Maths 2.5Lesson outlineLesson 5 of 5 in Module 2
Shapes and Distance in d Dimensions
Write the shapes that appear as decision regions and constraints, and see why distance behaves differently as the dimension grows.
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In this lesson9 parts
- 01Nearest and farthest neighbors as the dimension grows
- 02Spheres and balls in any dimension
- 03Hypercubes and axis-parallel boxes
- 04The ball inside the cube
- 05Checkpoint: shapes in d dimensions
- 06Distance concentration in a simulation
- 0764-pixel digit images against uniform random points
- 08The simulation and the digits in NumPy and SciPy
- 09Recap, challenge, and next: matrices
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- hypersphere
- every point at distance
rfrom a centre, inddimensions |‖x − c‖₂ = r - ball
- every point within distance
rof a centre |‖x − c‖₂ ≤ r - hypercube
- every component inside the same interval | the unit cube
[0, 1]ᵈ - axis-parallel box
- one interval per feature; the region a tree's splits describe |
12of 14 on the ward - distance concentration (curse of dimensionality)
- in high dimensions, the nearest and farthest neighbours are almost equally far away | ratio
1.123atd = 1000
Quiz
Your first pick on each question is the one that counts. Getting one wrong here is how the lesson sticks.