Tag: mathematics
All the articles with the tag "mathematics".
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Covering the sphere with ε-nets
Posted on:An ε-net replaces the sphere by finitely many directions, so a supremum over all unit vectors reduces to a finite union bound. Worked out on the operator norm of a random matrix.
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Sudakov minoration, or how big a maximum must be
Posted on:Sudakov minoration lower-bounds the expected maximum of many Gaussians: if no two are too alike, the maximum is at least of order ε√(log N). The engine behind many impossibility proofs.
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Bias-variance is a Pythagorean decomposition
Posted on:MSE = bias² + variance is the Pythagorean theorem in L²: a constant bias and a mean-zero residual are orthogonal, so their squared lengths add.
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Voronoi tessellations and Lloyd's algorithm
Posted on:Generators partition the plane into cells, each closer to its generator than to any other. Lloyd's algorithm moves each generator to its cell's centroid and converges to k-means.
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Pólya's recurrence theorem
Posted on:Simple random walk returns to the origin with probability one in one and two dimensions, but not in three. The transition reduces to the convergence of a single series.
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High-dimensional Gaussians live on a sphere
Posted on:The bell-curve picture says Gaussian samples live near the mean. In high dimensions almost all the mass lies in a thin shell at radius √d: density and mass are not the same thing.
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Central Limit Theorem - why sums become Gaussian
Posted on:Adding random variables convolves their densities, and convolution smooths. A geometric look at the central limit theorem, with a slider over the number of summands.
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Why hypercubes look spiky
Posted on:Counting the corners of an n-cube by Hamming weight gives a binomial distribution; plotted as a cross-section, it recovers the spiky cube of the textbooks.