high-dimensional_gaussian_is_like_sphere

Provides Model Context Protocol (MCP) server endpoints that expose mathematical operations for sampling from and analyzing high-dimensional Gaussian distributions, with focus on demonstrating how Gaussian distributions behave like spheres in high-dimensional spaces. Implements direct computation of probability density, sampling algorithms, and geometric properties through standardized MCP tool bindings that Claude and other MCP clients can invoke.

Computes and demonstrates the mathematical property that high-dimensional Gaussian distributions concentrate their probability mass on spherical shells rather than filling the entire space. Calculates metrics such as the radius of maximum probability density, concentration bounds, and the ratio of probability mass at different radii to quantify this phenomenon. Uses analytical formulas and numerical integration to verify concentration behavior across varying dimensions.

Generates random samples from multivariate Gaussian distributions with user-specified mean vectors and covariance matrices. Implements efficient sampling via Cholesky decomposition of the covariance matrix, enabling generation of correlated high-dimensional samples. Supports both diagonal (independent dimensions) and full covariance structures, with validation of covariance matrix positive-definiteness before sampling.

Computes the probability density function (PDF) values for points in a high-dimensional Gaussian distribution. Implements numerically stable log-PDF computation to avoid underflow in high dimensions, using the formula: log p(x) = -0.5 * (log(det(Σ)) + (x-μ)ᵀΣ⁻¹(x-μ)) + constant. Supports both dense and sparse covariance matrices, with efficient matrix inversion via Cholesky or LU decomposition.

Analyzes how geometric properties of Gaussian distributions change with dimensionality, computing metrics such as expected distance from origin, volume of probability mass shells, and the ratio of surface area to volume in high-dimensional spaces. Provides both theoretical predictions (via chi-squared distribution properties) and empirical measurements from samples, enabling comparison of theory vs. practice across dimensions.