The blog post discusses the author's recent paper on the randomized block Krylov iteration (RBKI) algorithm for low-rank approximation, highlighting the efficiency of RBKI for any block size, particularly intermediate sizes. It de...
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Ethan N. Epperly specializes in computational methods for science, machine learning, and quantum information.
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Markov's inequality reveals the wiggliness of polynomials, particularly highlighting Chebyshev polynomials' superiority over power functions in approximation tasks.
The post presents an alternate proof of the variance bound for tensor-product trace estimators, highlighting insights from Meyer and Avron's research in matrix analysis.
The blog post discusses the Gaussian integration by parts formula, a useful tool for computing moments of standard Gaussian random variables. It explains how to compute the second and fourth moments effortlessly using this formula...
Ethan N. Epperly reflects on his five-year journey of blogging about applied mathematics, inspired by his mentor and mathematical communicators. He shares his initial doubts, the success of his blog, and the connections he has mad...
The text discusses the randomized Kaczmarz method for solving systems of linear equations, focusing on the selection probabilities and error bounds for the method. It compares the performance of standard and uniform sampling strat...
The text discusses the Polar Express algorithm, which is an optimal matrix sign method and its application to the Muon algorithm. It explores the computation of the singular value transformation of a matrix using polynomials and t...
The text discusses Poincaré inequalities and their relation to the mixing properties of reversible Markov chains. It explains the variance and local variance of a function, and how Poincaré inequalities provide bounds on the eigen...
The post presents proofs for the Schur product theorem, which states that the entrywise product of two positive semidefinite matrices is also positive semidefinite. It includes proofs using trace formula, Gram matrix, covariances,...
The text discusses the sketch-and-solve method for solving an overdetermined linear least-squares problem. It presents a bound for the residual of the sketch-and-solve solution and provides a proof for the same. The author aims to...
0My Favorite Proof of the Cauchy–Schwarz Inequality
2024-12-10 •
The text discusses the Cauchy–Schwarz inequality and presents a proof from Chapter 3 of The Schur complement and its applications. It explains the proof and its significance, and also mentions another example of the matrix techniq...
The author shares a paper published online in Communications on Pure and Applied Mathematics. The paper discusses low-rank approximation algorithms, projection approximation, Nyström approximation, and the Gram correspondence. The...
The author shares that their paper on Randomized Kaczmarz with tail averaging has been posted to arXiv. They provide a different perspective on the results of the paper and discuss the randomized Kaczmarz method for solving system...
The author answers a question about the sketch-and-solve least-squares solution for a Gaussian sketch, and computes the expected least-squares residual. The author proves that the Gaussian sketch-and-solve is an unbiased estimate ...
The text discusses the Lanczos method and stochastic trace estimation, focusing on random vectors on the sphere. It provides a mathematical analysis of the approach and compares it to standard Gaussian vectors. The post evaluates ...
0Rejection Sampling
2024-10-08 •
The author discusses the concept of rejection sampling, providing an accessible introduction to the technique and its applications in solving linear systems of equations and accelerating the randomly pivoted Cholesky algorithm. Re...
The text discusses the computation of eigenvalue decompositions of Hermitian, non-Hermitian, and normal matrices in MATLAB. It introduces the concept of normal matrices and presents an algorithm called RandDiag, developed by He an...
0Neat Randomized Algorithms: Randomized Cholesky QR
2024-06-25 •
The text discusses the randomized Cholesky QR algorithm, which is a fast alternative to the classical Householder QR algorithm for computing a QR factorization. It explains the procedure for computing a QR factorization using Chol...
0Note to Self: Gaussian Hypercontractivity
2024-02-04 •
The note describes the Gaussian hypercontractivity inequality and its application to obtain a weaker version of the Hanson–Wright inequality. It discusses the noise operator, random inputs, moments and tails, Gaussian contractivit...
0Don’t Use Gaussians in Stochastic Trace Estimation
2024-01-28 •
The post discusses the use of Gaussian and sphere distributions for trace estimation. It argues that the sphere distribution is preferable to the Gaussian distribution for trace estimation, and explains the reasons behind this arg...