Lieve Göbbels
Data Mining (JBI030)
Semester 2, 2020-2021
Data Mining (JBI030)
Linear Algebra 2
Vector Spaces 2
Normed Vector Spaces 5
Matrix Norms 6
Singular Value Decomposition (SVD) 7
Optimization 9
Unconstrained optimization problems 9
Constrained Optimization Problems 10
Analyzing the Optimization Problem 11
Properties of Convex Functions 12
Regression 14
Formalizing the Regression Task 14
Function Families and Classes 14
Measuring the Fit 16
The Design Matrix 16
The Regression Task 16
Recommender Systems 19
The Rank-r Matrix Factorization Problem 19
The Truncated SVD 19
Principle Component Analysis (PCA) 20
K-Means Clustering 23
The Cluster Model 23
The Objective 23
Lloyds K-Means Algorithm 24
Indicating Clusters by a Binary Matrix 24
The Centroid Matrix 25
Theorems 25
Notations and Symbols Cheatsheet 27
, Linear Algebra
In short:
• Vector Spaces
• Normed Vector Spaces
• Matrix Norms
• SVD (Singular Value Decomposition)
Vector Spaces
Introduction to properties and operations of a vector space
A vector space is a set of vectors V with two operations: + and · , such that the following properties
hold:
- Addition: for v, w we have v + w ∈ V. The set of vectors with addition (V, + ) is an abelian group.
- Scalar multiplication: for α ∈ ℝ and v ∈ V, we have α v ∈ V, such that the following properties
hold:
‣ α(β v) = (αβ )v for α, β ∈ ℝ and v ∈ V
‣ 1v = v for v = V
- Distributivity: the following properties hold:
‣ (α + β )v = α v + β v for α, β ∈ ℝ and v ∈ V
‣ α(v + w) = α v + α w for α ∈ ℝ and v, w ∈ V
The following operations are well-de ned (allowed):
v 1
- = v for α ≠ 0
α α
- v −w
The following properties are ill-de ned (not allowed):
- v⋅w
α
-
v
The elements of the vector space ℝd are d-dimensional vectors
v1
v = ⋮ , vi ∈ ℝ for 1 ≤ i ≤ d
vd
For vectors, the addition between vectors and the scalar multiplication are de ned for
v, w ∈ ℝd and α ∈ ℝ as:
v1 + w1 α v1
v +w = ⋮ and α v = ⋮
vd + wd α vd
Data Mining (JBI030)
Semester 2, 2020-2021
Data Mining (JBI030)
Linear Algebra 2
Vector Spaces 2
Normed Vector Spaces 5
Matrix Norms 6
Singular Value Decomposition (SVD) 7
Optimization 9
Unconstrained optimization problems 9
Constrained Optimization Problems 10
Analyzing the Optimization Problem 11
Properties of Convex Functions 12
Regression 14
Formalizing the Regression Task 14
Function Families and Classes 14
Measuring the Fit 16
The Design Matrix 16
The Regression Task 16
Recommender Systems 19
The Rank-r Matrix Factorization Problem 19
The Truncated SVD 19
Principle Component Analysis (PCA) 20
K-Means Clustering 23
The Cluster Model 23
The Objective 23
Lloyds K-Means Algorithm 24
Indicating Clusters by a Binary Matrix 24
The Centroid Matrix 25
Theorems 25
Notations and Symbols Cheatsheet 27
, Linear Algebra
In short:
• Vector Spaces
• Normed Vector Spaces
• Matrix Norms
• SVD (Singular Value Decomposition)
Vector Spaces
Introduction to properties and operations of a vector space
A vector space is a set of vectors V with two operations: + and · , such that the following properties
hold:
- Addition: for v, w we have v + w ∈ V. The set of vectors with addition (V, + ) is an abelian group.
- Scalar multiplication: for α ∈ ℝ and v ∈ V, we have α v ∈ V, such that the following properties
hold:
‣ α(β v) = (αβ )v for α, β ∈ ℝ and v ∈ V
‣ 1v = v for v = V
- Distributivity: the following properties hold:
‣ (α + β )v = α v + β v for α, β ∈ ℝ and v ∈ V
‣ α(v + w) = α v + α w for α ∈ ℝ and v, w ∈ V
The following operations are well-de ned (allowed):
v 1
- = v for α ≠ 0
α α
- v −w
The following properties are ill-de ned (not allowed):
- v⋅w
α
-
v
The elements of the vector space ℝd are d-dimensional vectors
v1
v = ⋮ , vi ∈ ℝ for 1 ≤ i ≤ d
vd
For vectors, the addition between vectors and the scalar multiplication are de ned for
v, w ∈ ℝd and α ∈ ℝ as:
v1 + w1 α v1
v +w = ⋮ and α v = ⋮
vd + wd α vd
