Why the L1 unit ball is a diamond shape and the L2 unit ball is a circle

How does the choice of norm affect the shape of the unit ball in a given vector space, specifically comparing the properties of L1 and L∞ norms?

L1 norms create diamond-shaped unit balls, while L∞ norms yield cube-shaped unit balls

Why L1 distance is called Manhattan distance — grid-like paths

L1 distance mimics grid-like city blocks, hence "Manhattan" distance

Why L1 regularization produces sparse solutions — the diamond corners touch axes

L1 regularization promotes sparsity by penalizing non-zero coefficients, effectively driving some to zero

Write the formula for Mahalanobis distance

D^2 = (x - μ)^T Σ^(-1) (x - μ)

Why the curse of dimensionality makes nearest neighbor search unreliable

High-dimensional spaces increase distance ambiguity, reducing nearest neighbor search reliability

How does the curse of dimensionality affect the performance and accuracy of clustering algorithms in high-dimensional datasets?

High-dimensional data can lead to sparse clusters, reducing clustering accuracy due to increased distance between points