As dimensions increase, the volume of a unit ball approaches zero due to the formula V = (π^(d/2) / Γ(d/2 + 1))^(1/d), where d is the dimension
Image: Pcharito, CC BY-SA 4.0, via Wikimedia Commons
As dimensions increase, the volume of a unit ball approaches zero due to the formula V = (π^(d/2) / Γ(d/2 + 1))^(1/d), where d is the dimension
the Lp norm ball shape changes as p goes from 1 to 2 to infinity
As p increases from 1 to 2 to infinity, the Lp norm ball becomes more spherical
Surface tension
High surface tension in water
List of unsolved problems in mathematics
Random points in high dimensions are nearly equidistant due to the uniform distribution of volume in high-dimensional space
the determinant tells you about volume scaling under a linear transformation
The determinant of a matrix representing a linear transformation indicates the factor by which volumes are scaled
Norm (mathematics)
L∞ norm equals max absolute value
the L1 unit ball is a diamond shape and the L2 unit ball is a circle
L1 ball: Manhattan distance, L2 ball: Euclidean distance. Diamond shape for L1 reflects Manhattan geometry, circle for L2 reflects Euclidean geometry
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