Tag: mathematics
All the articles with the tag "mathematics".
-
Sudakov minoration, or how big a maximum must be
Posted on:Averages shrink, but maxima grow. Sudakov's minoration inequality is the clean tool for the harder direction: a lower bound on the expected maximum of many Gaussians. As long as no two of them are too alike, that maximum is at least ε times the square root of log N. This is the engine behind a lot of impossibility proofs.
-
Voronoi tessellations and Lloyd's algorithm
Posted on:A set of generators in the plane partitions it into regions, each closer to one generator than to any other. Lloyd's algorithm iterates "move each generator to the centroid of its region" and converges to a centroidal Voronoi tessellation (k-means).
-
Pólya's recurrence theorem
Posted on:Simple random walk on the integer lattice returns to the origin with probability one in 1D and 2D. In 3D and higher, there is a positive probability of never returning. The transition is exact, dimension-dependent, and reduces to convergence of a single harmonic-style series.
-
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.