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Everyone Should Learn Optimal Transport, Part 1

12 minute read

In my opinion, optimal transport (OT) is a seriously underrated topic. I think part of the reason is the way OT is often introduced: as an optimization problem or a metric on probability distributions. While these are interesting to study for their own sake, OT presents a tool much more powerful and arguably even fundamental to the study of probability. In a series of two posts, I’m going to present a message that is well understood by experts but often missed by the uninitiated:

An Unusally Clean Proof: Dyson Brownian Motion via Conditioning on Non-intersection

9 minute read

Dyson Brownian motion [Dy62] is best known to characterize the eigenvalues of special random matrices [Ta12]. Most interestingly, it is also equal in distribution to \(n\) independent Brownian motions conditioned to not intersect [Gr99]. In a topics course by Bálint Virág, I came across a proof of this result that is just too clean for this type of calculations. After picking up my jaw from the ground months later, I finally decided to write up this surprisingly elegant proof.

On Escape Time, Lyapunov Function, Poincaré Inequality, and the KLS Conjecture Beyond Convexity

15 minute read

Nobody has time to read an 80 page paper [LE20]. Therefore I doubt most readers realized the manifold Langevin algorithm paper actually contains a novel technique for establishing functional inequalities. And I really doubt anyone had time to interpret the intuitive consequences of such results on perturbed gradient descent, and definitely not extending the Kannan-Lovász-Simonovits (KLS) conjecture [LV18] - which brings me to write this blog post.

The Auffinger-Chen Representation

14 minute read

Equivalent representation results contribute not only a connection between different concepts, but also a new set of proof techniques. Indeed, stochastic analysis has offered a number of alternative proofs to many problems. Occasionally the proof can simplify drastically. In this post, we will discuss a particularly elegant application by Auffinger and Chen (2015), for an otherwise very difficult problem in spin glass.

Stone-Weierstrass and an Alternative Proof of Itô’s Lemma

15 minute read

In a similar sense to line integrals, stochastic calculus extends the classical tools to working with stochastic processes. One of the most elegant and useful result is the change of variable formula for stochastic integrals, commonly known as Itô’s Lemma (see end of this post for a discussion on Doeblin’s contribution). While this lemma is quite easy to use, the proof usually relies heavily on technical lemmas, hence difficult to develop intuition, especially for the first time reader.

Connected by Poincaré Inequality

11 minute read

While studying two seemingly irrelevant subjects, probability theory and partial differential equations (PDEs), I ran into a somewhat surprising overlap: the Poincaré inequality. On one hand, it is not out of the ordinary for analysis based subjects to share inequalities such as Cauchy-Schwarz and Hölder; on the other hand, the two forms of Poincaré inequality have quite different applications.

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Teaching experience 1

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

This is a description of a teaching experience. You can use markdown like any other post.