Three-Toed Sloth

Books to Read While the Algae Grow in Your Fur, September 2024

Attention conservation notice: I have no taste, and no qualifications to opine on world history, or even on random matrix theory. Also, most of my reading this month was done at odd hours and/or while chasing after a toddler, so I'm less reliable and more cranky than usual.

Marc Potters and Jean-Philippe Bouchaud, A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists, doi:10.1017/9781108768900

I learned of random matrix theory in graduate school; because of my weird path, it was from May's Stability and Complexity in Model Ecosystems, which I read in 1995--1996. (I never studied nuclear physics and so didn't encounter Wigner's ideas about random Hamiltonians.) In the ensuing nearly-thirty-years, I've been more or less aware that it exists as a subject, providing opaquely-named results about the distributions of eigenvectors of matrices randomly sampled from various distributions. It has, however, become clear to me that it's relevant to multiple projects I want to pursue, and since I don't have one student working on all of them, I decided to buckle down and learn some math. Fortunately, nowadays this means downloading a pile of textbooks; this is the first of my pile which I've finished.

The thing I feel most confident in saying about the book, given my confessed newbie-ness, is that Potters and Bouchaud are not kidding about their subtitle. This is very, very much physicists' math, which is to say the kind of thing mathematicians call "heuristic" when they're feeling magnanimous *. I am still OK with this, despite years of using and teaching probability theory at a rather different level of rigor/finickiness, but I can imagine heads exploding if those with the wrong background tried to learn from this book. (To be clear, I think more larval statisticians should learn to do physicists' math, because it is really good heuristically.)

To say just a little about the content, the main tool in here is the "Stieljtes transform", which for an $N\times N$ matrix $\mathbf{A}$ with eigenvalues $\lambda_1, \ldots \lambda_N$ is a complex-valued function of a complex argument $z$, \[ g^{\mathbf{A}}_N(z) = \frac{1}{N}\sum_{i=1}^{N}{\frac{1}{z-\lambda_i}} \] This can actually be seen as a moment-generating function, where the $k^{\mathrm{th}}$ "moment" is the normalized trace of $\mathbf{A^k}$, i.e., $N^{-1} \mathrm{tr}{\mathbf{A}^k}$. (Somewhat unusually for a moment generating function, the dummy variable is $1/z$, not $z$, and one takes the limit of $|z| \rightarrow \infty$ instead of $\rightarrow 0$.)

The hopes are that (i) $g_N$ will converge to a limiting function as $N\rightarrow\infty$, \[ g(z) = \int{\frac{\rho(d\lambda)}{z-\lambda}} \] and (ii) the limiting distribution $\rho$ of eigenvalues can be extracted from $g(z)$. The second hope is actually less problematic mathematically **. Hope (i), the existence of a limiting function, is just assumed here. At a very high level, Potters and Bouchaud's mode of approach is to derive an expression for $g_N(z)$ in terms of $g_{N-1}(z)$, and then invoke the assumption (i), to get a single self-consistent equation for the limiting $g(z)$. There are typically multiple solutions to these equations, but also usually only one that makes sense, so the others are ignored ***.

At this very high level, Potters and Bouchaud derive limiting distributions of eigenvalues, and in some cases eigenvectors, for a lot of distributions of matrices with random entries: symmetric matrices with IID Gaussian entries, Hermitian matrices with complex Gaussian entries, sample covariance matrices, etc. They also develop results for deterministic matrices perturbed by random noise, and a whole alternate set of derivations based on the replica trick from spin glass theory, which I do not feel up to explaining. These are then carefully applied to topics in estimating sample covariance matrices, especially in the high-dimensional limit where the number of variables grows with the number of observations. This in turn feeds in to a final chapter on designing optimal portfolios when covariances have to be estimated by mortals, rather than being revealed by the Oracle.

My main dis-satisfaction with the book is that I left it without any real feeling for why the eigenvalue density of symmetric Gaussian matrices with standard deviation $\sigma$ approaches $\rho(x) = \frac{\sqrt{4\sigma^2 - x^2}}{2\pi \sigma^2}$, but other ensembles have different limiting distributions. (E.g., why is the limiting distribution only supported on $[-2\sigma, 2\sigma]$, rather than having, say, unbounded support with sub-exponential tails?) That is, for all the physicists' tricks used to get solution, I feel a certain lack of "physical insight" into the forms of the solutions. Whether any further study will make me happier on this score, I couldn't say. In the meanwhile, I'm glad I read this, and I feel more prepared to tackle the more mathematically rigorous books in my stack, and even to make some headway on my projects. §

*: As an early example, a key step in deriving a key result (pp. 21--23) is to get the asymptotic expected value of such-and-such a random variable. Using a clever trick for computing the elements of an inverse matrix in terms of sub-matrices, they get a formula for the expected value of the reciprocal of that variable. They then say (eq. 2.33 on p. 22) that this is clearly the reciprocal of the desired limiting expected value, because after all fluctuations must be vanishing. ^

**: We consider $z$ approaching the real axis from below, say $z=x-i\eta$ for small $\eta$. Some algebraic manipulation then makes the imaginary part of $g(x-i\eta)$ look like the convolution of the eigenvalue density $\rho$ with a Cauchy kernel of bandwidth $\eta$. A deconvolution argument then gives $\lim_{\eta \downarrow 0}{\mathrm{Im}(gx-i\eta)} = \pi \rho(x)$. This can be approximated with a finite value of $N$ and $\eta$ (p. 26 discusses the numerical error). ^

***: There is an interesting question about physicists' math here, actually. Sometimes we pick and choose among options that, as sheer mathematics, seem equally good, we "discard unphysical solutions". But sometimes we insist that counter-intuitive or even bizarre possibilities which are licensed by the math have to be taken seriously, physically (not quite "shut up and calculate" in its original intention, but close). I suspect that knowing when to do one rather than the other is part of the art of being a good theoretical physicist... ^

Fernand Braudel, The Perspective of the World, volume 3 of Civilization and Capitalism, 15th--18th Century

This is the concluding volume of Braudel's trilogy, where he tries (as the English title indicates) to give a picture of how the world-as-a-whole worked during this period. It's definitely the volume I find least satisfying. Braudel organizes everything around a notion of "world economies" borrowed from Immanuel Wallerstein (an unfortunate choice of guide), postulating that these are always centered on a single dominating city, and spends a lot of his time tracking the shifts of what he says is the dominating city of the European world economy. But by his own definition of world economy, I don't see how there was more than one during his period, because all his other "world economies (East Asia, India, sub-Saharan Africa, the Americas, etc., etc.) were all tied in to the same economic system as western-and-central Europe. In fact, Braudel goes on at great length about these ties! (At most, Australia and Oceania might have been outside the world economy during this period.) This is also the volume where the, let us say, eccentricity of Braudel's economic thought began to press on me *. It was his discussions of cycles, "the conjuncture" and time-series decomposition which however truly irritated me. Or, rather, it made me want to sit him down and give him a lecture on the Yule-Slutsky effect, because I am quite certain he was smart enough to grasp it **. --- All these remarks are, of course, the height of presumption on my part. §

Previously.

*: After quoting a detailed passage from Ricardo about how both Portugal and England are better off if the former grows wine and the latter grows wheat and they exchange, Braudel spends many pages going over how Portuguese wine-growers came to rely on credit from English merchants. Stipulating that this is all true, and even stipulating that in some sense those English merchants dominated the Portuguese vintners, it does not refute Ricardo! (The cooperative socialist commonwealth will care very much about comparative advantage.) Or, again, Braudel repeatedly talks about how certain cities or countries were dis-advantaged by their high wages, without ever considering that some employers there must have felt those wages were worth paying. Indeed many employers there must have, or those would not have been the prevailing wages. --- In general, I sympathize with wanting to rescue older perspectives, here those of the mercantilists, from the condescension of posterity, but I think Braudel takes that too far, to the detriment of his understanding of his material. ^

**: To be fair, there are some hints in those passages that Braudel might have been happy to accept Slutsky's perspective on the effect. Namely: the appearance of low-frequency cycles is the natural consequence of high-frequency noise (what Braudel would call "events") whose effects just take time to work their way through the economic system. (This reminds me that I need to actually read Barnett's biography of Slutsky one of these years.) ^

Tamim Ansary, The Invention of Yesterday: A 50,000-Year History of Human Culture, Conflict, and Connection

It's not quite true to say that this is an attempt to write Marshall Hodgson's never-completed world history as a volume of pop history. This is not quite true because it is also, and equally, inspired by McNeill and McNeill's The Human Web. The result is extremely engaging, and while I didn't particularly learn from it, I daresay most of the prospective audience will not, in fact, have read as widely in Ansary's sources as I happen to have done. §

Errata: When describing Mesopotamian civilizations, Ansary repeatedly refers to Sumerian as a Semitic language, which is wrong. This is not particularly consequential, and I didn't notice any other errors of fact.

Disclaimer: My grandfather and Ansary's father were friends, so he's a family connection.

Books to Read While the Algae Grow in Your Fur; Writing for Antiquity; The Great Transformation; Mathematics; Enigmas of Chance; The Dismal Science