Jean Bourgain | Hacker News Reader

Jean Bourgain

terrytao.wordpress.com

104 pointsfurcyd19 comments

Tangential, but Tao is such a great teacher of mathematics and person. A rare combination of skill and modesty. Highly recommend his analysis and measure theory textbooks.

> [..] Tao is such a great teacher of mathematics and person.

And a good writer too. I could read his blog posts all day long. In this post I particularly like how he describes his feelings when reading through one of Jean's papers.

>> It’s hard to describe (especially in lay terms) the experience of reading through (and finally absorbing) the sections of this paper one by one; the best analogy I can come up with would be watching an expert video game player nimbly navigate his or her way through increasingly difficult levels of some video game, with the end of each level (or section) culminating in a fight with a huge “boss” that was eventually dispatched using an array of special weapons that the player happened to have at hand.

> A rare combination of skill and modesty.

That's what I was thinking when reading the following paragraph. He is one of the greatest mathematicians of our times but what he writes is so relatable.

>> While I was a graduate student at Princeton, Jean worked at the Institute for Advanced Study which was just a mile away. But I never actually had the courage to set up an appointment with him (which, back then, would be more likely done in person or by phone rather than by email). I remember once actually walking to the Institute and standing outside his office door, wondering if I dared knock on it to introduce myself.

Fascinating video. I use Fourier analysis all the time. It's a foundational tool of controls engineering, which is my field. But I never knew it was invented to describe thermal dynamics problems.

Depending on where you draw the line (heh), Lagrange's vibrating string work predates the heat equation work by a few years.

I learned so much from Bourgain's work. While most of his work was in Fourier analysis, and later in number theory, he really was a very versatile analyst. Just two of his early papers, on embeddings of metric spaces, and on concentration for covariance matrices started whole fields later. They are very useful now in theoretical machine learning.

Interesting! Could you share examples of works in theoretical ML using these ?

Well, lets try. For the embeddings, you'll have to search the internet for details, and this may be not a "proper" ML. But the general idea is that if you have some nice euclidean algorithm, like k-means, and you want to cluster some data w.r.t a different metric, you could try putting the data in the euclidean space without distorting the distances too much and then apply the algo. there. For k-means in particular the nearest neighbor search in Euclidean space can be made efficient, I think. This didn't take off much because most of the results were unfortunately negative. Although there were some recent positive results, even with discussion here [1]. Bourgain had some early results on the embeddability of finite metric spaces into classical spaces.

The concentration of covariance matrices is fundamental for most "spectral" methods, in particular spectral clustering. See for instance [2]. People proved and reproved these concentration results a lot of times, but Bourgain's was the earliest with N\log N sample complexity for log concave measures, improving on a result of Lovazs et al. (!!!) More importantly, the method with which he proved this, controlling the concentration by decomposing into bounded and "rarely non-zero" parts is now all around in statistical learning theory. So each time people want to prove uniform law of large numbers and nothing standard works, people try to find a good decomposition.

[1] https://news.ycombinator.com/item?id=17752836 [2] Clustering with Spectral Norm and the k-means Algorithm, Amit Kumar and Ravindran Kannan (Section 6...)

Bourgain's paper on monotone expanders is the most difficult paper I have tried to read, since it has some implications on open questions in computational complexity. Spoiler alert: I have still not understood it.

Is this the paper you're referencing? https://arxiv.org/abs/1108.2555

Yes, that's the one.

> I began to realise that Jean had a certain collection of tools, heuristics, and principles that he regarded as “basic”, such as dyadic decomposition and the uncertainty principle, and by working “modulo” these tools (that is, by regarding any step consisting solely of application of these tools as trivial), one could proceed much more rapidly and efficiently.

Does anybody know what those "basic" tools and heuristics are? Has anybody created a list of them somewhere for others to understand the papers more easily?

The answer is in the blog comments:

>Dear Terry,

>Have you ever considered writing a blog post on “Bourgain’s toolbox” summarizing the kinds of techniques used continually (without explanation) in his papers?

>[...]

Tao's final conclusion, besides other hints like http://www.tricki.org/ is:

> I think that ultimately the best way to acquire these tools is to constantly be exposed to interesting new mathematics, whether it is through reading a challenging paper from an expert, or listening to a talk on a topic outside your own specialty, or by working (particularly with collaborators) on a problem outside of your own comfort zone. Bourgain is not the only author whose writings happen to be particularly worth investing the effort to properly read; I for instance found reading Szemeredi’s work or Perelman’s work to be a broadly similar experience, and I am sure that there are several other mathematicians with this property also.

Though:

>Eventually, though, and with the help of Eli Stein and Tom Wolff, I managed to decode the steps which had mystified me – and my impression of the paper reversed completely.

If he hasn't managed to do this on his own, how should anybody else acquire those tools just by reading challenging papers?

That is a beautiful memorial. Thank you Terry for writing it, and helping me appreciate Jean's life and work better.

Also off-topic: Why are 6 of the 8 'comments' on that page autocomments merely saying that it's a top story on HN? That seems a bit excessive.

edit: Ah, downvoted within seconds. This is a Displeasing Question apparently. Sorry for whyever that is. But I'm curious. ..and won't be bullied into deleting the question.

Upvoted you, I asked myself the same question. Looks like a bot that annoys bloggers by notifying them that their writing is trending on various aggregators.

The "pingbacks" are common on WordPress blog using WordPress comments. As you mentioned they simply appear when another website provides a backlink.

This is default behavior, but most disable it.

These comments link back to blogs that have links to "top stories on HN", linking directly to the stories.

Please don't break the site guidelines by going on about downvotes. As they say, it never does any good, and it makes boring reading.

https://news.ycombinator.com/newsguidelines.html

An additional consideration is that corrective upvotes usually come along after awhile, rendering the complaint not only boring but factually inaccurate.

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