Remembering Bill Thurston

August 22, 2012

Bill Thurston passed away yesterday at the far-too-young age of 65. My thoughts and prayers (for what they’re worth) are with his family.

I never read any of Dr. Thurston’s mathematical work in great detail (geometric topology never really being my thing), but from the popular surveys I read, he was of course brilliant. I can’t really do justice to his mathematical accomplishments, but I’ll quickly say for non-mathematical readers that he was probably best known for his geometrization conjecture, which helped pave the way for the proof last decade of the Poincaré conjecture.

What I did read were some of his writings on the philosophy of math; not the boring questions of how “real” mathematical entities are, but thoughts on how we do mathematics. I am incredibly grateful that I had the opportunity to let Dr. Thurston know how much those writings meant to me a couple of years ago. It’s nice to think that I contributed, even infinitesimally, to the net happiness in his life.

[ETA: Terry Tao has a much more mathematically complete obituary.]

Tonal whiplash

July 26, 2012

People who know me well know that I don’t have any particular objection to targeted advertising. Like most things, it can be used for good purposes or bad purposes, but I think on balance, it’s probably better than untargeted advertising. Either way, it’s more efficient for many advertisers, which is good, because that means they can invest more capital into developing and improving products.

I’m not sure whether YouTube targets the ads that sometimes play before videos, either based on the user or the video. But if so: YouTube, I can’t imagine that there are that many people who want to see something about “The Purina Cat Chow Real Stories Project” when they’re looking for a Nick Cave song.

Well, well, well. Look who’s come crawling back

July 22, 2012

I’m not sure how one returns to blogging after a two-year hiatus. Probably it should be with something more substantial than this post, which is about regularization in statistics and how I came to understand it. (This account comes from Vapnik’s Nature of Statistical Learning Theory, which I should have finished by now but I keep getting distracted by “classes” and “research” and also “television” and “movies.”)

One of the most basic questions I had when initially learning statistics was: Why do we need to regularize least-squares regression? The answer, I was told, was to prevent overfitting. But this wasn’t totally satisfactory; in particular, I never learned why overfitting is a phenomenon that should be expected to occur, or why ridge regression solves it.

The answer, it turns out, is right there on the Wikipedia page.  Speaking very generally: we have some data, say A, b, and a class of functions f from which we want to draw our model. The goal is to minimize the distance between f(A) and b — if we can measure this distance, we have a functional R(f) = d(f(A), b), which we seek to minimize.

Now we want this functional to have the property that if the data of the response variable b changes a tiny bit, then the f at which our functional is minimized only changes a tiny bit. (Because we expect the data we measure to have some level of noise.) But this is not the case in general — actually, it’s not even the case when our functional is simply R(f) = |Af - b|^2, i.e., when we’re doing least-squares linear regression. However, if we instead consider the related functional R^*(f) = |Af - b_\delta|^2 + c_\delta\Omega(f), then this does have the desired property — our problem is well-posed. And in general, we can regularize many implicit functionals in many paradigms in the same way.

I also learned — though not from Vapnik — that you can derive ridge regression by putting a normal prior on the parameters in your model and then doing standard Bayesian things. But this doesn’t really explain why overfitting is a problem in the way the above account does — at least I don’t think so.

Question: Is there a natural correspondence between well-posed problems and “good” prior distributions on the parameters? (Meta-question: Is there a way to make that question more precise?)

(The title of this post is a Simpsons reference. Some things never change.)

I’m in love. What’s that song?

March 29, 2010

I’ve been on spring break for the past week, so this is even older news than it would have been, but Alex Chilton died earlier this month. If you don’t know who Alex Chilton is, he started off as the singer of the Box Tops, who scored a late-’60s hit with “The Letter.” But in certain circles he’s better known as the lead singer and songwriter for the defining power pop band, Big Star. Big Star’s music was a formative influnce on me as a person, if not as a mathematician, so I’m breaking from my usual routine of crackpot mathematics and expository articles to pay tribute.

Most of what I could say has already been said better, so I’ll link to some other tributes shortly. People have talked about how Big Star were a major influence on all sorts of bands in the ’80s and ’90s up to the present, from R.E.M. to the Replacements to Elliott Smith. People have talked about the timeless nature of Big Star’s songs, about how “Thirteen” is maybe the ultimate expression of what it’s like to be that age. But one thing that’s perhaps been lost in the chatter is the fact that so much of Chilton’s work with Big Star is just plain wonderful music. Witness “Nighttime” and “Blue Moon,” back-to-back tracks from Big Star’s final album, Third (also known as Sister Lover) — to my ears, two of the most achingly beautful pop songs ever recorded.

The best tribute I’ve read — maybe the definitive one — is Paul Westerberg’s in the New York Times. (Westerberg wrote and performed, with the Replacements, the song that gave this post its title.) But the best possible tribute, in my mind, would be if this blog post turned one person on to the music of the late, great Alex Chilton. It might change your life. More likely it won’t. But either way, it’s great damned music, and I hope — and expect — that it’ll live on far after I’m gone.

The extremal utility belt: Cauchy-Schwarz

March 10, 2010

I found this little gem in an old post on Terry Tao’s blog. There’s not really enough content in it to merit an entire Extremal Toolbox post, but it’s too cool not to point out.

Theorem. Graphs of order n and girth at least 5 have o(n^2) edges.

Proof. Suppose that G has 1/2*c*n^2 edges. Define the function A: V^2 \rightarrow \{0, 1\} to be the “adjacency characteristic function.” Now, by Cauchy-Schwarz:

n^4 \Sigma A(x_1, y_1)A(x_1, y_2)A(x_2, y_1)A(x_2, y_2)

\geq (n \Sigma A(x, y_1) A(x, y_2))^2

\geq (\Sigma A(x, y))^4 = \frac{1}{16} c^4 n^8.

Clearly for c fixed, \frac{1}{16} c^4 n^8 is unbounded as n \rightarrow \infty. But the first expression is n^4 times the number of (possibly degenerate) 4-cycles in the graph; it’s easy to check that there are O(n^3) degenerate 4-cycles, so as n is unbounded our graph must contain a 4-cycle. QED

Now, it’s possible to get better bounds by cleverly doing “surgery” on the graph and just using pigeonhole. (See here for details.) But it’s tricky and far less beautiful than the argument with Cauchy-Schwarz, which really demonstrates one way in which C-S can be thought of as “strengthening” pigeonhole.

2^n \geq n: The graph theory proof?

February 15, 2010

Theorem. For every positive integer n, 2^n \geq n.

Proof. Consider a tree on n vertices T = (V, E) with root v_0. Assign to each edge \{v, w\} an element of the vector space GF(2)^V, obtained by setting 1s in the the coordinates corresponding to v and w and 0s elsewhere. I claim that these vectors are linearly independent; for suppose otherwise, and letthe vectors corresponding to S \subset E sum to 0. There is a natural “distance” function on E w/r/t v_0; let e_0 = \{s, t\} have maximal distance in S, and suppose WLOG that t is farther than s from v_0. Then the coordinate corresponding to t is nonzero for exactly one element of S, and the sum over all elements of S must therefore be nonzero. This is a contradiction. So in particular these |E| = n-1 elements are distinct and nonzero, which means (by Pigeonhole) that there are at most 2^n-1 of them.

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Is the Super Bowl a Z/2Z-graded bowl?

February 8, 2010

And if so, what’s just a regular ungraded bowl?

(Hat tip to Vladimir S. for the bad pun; new post coming soon, really, I promise.)

On bias

January 23, 2010

[From Homer the Smithers, Mr. Burns sends Smithers on a forced vacation and tasks him with finding a temporary replacement]

Smithers: I’ve got to find a replacement who won’t outshine me. Perhaps if I search the employee evaluations for the word ‘incompetent…’

[Computer beeps, screen displays "714 Matches Found']

Smithers: 714 names?! Heh. Better be more specific. ‘Lazy,’ ‘clumsy,’ ‘dimwitted,’ ‘monstrously ugly.’

[After a couple seconds, computer beeps, screen displays '714 Matches Found']

Smithers: Ah, nuts to this, I’ll just go get Homer Simpson.

Actually, I tried to find the statistical nomenclature for this kind of thing, but couldn’t. Anyone have any idea what this is? (I want to say selection bias, but that’s not quite it…)

New Extremal Toolbox post should be coming later this weekend.

Someday we’ll find it

January 21, 2010

Apparently it’s used to model some password-protected networks, but I’m still pretty sure that the rainbow connection number of a graph can only have been invented (or at least named) as a joke.

Full HAP tables for the second 1124 sequence

January 12, 2010

Since it’s not yet on Thomas’ blog, and I need to use the data, I thought I’d make it publicly available. All credit goes to Thomas Sauvaget for the sequence-to-HTML code, and to Polymath for the sequence itself. (Probably someone specific discovered it, but I don’t know who.) Tables below the fold.

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