BIZNIQUE / LINEAR ALGEBRA / 13
Distance in many dimensions
Distance in many dimensions
Watch a small per-feature difference accumulate across many coordinates.
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TRY IT YOURSELF
A quick check.
A quick check.
If 100 features each differ by 0.1, what is the Euclidean distance?
Check your reasoning
Check your reasoning
√(100×0.1²) = 1.
THE IDEA
Read the example.
Read the example.
When each coordinate differs by the same amount δ, Euclidean distance is |δ|√d. Many weak or irrelevant differences can therefore contribute substantially to a distance. Feature selection and representation choices matter alongside scaling.
‖x − y‖₂ = √Σⱼ(xⱼ − yⱼ)²
Use it well
Use it well
Distance concentration depends on the data distribution and metric. More dimensions do not automatically make every distance useless.
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