = 𝑏2−𝑎23𝑥3 𝑎22
𝑥 =𝑏1−𝑎13𝑥3−𝑎12𝑥2
𝑏 1
1
𝑎11
23

Example: Gaussian Elimination
➢Gaussian elimination uses elementary row operations to transform a linear system into a triangular system:
➢ Add -2 times first row to second ➢ Add 1 times first row to third
➢ Add 1 times second row to third
2 1
0 −8
0 8 3 14
2𝑥1+𝑥2+𝑥3=5 4𝑥1−6𝑥2=−2 −2𝑥1+7𝑥2+2𝑥3=9
2𝑥1 + 𝑥2 + 𝑥3 = 5 −8𝑥2 − 2𝑥3 = −12 8𝑥2 + 3𝑥3 = 14
2𝑥1 + 𝑥2 + 𝑥3 = 5 −8𝑥2 −2𝑥3 =−12
2 1 15
4 −6 0อ−2 −2 7 2 9
0 −8
𝑥3=2 0012
2 1 1 5
Row Echelon form
24
1
−2อ −12
−2อ−12
5

Row Echelon Form (REF)
• The first non-zero coefficient from the left (the “leading coefficient”) is always to the right of the first non-zero coefficient in the row above.
• Rows consisting of all zero coefficients are at the bottom of the matrix.
2115 0 −8 −2อ −12 0012
Row Echelon form
25

Example: Reduced Echelon Form
➢We can simplify this even further:
➢ Divide first row by 2 ➢ Divide 2nd row by -8
2𝑥1 + 𝑥2 + 𝑥3 = 5 −8𝑥2 − 2𝑥3 = −12 𝑥3 = 2
𝑥1+0.5𝑥2+0.5𝑥3=2.5 𝑥2 +0.25𝑥3 =1.5
𝑥3 = 2
2115 0 −8 −2อ −12 0012
1 0.5 0.5 2.5 0 1 0.25อ1.5 0012
➢ Add -0.25 times third row to second row ➢ Add -0.5 times third row to first row
➢ Add -0.5 times second row to first row
𝑥1=1 𝑥2=1 𝑥3=2
1 0 01 0 1 0อ1 0 0 12
Reduced row Echelon form
26

The coefficient matrix could be non-square Example 2.6 from MML book (reading assignment 4):
Four equations and five unknowns

Alternative Method: Inverse Matrix
➢We can also solve linear systems of equations is by applying the inverse.
➢The solution to 𝐴𝑥 = 𝑏 can be obtained by multiplying by 𝐴−1 to isolate for x.
𝐴𝑥 = 𝑏 𝐴−1𝐴𝑥 = 𝐴−1𝑏
−1 𝑥=𝐴 𝑏
Note that 𝐴−1will cancel out 𝐴 only if multiplied from the left-hand side, otherwise we have 𝐴−1𝑥𝐴
𝐼 𝑥 = 𝐴 𝑏
𝑛
−1
33

Calculating an Inverse Matrix
➢ To determine the inverse of a matrix A ➢ Write down the augmented matrix with
the identity on the right-hand side
➢ Apply Gaussian elimination to bring it into reduced row-echelon form. The desired inverse is given as its right-hand side:
➢ We can verify that this is indeed the inverse by
performing the multiplication 𝐴𝐴−1 and
observing that we recover 𝐼 . 𝑛
𝐴=
1020 1100 1201 1111
10201000 1 1 0 0ተ0 1 0 0 12010010 11110001
1 0 0 0 −1 2 −2 2 0 1 0 0ተ 1 −1 2 −2 00101 −1 1 −1 0 0 0 1 −1 0 −1 2
−1 2 −2 2 1 −1 2 −2 1 −1 1 −1
−1 0 −1 2
𝐴−1 =
34

What can go wrong?
➢ Applying Gaussian Elimination (row reduction) does not always lead to a solution.
➢ Singular Case: When we have a 0 in a pivot column. This is an example of a matrix that is not invertible.
➢ For example:
2111 1005 0 0 3อ−2 0 1 0อ4 0042 0000
singular
singular
35

Several Interpretations
➢Given A and b, we want to solve for x:
𝐴𝑥=𝑏 2−1𝑥=1
➢This can be given several interpretations: ➢By rows: x is the intersection of hyper-planes:
2𝑥 − 𝑦 = 1 𝑥+𝑦=5
➢By columns: x is the linear combination that gives b: 𝑥 2 +𝑦 −1 = 1
115
➢Transformation: x is the vector transformed to b: 𝑇𝑥=𝑏
37
11𝑦5

Geometry of Linear Equations
➢By Rows:
Find intersection of
hyperplanes
Hyperplane from row 1
Solution(x) x=[1.7,2.2]T
Hyperplane from row 2
➢By Columns:
Find linear combination of
columns
𝑥 2 +𝑦 −1 = 1 115
b=1.7*Column1+ 2.2*Column2
(so x=[1.7,2.2]T
Column 1
Column 2
38

What can go wrong?
➢By rows:
Hyperplane from row 2
Hyperplane from row 1
Hyperplane from row 1
Hyperplane from row 2
No intersection Infinite intersection
39

One unique solution
$1000 USD
$10 CAD
Want to buy a phone: $1000USD: phone +$10 CAD shipping
$1 USD
1000x 10x
$1 CAD
40

No solution
Want to buy a phone: $1000USD: phone + 10 Euro shipping
$1000USD €10EURO
$1 USD $1 CAD 1000x 10x
41

No solution
➢By columns:
Vector not in column
space (no solution)
Column-space
Columns of matrix
42

Infinite solution
$1000 USD
$10 CAD
Want to buy a phone: $1000USD: phone +$10 CAD shipping
USD: 1000 CAD: 10 BTC: 0
USD: 999 CAD: 10 BTC: 0.0001
USD: 999 CAD: 9
BTC: 0.00019
$1 USD
$1 CAD
USD: 998.2 CAD: 8.6
BTC: 0.000216
BTC
43

Independent Columns
➢By columns:
Column-space
Columns of matrix
Vector in column space (infinite solutions)
44

Solutions to Ax=b
➢Q: In general, when does Ax=b have a unique solution?
➢A: When b is in the column- space of A, and the columns of A are linearly independent
➢Q: What does it mean to be independent?
Vector not in column space (no solution)
Column-space
Columns of matrix
Vector in column space (infinite solutions)
45

Linear Dependence
➢A set of vectors is either linearly dependent or linearly independent.
➢A vector is linearly dependent on a set of vectors if it can be written as a linear combination of them:
𝑐=𝛼𝑏+𝛼𝑏+…𝛼𝑏 1122𝑛𝑛
➢We say that c is “linearly dependent” on {b1, b2, …, b3}, and that the set {c,b1, b2, …, b3} is “linearly dependent”
➢A set is linearly dependent iff the zero vector can be written as a combination of the vectors {b1, b2, …, b3}:
∃ ≠0,𝑠.𝑡.0=𝛼 𝑏 +𝛼 𝑏 + …𝛼 𝑏 => 𝑏 ,𝑏 ,…,𝑏 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝛼 1122𝑛𝑛12𝑛
46

Linear Independence
➢If a set of vectors is not linearly dependent, we say it is linearly independent
➢The zero vector cannot be written as a combination of independent vectors unless all coefficients 𝛼 are set to zero:
0 = 𝛼 𝑏 + 𝛼 𝑏 + … 𝛼 𝑏 => 𝛼 = 0 ∀ 1122𝑛𝑛𝑖𝑖
➢If the vectors are independent, then there is no way represent one of the vectors as a combination of the others.
47

Linear Dependence vs Independence
➢Independence in R2:
Dependent
Independent
Independent Dependent
48

Linear Independence
➢ Consider we have a set of three vectors {𝑥1, 𝑥2 , 𝑥3} 𝜖 R4 ➢ To check whether they are linearly dependent, we
solve: 𝜆1𝑥1 + 𝜆2𝑥2 + 𝜆2𝑥3 = 0
➢We write the vectors 𝑥𝑖,𝑖 = 1,2,3, as the columns of a matrix and apply elementary row operations until we identify the pivot columns.
➢ All column vectors are linearly independent if and only if all columns are pivot columns. If there is at least one non-pivot column the vectors are linearly dependent.
1 1 −1
𝑥 = 2 ,𝑥 = 1 ,𝑥 = −2 1 −3 2 0 3 1
421 𝜆1𝑥1 +𝜆2𝑥2 +𝜆2𝑥3 =0
1 1 −1
2 1 −2 −3 0 1 421
…
1 1 −1 010 001 000
49

Vector Space
➢A vector space is a set of objects called “vectors”, with closed operations “addition” and “scalar multiplication” satisfying certain axioms:
1. 𝑥+𝑦=𝑦+𝑥
2. 𝑥+𝑦+𝑧=𝑥+𝑦+𝑧
3. exists a zero vector “0” s.t. ∀𝑥,𝑥 + 0 = 𝑥
4. ∀𝑥,𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑛𝑎𝑑𝑑𝑖𝑡𝑖𝑣𝑒𝑖𝑛𝑣𝑒𝑟𝑠𝑒“−𝑥”,𝑠.𝑡. 𝑥+ −𝑥 =0
5. 1𝑥=𝑥
6. 𝑐1𝑐2 𝑥 = 𝑐1(𝑐2𝑥)
7. 𝑐𝑥+𝑦=𝑐𝑥+𝑐𝑦
8. 𝑐1+𝑐2 𝑥=𝑐1𝑥+𝑐2𝑥
➢ Examples: R, R2, R𝑛
50

Subspace
➢Subspaces generated in R2:
span({x1,x2})=line
span({x1,x2})=R2
set of vectors 𝒜 = {𝑥1,…,𝑥𝑘} ⊆ 𝒱
The set of all linear combinations of vectors in 𝒜 is called the span of 𝒜. If 𝒜 spans the vector space V, write 𝑉 = 𝑠𝑝𝑎𝑛[𝒜] or
𝑉 = 𝑠𝑝𝑎𝑛[𝑥1,…,𝑥𝑘]
span({x1,x2})=R2
span({x1,x2,x3})=R2
51

Basis
➢The vectors that span a subspace are not unique
➢However, the minimum number of vectors needed to span a subspace is
unique
➢This number is called the dimension or rank of the subspace
➢A minimal set of vectors that span a subspace is called a basis for the space
➢The vectors in a basis must be linearly independent, otherwise we could remove one and still span space
52

Basis
➢Basis in vector space V ∈ R2:
Not a
basis
Basis
Every linearly independent set of vectors that span V is called a basis of V
Basis
Not a
basis
53

Example Bases
➢ In R3 , the canonical/standard basis is: 100
B=൝0,1,0ൡ 001
➢ Two different bases of R3are:
1 1 1 0.5 1.8 −2.2
B =൝0,1,1ൡ B=൝0.8,0.3,−1.3ൡ 12
0 0 1 0.4 0.3 3.5
54

Linear Mapping/Transformation
➢Earlier, we saw that vectors are objects that can be added together and multiplied by a scalar, and the resulting object is still a vector
➢Now, we do the same for vector spaces
➢ Linear Mapping: For vector spaces V, W, a mapping 𝜙: 𝑉 → 𝑊 is called a linear
mapping (or linear transformation) if:
∀𝑥,𝑦∈𝑉 ∀𝜆,𝜓∈R: Φ 𝜆𝑥+𝜓𝑦 =𝜆Φ 𝑥 +𝜓Φ(𝑦)
➢It turns out that we can represent linear mappings as matrices. Recall that we can also collect a set of vectors as columns of a matrix. When working with matrices, we have to keep in mind what the matrix represents: a linear mapping or a collection of vectors.
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Linear Mapping/Transformation
➢A vector has different coordinate representations depending on which coordinate system or basis is chosen.
➢ Example: two different coordinate systems defined by two sets of basis vectors.
𝑥
𝑥 = − 1 𝑏1 + 5 𝑏2 22
two different bases
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𝑒2
𝑏2
𝑒1 𝑏1
𝑥 = 2𝑒1 + 3𝑒2

Example: Change of Basis Matrix
v2
f u1
U = [2 3]T[4 5]T [f]v=[2 4]T [f]u=?
u2
v1
Source:

Examples of Transforms
➢Scale
➢Rotation
➢Horizontal Mirror
➢Vertical Mirror
➢Combination of Transformations
Source: mathisfun.com
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Part 2 Analytical Geometry
Readings:
• Chapter 3.1-5,8,9 MML Textbook

Norms
➢A norm is a scalar measure of a vector’s length.
➢The most important norm is the Euclidean norm and for 𝑥 ∈
R𝑛 is defined as:
𝑛
𝑥2≔ ෍𝑥2= 𝑥𝑇𝑥 𝑖
𝑖=1
computes the Euclidian distance of x from the origin.
Euclidean norm is also known as the L2 norm
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Norms
➢For different norms, the red lines indicate the set of vectors with norm 1.
1
𝑥1=1
1
1 𝑥2=1
Manhattan norm
Euclidean distance
1
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Dot product
➢Dot product:
𝑎 ∙𝑏 = 11
𝑛
𝑥𝑇𝑦 = ෍𝑥𝑖𝑦𝑖 1 3 𝑖=1
∙ = 1∙3 + 7∙5 =38 75
➢Commonly, the dot product between two vectors a, b is denotedby𝑎𝑇𝑏or 𝑎,𝑏.
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Lengths and Distances
➢Consider an inner product space. ➢ Then
𝑑𝑥,𝑦≔𝑥−𝑦= 𝑥−𝑦,𝑥−𝑦 is called the distance between x and y for 𝑥, 𝑦 ∈ 𝑉.
➢If we use the dot product as the inner product, then the distance is called Euclidean distance.
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Angles
➢ The angle 𝜽 between two vectors 𝒙, 𝒚 is computed using the inner product. ➢ For Example: Let us compute the angle between
𝑥 = [1,1]𝑇∈ R2 and 𝑦 = [1,2]𝑇∈ R2
➢ Using the dot product as the inner product we get:
cos𝜃= 𝑥,𝑦 =𝑥𝑇𝑦=3 𝑥,𝑥 𝑦,𝑦 𝑥𝑇𝑥𝑦𝑇𝑦 10
y
1
➢ Then the angle between the two vectors is cos−1( 3 ) ≈ 0.32𝑟𝑎𝑑 , which corresponds to
0
𝜃x
1
10 approximately 18°.
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Orthogonality
➢Orthonormal = Orthogonal and unit vectors
➢Orthogonal Matrix: A square matrix 𝐴 ∈ R𝑛×𝑛 is an orthogonal
matrix if and only if its columns are orthonormal so that 𝐴𝐴𝑇 = 𝐼 = 𝐴𝑇𝐴,
➢which implies that
i.e., the inverse is obtained by simply transposing the matrix.
𝐴−1 = 𝐴𝑇,
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Orthonormal Basis
➢In n-dimensional space, we need n basis vectors that are linearly independent, if these vectors are orthogonal, and each has length 1, it’s a special case: orthonormal basis
➢ Consider an n-dimensional vector space V and a basis 𝑏,𝑏 =0𝑓𝑜𝑟𝑖≠𝑗
𝑏 , … , 𝑏 1𝑛
of V. If
𝑖𝑗
𝑏𝑖,𝑏𝑖 =1
for all 𝑖, 𝑗 = 1, … , 𝑛 then the basis is called an orthonormal basis (ONB). Note that 𝑏𝑖 , 𝑏𝑖 = 1 implies that every basis vector has length/norm 1.
➢ If only 𝑏 , 𝑏 = 0 𝑓𝑜𝑟 𝑖 ≠ 𝑗 is satisfied, then the basis is called an orthogonal 𝑖𝑗
basis.
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Orthonormal Basis
➢The canonical/standard basis for a Euclidean vector space R𝒏 is an orthonormal basis, where the inner product is the dot product of vectors.
➢Example: In R2, the vectors: 𝑏=11,𝑏=11,
1 21 2 2−1
form an orthonormal basis since 𝑏𝑇𝑏 = 0 and 𝑏 = 1 = 𝑏 1212
.
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Orthogonal Projections
➢ Projections are linear transformations, project to lower dimensional feature space 68

Orthogonal Projections
➢ The projection is defined
𝒃𝑇𝒙 𝒃𝒃𝑇
𝜋𝑈 𝒙 = 𝜆𝒃 = 𝒃 𝒃 2 = 𝒃 2 𝒙
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Example: Orthogonal Projections
1
0
b
1
x
➢ Compute the projection of 𝑥
= [1,2]𝑇 ∈ R2
onto 𝑏
= [1,1]𝑇∈ R2
𝒃𝒃𝑇 𝜋𝑈 𝒙 = 𝒃 2 𝒙
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Projection Matrix
➢ We can also use a projection matrix, which allows us to project any vector x onto the subspace defined by 𝜋.
𝑃= 𝜋
➢ Note that 𝒃𝒃𝑇 will be a symmetric matrix
𝜋𝑈𝒙= 𝒙
𝒃𝒃𝑇
2
𝒃
𝒃𝒃𝑇
2
𝒃
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Example: Applying Projection Matrix
➢Compute the projection matrix for 𝑏
= [1,1]𝑇∈ R2
𝒃𝒃𝑇
𝑃= 𝜋
1
𝒃
2
0
b
1
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Part 3
Data Augmentation

Non-Representative Data
➢Everything our algorithms learn comes form the data used to train them.
➢If the data is of poor quality, unbalanced or not representative of the task we want to solve, then how are our algorithms going to learn to generalize?
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Capacity and Training
➢Deep learning algorithms have the capacity to classify real images in various orientations and scales.
➢If you train your algorithms on perfectly processed samples, then they won’t know how to predict anything but perfectly cropped images.
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Data Augmentation
➢Use linear algebra to perform common transformations to supplement datasets
➢ Translation, Scaling, Rotation, Reflection ➢ Noise, Light and Colour Intensity
➢ Many more…
GAN Fake Celebrities
Source: Viridian Martinez
Source: kaggle.com
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➢ Advanced:
➢Generative models (i.e., Deep learning) to create new images with similar characteristics

Test Time Data Augmentation
➢You can also apply data augmentation to better evaluate your performance on test examples.
➢Great way to assess limitations of your model to images of different rotations, scales, noise, etc.
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Next Time
➢Oct 7th: Week 5 Tutorial 2 on Anomaly Detection ➢Project 2 Overview
➢Oct 11-15 Reading Week (Q&A Session on Thursday Oct 14th, no lecture) ➢Oct 18-19 Midterm (Q&A Session on Thursday Oct 21st, no lecture)
➢Project 2 is due on Oct 22nd
➢Lecture 7 – Dimensionality Reduction (Oct 26th) ➢Curse of Dimensionality
➢ Eigendecomposition ➢Singular Value Decomposition ➢Principle Component Analysis
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Google Colab

CS代考 APS1070 – cscodehelp代写

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