Tag: mathematics
All the articles with the tag "mathematics".
-
Stein's paradox
Posted on:In three or more dimensions, the sample mean is dominated everywhere by a shrinkage estimator. The geometric reason is the Gaussian shell: noise pushes you outward, and pulling back is uniformly better. A precursor of ridge regression and most modern regularization.
-
Nearest neighbor breaks in high dimensions
Posted on:In high dimensions, all pairwise distances become essentially equal. Nearest and farthest neighbor are no longer meaningfully different. A short geometric tour of the curse of dimensionality.
-
High-dimensional Gaussians live on a sphere
Posted on:The bell-curve picture says Gaussian samples live near the mean. In high dimensions that picture is catastrophically wrong: almost all the mass lies in a thin spherical shell at radius √d. Density and mass are not the same thing.
-
Central Limit Theorem - why sums become Gaussian
Posted on:A geometric look at the central limit theorem. Adding random variables is convolving their densities. Convolution smooths. Watch a Bernoulli, a die roll, or a bimodal distribution become Gaussian as you slide the number of summands.
-
Why hypercubes look spiky
Posted on:Counting the corners of an n-dimensional cube by Hamming weight gives a binomial distribution. Plot it as a vertical cross-section and you recover the spiky cube shape that high-dimensional textbooks love to draw.