Machine Learning

Differentiation of Univariate Functions #

Differentiation measures rate of change.

For a function f(x), the derivative measures the rate of change.

$[ f'(x) = $lim_{h $to 0} $frac{f(x+h)-f(x)}{h} $]

Interpretation:

Slope of tangent
Instantaneous rate of change

Home | Vector Calculus

Partial Differentiation and Gradients

AI, ML

Machine Learning, Mathematics, Vector Calculus

Partial Differentiation and Gradients #

For f(x1, x2, …, xn):

[ \frac{\partial f}{\partial x_i} ]

Gradient vector:

[ \nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \ \vdots \ \frac{\partial f}{\partial x_n} \end{bmatrix} ]

Gradient points in direction of steepest ascent.

flowchart LR
    Input --> Function
    Function --> Gradient
    Gradient --> Optimisation

Home | Vector Calculus

Linear Independence

AI, ML

Machine Learning, Mathematics, Linear Algebra

Linear Independence #

A set of vectors is linearly independent if none of them can be written as a linear combination of the others.

\[ c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \;\Rightarrow\; c_1=\cdots=c_k=0 \]

Independence means each vector adds new information.

Gradients of Vector-Valued and Matrix Functions

AI, ML

Machine Learning, Mathematics, Vector Calculus

Gradients of Vector-Valued and Matrix Functions #

Covers gradients when outputs or parameters are vectors/matrices.

If f: R^n -> R^m, the derivative is the Jacobian.

[ J = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_n} \ \vdots & \ddots & \vdots \ \frac{\partial f_m}{\partial x_1} & \dots & \frac{\partial f_m}{\partial x_n} \end{bmatrix} ]

For scalar f(x):

[ H = \nabla^2 f ]

Hessian captures curvature.

Reinforcement Learning

AI, ML

AI, Machine Learning, Reinforcement Learning

Reinforcement Learning (RL) #

RL is learning by trial and error.

Reinforcement Learning (RL) is a type of machine learning where an autonomous agent learns to make decisions by interacting with an environment.

Instead of being told the correct answer, the agent:

takes actions
observes outcomes
receives rewards or penalties
gradually learns a strategy that maximises long-term reward

Reinforcement Learning teaches an agent how to act, not what to predict.

Useful Gradient Identities

AI, ML

Machine Learning, Mathematics, Vector Calculus

Useful Gradient Identities #

[ \nabla (a^T x) = a ] [ \nabla (x^T A x) = (A + A^T)x ]

If A symmetric:

[ \nabla (x^T A x) = 2Ax ]

These are heavily used in optimisation.

Home | Vector Calculus

Inner Products and Dot Product

AI, ML

Machine Learning, Mathematics, Linear Algebra

Inner Products and Dot Product #

An inner product maps two vectors to a single scalar.

It allows us to measure:

similarity
vector length
projections
orthogonality

flowchart TD
T["Inner<br/>products<br/>(types)"] --> DOT["Euclidean<br/>Dot product"]
T --> WIP["Weighted<br/>inner product"]
T --> FN["Function-space<br/>(integral)"]
T --> HERM["Complex<br/>Hermitian"]
T --> MAT["Matrix<br/>inner product<br/>(Frobenius)"]

DOT --> Rn["Vectors in<br/>
<span>
  \( \mathbb{R}^n \)
  </span>

"]
WIP --> SPD["SPD matrix<br/>W"]
FN --> L2["L2 space<br/>functions"]
HERM --> Cn["Vectors in<br/>C^n"]
MAT --> Mnm["Matrices<br/>R^{m×n}"]

style T fill:#90CAF9,stroke:#1E88E5,color:#000

style DOT fill:#C8E6C9,stroke:#2E7D32,color:#000
style WIP fill:#C8E6C9,stroke:#2E7D32,color:#000
style FN fill:#C8E6C9,stroke:#2E7D32,color:#000
style HERM fill:#C8E6C9,stroke:#2E7D32,color:#000
style MAT fill:#C8E6C9,stroke:#2E7D32,color:#000

style Rn fill:#CE93D8,stroke:#8E24AA,color:#000
style SPD fill:#CE93D8,stroke:#8E24AA,color:#000
style L2 fill:#CE93D8,stroke:#8E24AA,color:#000
style Cn fill:#CE93D8,stroke:#8E24AA,color:#000
style Mnm fill:#CE93D8,stroke:#8E24AA,color:#000