Practice

Numerical experiments

8 interactive simulations. Each one states the question it answers, runs in your browser from a seeded generator, and says what the figure is — and is not — evidence for. Several exist specifically to show where a theorem's bound is loose.

Determinism is deliberate: the same parameters and seed reproduce the same figure on any machine, so a number you read off here is a number you can quote. A simulation is evidence about the regime you sampled. The proofs are on the topic pages.

High-Dimensional Probability

Theory

Random Matrix Theory

Theory

Marchenko–Pastur law
Eigenvalues of a sample covariance matrix against the Marchenko–Pastur density for aspect ratio λ = p/n.
⏱ 30 minrandom matricesspectrumcovariance

SGD & Optimization Dynamics

Theory

Gradient descent, momentum and SGD
Trajectories and suboptimality curves for GD, Nesterov acceleration, heavy ball and SGD on a quadratic.
⏱ 30 mingradient descentsgdmomentum

Statistical Learning Theory

Theory

Bias–variance decomposition
Squared bias, variance and total risk of polynomial regression as model capacity grows.
⏱ 30 minbias-varianceriskregression

Classical Machine Learning

Theory

Kernel ridge regression
The fitted function of kernel ridge regression as bandwidth and regularisation vary.
⏱ 30 minkernelsrkhsridge

RAG Mathematics

Theory

Embedding geometry and retrieval
Cosine similarity between random unit vectors in R^d, and the separation available to a retriever.
⏱ 30 minragembeddingscosine similarity