March 29, 2025
Principal Component Analysis
We discuss some very basic ideas from empirical processes and use them to motivate PCA.
March 29, 2025
Rescaling Errors
We investigate the appearance of the scaling factor √n / σ in the central limit theorem.
March 20, 2025
Failure of the Central Limit Theorem
We discuss why a naive application of a CLT approximation fails, motivating us to develop Hoeffding's inequality.
March 19, 2025
Monte Carlo and the Curse of Dimensionality
We use Monte Carlo integration as a backdrop to explore how introducing probablistic ideas help alleviate the curse of dimensionality.
March 16, 2025
The Russian Tail Bounds
We record a simple proof of Markov's inequality via indicator functions, then discuss how how to tighten it using various transformations.
March 14, 2025
The Dual Problem to SVM
We derive the dual problem to soft margin SVM.
March 5, 2025
Perturbations to Flow Networks
We discuss how to find a min cut with the minimum number of edges via a perturbative argument on flow networks.
February 23, 2025
Derivation of OLS
We formulate and solve for OLS as the solution to a minimization problem.
February 22, 2025
Bias-Variance, Randomness, and Gauss-Markov
We use the bias-variance decomposition in machine learning as a backdrop to explore some probability-theoretic ways of thinking. We also make a comparison between the bias in Gauss-Markov and in bias-variance.
February 10, 2025
Gauss Equation
We take two traces of the Gauss equation to get a scalar relationship between intrinsic and extrinsic curvature quantities of an embedding.
December 11, 2024
A Simple Proof of No Cloning
We provide a simple proof of the no cloning theorem from quantum computing.
October 26, 2024
A Geometer's Derivation of the Graph Laplacian
The graph Laplacian is central in spectral graph theory, but its link to the standard Laplacian is often unclear. We show both represent the difference between a function and its local average.