Rui-Yang Zhang - Blog

[Derivation Scribbles] Fourier Transform and FFT

Derivation Scribbles

Describes the Fourier transform, mentions the sampling theorem that connects discrete and continuous time Fourier transform, and presents the fast Fourier transform that efficiently applies the discrete Fourier transform to a uniformly sampled sequence.

Approximate GP Posterior Sampling via the Matheron Rule

Gaussian Process

The Matheron rule of conditional multivariate normal distribution offers a new way to approximately sample from a posterior Gaussian process.

Monte Carlo, Antithetic, and Quasi Monte Carlo

Sampling

Basics of Monte Carlo, Antithetic, Quasi Monte Carlo, Randomised Quasi Monte Carlo.

Upscale a Brownian Motion Path

Gaussian Process

Sampling

For a Brownian motion path observed at fixed locations, upscale the resolution using conditioning using the fact that a Brownian motion is a Gaussian process.

[Reading Notes] Deep Learning is Not So Mysterious or Different

Deep Learning

Reading Notes

Notes on reading Andrew Gordon Wilson’s ICML 2025 paper Deep Learning is Not So Mysterious or Different.

Toy Bayesian Data Assimilation

Data Assimilation

Bayesian data assimilation for a toy problem.

Identifiability Issues of Gaussian Processes

Gaussian Process

Describe the consistency and identifiability of Matérn GPs.

Spatial-Temporal GP (3)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. SDE Approach to Temporal GP Regression.

Expected Information Gain with Gaussian Process Surrogate Models

Active Learning

Gaussian Process

Computations and derivations of the expected information gain utility function of active learning when the surrogate model is a conjugate Gaussian process.

Summary of a Bivariate Gaussian Covariance Matrix

Active Learning

Gaussian Process

Various ways one could summarise a bivariate Gaussian’s covariance matrix.

[Derivation Scribbles] 3D-Var and 4D-Var

Data Assimilation

Derivation Scribbles

Derivations for 3D-Var and 4D-Var.

[Derivation Scribbles] Kalman Filter and Ensemble Kalman Filter

Data Assimilation

Derivation Scribbles

Derivations for Kalman filters and Ensemble Kalman Filter.

Spatial-Temporal GP (2)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. Exploiting the Kronecker structure of temporal GP regression with 1d space.

Spatial-Temporal GP (1)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. Temporal GP regression with 1d space.

[Derivation Scribbles] Basic GP Regression Formula

Gaussian Process

Derivation Scribbles

Derivations for the Gaussian process predictive distribution. Single-output GP, observe with additive Gaussian noise.

Why Should We Care About Gradient Flows?

Gradient Flow

Blog post on gradient flows in Euclidean and Wasserstein spaces.

Rui-Yang Zhang, Christopher Nemeth