Kiru Lab  /  Track

The Mathematics of Learning

Three ideas — direction, change, and uncertainty — carry the entire field.

You do not need a mathematics degree to do this work, but you do need three ideas held firmly: a vector as a direction in a space, a derivative as a rate of change, and a probability distribution as an honest statement of what you do not know. Every later track is these three ideas wearing different clothes. The Jacobian that moves a robot arm and the gradient that trains a transformer are the same object.

The question this track answers

Why does following a derivative downhill produce intelligence-shaped behavior?

By the end you can

  • Read matrix notation as a description of a function between spaces
  • Compute and interpret a gradient without fear
  • Apply the chain rule to a composition of several functions
  • Explain a loss function as a negative log-likelihood
  • Recognize the Jacobian as the general case of the derivative

Come here from

Foundations: Python as an Instrument

Scope

7 lessons, roughly 16 hours.

Module 1

The Linear Algebra You Actually Need

Vectors as meaning, matrices as functions, and the geometry of similarity.

Vectors, Dot Products, and Alignment
The dot product is the single most-used operation in machine learning. It measures agreement.
Concept · 30m
Matrices Are Functions
A matrix is not a grid of numbers. It is a machine that turns vectors of one kind into vectors of another.
Concept · 30m
Decomposition: Finding Structure in a Matrix
Eigenvectors and the SVD tell you what a transformation actually does, independent of coordinates.
Concept · 30m

Module 2

The Calculus of Change

Derivatives, gradients, the chain rule, and the first sighting of the Jacobian.

Derivatives and Gradients
A derivative answers one question: if I nudge this input, how much does the output move?
Concept · 30m
The Chain Rule as Plumbing
Backpropagation is the chain rule applied to a graph, cached so nothing is recomputed.
Concept · 30m

Module 3

Uncertainty and Loss

Probability as an honest account of ignorance, and loss functions as its consequence.

Distributions and Expectation
A distribution is a statement about what you do not know. Expectation is its center of mass.
Concept · 25m
Where Loss Functions Come From
Squared error and cross-entropy are not arbitrary. Both fall out of maximizing likelihood.
Concept · 30m
← Foundations: Python as an Instrument Classical Machine Learning →