Tag: mathematics
All the articles with the tag "mathematics".
-
The 100 prisoners problem
Posted on:100 prisoners must each find their own number among 100 randomly filled boxes, opening at most 50 each. Random guessing succeeds with probability one in a nonillion. A particular cycle-following strategy succeeds about 31% of the time. The reason is the cycle structure of a random permutation.
-
Marchenko-Pastur and the Wigner semicircle
Posted on:The eigenvalues of a large random matrix do not scatter around. They concentrate, as a histogram, on a deterministic shape. For sample covariance matrices the shape is Marchenko-Pastur; for symmetric matrices with i.i.d. entries it is the Wigner semicircle. Both shapes are computable, and they explain precisely why high-dimensional covariance estimation is biased.
-
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.
-
The Newton fractal
Posted on:Newton's method for finding roots of polynomials is a discrete dynamical system on the complex plane. Each starting point converges (almost always) to one of the roots. The basins of attraction have fractal boundaries, intricate to a degree the algebra of the polynomial gives no hint of.
-
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.