Kaizhao Liu

Latest Posts

  • Wasserstein Gradient Flows: Applications

    August 12, 2026

    In this post, we summarize several important applications of the theory of Wasserstein gradient flows in statistics and machine learning.

    Wasserstein Gradient Flows Optimal Transport Sampling Machine Learning

  • Wasserstein Geometry

    August 10, 2026

    We provide an introduction to Wasserstein geometry. Although it looks like a Riemannian geometry, $\cP_2(\RR^d)$ is not a smooth manifold.

  • Networks in Their Surrounding Contexts

    June 14, 2026

    How characteristics outside the nodes.

  • Strength of Weak Ties

    June 6, 2026

    The seminal paper of Mark Granovetter (1973).

  • Small-World Phenomenon

    May 30, 2026

    We can abstract a social network as a graph consist of nodes and edges. We describe some fundamental characteristic of such graphs.

  • Sequence Model

    March 5, 2026

    We illustrate basic inequality, noise complexity (e.g. Radamacher complexity, Gaussian complexity), and localization for the least square estimator in the simplest sequence model setting.

  • Sparse Linear Models

    July 19, 2025

    This is a note on sparse linear models in high dimension.

  • Dyson Series

    June 17, 2024

    Some quantum mechanical calculation.

  • Linear and Convex Programming

    June 5, 2024

    Consider the general constrained optimization problem $\eqref{P}$ shown below, where we have not assumed anything regarding the functions $f$, $h$, $l$ (like linearity or convexity).

  • Boolean Analysis

    June 2, 2024

    Denote

  • Point Estimation

    May 5, 2024

    In this article, I breifly review the theory of point estimation.

  • Classical Gradient Descent Analysis

    May 4, 2024

    In this article, I recap the classical gradient descent analysis.

  • Completely Positive Map

    March 18, 2024

    In this article, I describe $C^*$-algebra, complete positive map, and its relationship to quantum mechanics.

  • Diffusion Models

    March 7, 2024

    Let us first recall how to reverse a classical diffusion process.

  • Investigation in Plane Geometry

    February 10, 2018

    This article records some of my proofs in plane geometry.

  • New Geometry Problems

    October 4, 2017

    This article records some problems in plane geometry that I, my friend Yonggang Ren, and my middle school teacher Lingfeng Chen have discovered.

  • An IMO Problem Revisited

    September 30, 2017

    This article records some solutions of a problem in plane geometry that I discovered.