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The spectral norm of the infinite -regular tree is . We will see what this means and how to prove it.
When talking about the expansion of random graphs, abobut the construction of Ramanujan expanders, as well as about sparsifiers, community detection, and several other problems, the number comes up often, where is the degree of the graph, for reasons that tend to be related to properties of the infinite -regular tree.
A regular connected graph is Ramanujan if and only if its Ihara zeta function satisfies a Riemann hypothesis.
The purpose of this post is to explain all the words in the previous sentence, and to show the proof, except for the major step of proving a certain identity.
There are at least a couple of reasons why more computer scientists should know about this result. One is that it is nice to see a connection, even if just at a syntactic level, between analytic facts that imply that the primes are pseudorandom and analytic facts that imply that good expanders are pseudorandom (the connection is deeper in the case of the Ramanujan Cayley graphs constructed by Lubotzky, Phillips and Sarnak). The other is that the argument looks at eigenvalues of the adjacency matrix of a graph as roots of a characteristic polynomial, a view that is usually not very helpful in achieving quantitative result, with the important exception of the work of Marcus, Spielman and Srivastava on interlacing polynomials.
In preparation for the special program on spectral graph theory at the Simons Institute, which starts in a week, I have been reading on the topics of the theory that I don’t know much about: the spectrum of random graphs, properties of highly expanding graphs, spectral sparsification, and so on.
I have been writing some notes for myself, and here is something that bothers me: How do you call the second largest, in absolute value, eigenvalue of the adjacency matrix of a graph, without resorting to the sentence I just wrote? And how do you denote it?
I have noticed that the typical answer to the first question is “second eigenvalue,” but this is a problem when it creates confusion with the actual second largest eigenvalue of the adjacency matrix, which could be a very different quantity. The answer to the second question seems to be either a noncommittal “” or a rather problematic “.”
For my own use, I have started to used the notation , which can certainly use some improvement, but I am still at a loss concerning terminology.
Perhaps one should start from where this number is coming from, and it seems that its important property is that, if the graph is regular and has vertices, and has adjacency matrix A, this number is the spectral norm of (where is the matrix with ones everywhere), so that it measures the distance of from the “perfect -regular expander” in a norm that is useful to reason about cuts and also tractable to compute.
So, since it is the spectral norm of a modification of the adjacency matrix, how about calling it adjective spectral norm? I would vote for shifted spectral norm because I would think of subtracting as a sort of shift.
Please, do better in the comments!
This year, perhaps because of a mistake, the winners of the Field Medals and the Nevanlinna prize were made public before the opening ceremony of the ICM.
Congratulations to my former colleague Maryam Mirzakhani for being the first Fields Medals winner from Iran, a nation that can certainly use some good news, and a nation that has always done well in identifying and nurturing talent in mathematics and related fields. She is also the first woman to receive this award in 78 years.
And congratulations to Subhash Khot for a very well deserved Nevanlinna prize, and one can read about his work in his own words, in my words, and about the latest impact of his work in the the words of Barak and Steurer.
The Simons foundations has excellent articles up about their work and the work of Artur Avila, Manjul Bhargava, and Martin Hairer, the other Fields Medal recipient. An unusual thing about Manjul Bhargava’s work is that one can actually understand the statements of some of his results.
The New York Times has a fascinating article according to which the Fields Medal got its current status because of Steve Smale and cold war paranoia. I don’t know if they are overstating their case, but it is a great story.
A few weeks ago, the Proceedings of the National Academy of Science published an article on a study conducted by a group of Cornell researchers at Facebook. They picked about 600,000 users and then, for a week, a subset of them saw fewer “negative” posts (up to 90% were filtered) than they would otherwise see, a subset saw fewer “positive” posts (same), and a control group got a random subset.
After the week, the users in the “negative” group posted fewer, and more negative, posts, and those in the “positive” group posted more, and more positive, posts.
Posts were classified according to an algorithm called LIWC2007.
The study run contrary to a conventional wisdom that people find it depressing to see on Facebook good things happening to their friends.
The paper has caused considerable controversy for being a study with human subjects conducted without explicit consent. Every university, including of course Cornell, requires experiments involving people to be approved by a special committee, and participants must sign informed consent forms. Facebook maintains that the study is consistent with its terms of service. The highly respected privacy organization EPIC has filed a complaint with the FTC. (And they have been concerned with Facebook’s term of service for a long time.)
Here I would like to explore a different angle: almost everybody thinks that observational studies about human behavior can be done without informed consent. This means that if the Cornell scientists had run an analysis on old Facebook data, with no manipulation of the feed generation algorithm, there would not have been such a concern.
At the same time, the number of posts that are fit for the feed of a typical user vastly exceed what can fit in one screen, and so there are algorithms that pick a rather small subset of posts that are evaluated to be of higher relevance, according to some scoring function. Now suppose that, if N posts fit on the screen, the algorithm picks the 2N highest scoring posts, and then randomly picks half of them. This seems rather reasonable because the scoring function is going to be an approximation of relevance anyway.
The United States has roughly 130 million Facebook subscriber. Suppose that the typical user looks, in a week, at 200 posts, which seems reasonable (in our case, those would be a random subset of roughly 400 posts). According to the PNAS study, roughly 50% of the posts are positive and 25% are negative, so of the initial 400, roughly 200 are positive and 100 are negative. Let’s look at the 100,000 users for which the random sampling picked the fewest positive posts: we would be expecting roughly 3 standard deviations below the mean, so about 80 positive posts instead of the expected 100; the 100,000 users with the fewest negative posts would get about 35 instead of the expected 50.
This is much less variance than in the PNAS study, where they would have got, respectively, only 10 positive and only 5 negative, but it may have been enough to pick up a signal.
Apart from the calculations, which I probably got wrong anyway, what we have is that in the PNAS study they picked a subset of people and then they varied the distribution of posts, while in the second case you pick random posts for everybody and then you select the users with the most variance.
If you could arrange distributions so that the distributions of posts seen by each users are the same, would it really be correct to view one study as experimental and one as observational? If the PNAS study had filtered 20% instead of 90% of the positive/negative posts, would it have been ethical? Does it matter what is the intention when designing the randomized algorithm that selects posts? If Facebook were to introduce randomness in the scoring algorithm with the goal of later running observational studies would it be ethical? Would they need to let people opt out? I genuinely don’t know the answer to these questions, but I haven’t seen them discussed elsewhere.
I have always known that I don’t completely understand how fiat money works, however I have recently realized that I don’t understand it at all! Maybe some of my readers can clarify my confusion.
So let’s start a national economy from scratch, without international trade: so a group of people move to a deserted island, declare independence, the people have all kind of skills, they bring all kind of materials and machineries with them, the island has all kinds of natural resources and maybe a lottery system gives some people ownership of various plots of lands and mining rights.
Now some would-be entrepreneurs would like to begin hiring people with the right skills, buying or renting various stuff, and start some businesses. It seems that there is no loss of generality if I think of the entrepreneurs as just one person for the sake of what I want to think about. She is going to need to get a loan to start her business(es). Meanwhile, the new island government created a central bank, which issues theory dollars, or thollars, which are the currency of the island. The central bank “creates” thollars and lends them to banks, then the banks keep a fractional reserve and lend to the entrepreneur. Again, for the sake of what I want to say, there is no loss of generality if I identify the central bank and the other banks as just one entity.
So the central bank lends thollars to the entrepreneur, and she uses the money to start the business and being to pay people. At this point money starts circulating in the economy, and people will hire gardeners and babysitters, and lawyers, they will give to charities, they will hire computer science theory tutors for their kids, they will buy and sell houses to each others, and, crucially, will buy whatever goods and services the entrepreneur is selling. Now she is making a profit and she can pay back the loan to the central bank and invest in more … wait, she can never pay back the loan!
That’s because all the thollars in circulation are the ones that the central bank lent her, so there is no way that, as the money circulates, she can ever make more money that she owes!
Ok, so maybe people will also take loans to buy houses and stuff, so that’s more money that circulates, but is it really the case that, overall, it is impossible for everybody to be debt-free? That all the cash that any debt-free person has needs to be compensated by an equivalent amount of debt from other people? This is not how things seem to be in practice.
Now, clearly it is possible for everybody to have positive net worth, because, at the start, people have stuff and that stuff is worth money, but it seems strange that not everybody can be debt-free.
Maybe the problem is deflation? That if the entrepreneur borrowed money to create a business that creates new wealth (because it makes stuff that people find more valuable than the value of the raw material and the value of the work that went into it), but the amount of circulating money stays the same, then there is deflation, and her debt is spiraling out of control in deflation-adjusted terms, even if the interest rate is zero?
It seems that there are only two ways in which you can have everybody be debt-free and have a positive amount of cash: (i) the central bank starts buying stocks of private companies, (ii) the government runs a deficit, and the central bank buys government debt.
Is this correct? Normally, we think of people and companies being debt-free (or having more cash than debt) as ideal, and a government running no deficit as ideal, and usually central banks don’t buy stocks (they only buy bonds, which is formally equivalent to lending), so are these three conditions contradictory?
Where you least expect them:
a common [definiton] for “population” is a geographical cluster of people who mate more within the cluster than outside of it
The Simons Institute for the Theory of Computing at Berkeley has started operations a couple of months ago, it has been off to a great start. This semester there is a program on applications of real analysis to computer science, in which I am involved, and one on “big data,” whose workshops have been having such high attendance that they had to be organized offsite.
The institute itself is housed in a beautiful circular three-story building, half of whose second floor is an open space with sofas, whiteboards, tall ceilings, big windows, and exposed pipes on the ceiling, for an added loft-like look. If it had a ball pit, a foosball table, and free sushi it would look like the offices of a startup.
Next year, there will be programs on spectral graph theory, on applications of algebraic geometry, and on information theory.
Junior people (senior graduate students, postdocs, and junior assistant professors who have received their PhD no longer than six years ago) can participate in the program of the institute as “fellows.” Information on the fellowships is at simons.berkeley.edu/fellows2014; the deadline to apply is December 15.
This year is the centennial of Paul Erdős’s birth. Erdős lived most of his adult life as a traveling mathematician, “couchsurfing,” as we would say now, from place to place and from mathematical conference to mathematical conference. He wrote more than 1,500 papers with more than 500 different coauthors, introduced the probabilistic method and was the defining figure of the “Hungarian approach” to combinatorics. He died at age 83 while attending a mathematical conference.
Last year, we celebrated the centennial of Alan Turing’s birth. Turing and Erdős have become such iconic figures both for the impact of their work and for the fascinating facts of their lives. I would like to argue that the cultural archetype through which we interpret their lives is that of the saint. It is clearly that of the martyr saint in the case of Turing, while Erdős gave up material possessions and devoted his life to others, traveling everywhere and “preaching” to everybody, much in the mold of Saint Francis.
(A comparison of the Turing centennial celebration and Erdős’s, and a look at the frescoes of Medieval Catholic churches will show which kind of saint people are more interested in.)
The first step to become a saint of the Catholic church is to establish that the person exhibited “heroic virtues,” which is a great expression. This is an archetype that is not restricted to religion: you see it occurring in communist propaganda (Stakhanov, Lei Feng) and in every civil rights movement.
Saints were the “celebrities” of the Middle Ages, those whose life people liked to talk about. But contemporary celebrities come from a totally different archetype, that of the Greek God. Greek (and Roman) gods were petty and jealous, they cheated on their spouses, they were terrible parents, but there were good stories to be told about them. We don’t want (at least, I don’t) to live the life of a saint, but thinking about them is certainly inspirational and it makes us think that if someone can be so much better than us, maybe we can be a little better ourself in the practice of “virtues”, whatever this may mean to us. And we don’t admire gods, but, well, it’s probably fun to be one.
As usual, I have lost track of what I was trying to say, but I think that it speaks well of the academic community that we are more interested in saints than in gods, I will close by saying that my favorite saint of complexity theory is Avi Wigderson, I will keep to myself who my favorite god of complexity theory is, and I will leave it to the readers to contribute their picks.
If is a positive semidefinite matrix (a symmetric matrix all whose eigenvalues are nonnegative), then the power method is simply to pick a random vector , and compute . If is of the order of , then one has a constant probability that
where is the largest eigenvalue of . If we are interested in the Laplacian matrix of a -regular graph, where is the adjacency matrix of the graph, this gives a way to compute an approximation of the largest eigenvalue, and a vector of approximately maximum Rayleigh quotient, which is useful to approximate Max Cut, but not to apply spectral partitioning algorithms. For those, we need a vector that approximates the eigenvector of the second smallest eigenvalue.
Equivalently, we want to approximate the second largest eigenvalue of the adjacency matrix . The power method is easy to adjust to compute the second largest eigenvalue instead of the largest (if we know an eigenvector of the largest eigenvalue): after you pick the random vector, subtract the component of the vector that is parallel to the eigenvector of the largest eigenvalue. In the case of the adjacency matrix of a regular graph, subtract from every coordinate of the random vector the average of the coordinates.
The adjacency matrix is not positive semidefinite, but we can adjust it to be by adding a multiple of the identity matrix. For example we can work with . Then the power method reduces to the following procedure: pick randomly , then subtract from every entry of , then repeat the following process times: for every entry , assign , that is, replace the value that the vector assigns to vertex with a convex combination of the current value and the current value of the neighbors. (Note that one iteration can be executed in time .
The problem is that if we started from a graph whose Laplacian matrix has a second smallest eigenvalue , the matrix has second largest eigenvalue , and if the power method finds a vector of Rayleigh quotient at least for , then that vector has Rayleigh quotient about for , and unless we choose of the same order as we get nothing. This means that the number of iterations has to be about , which can be quite large.
The video below (taken from this week’s lecture) shows how slowly the power method progresses on a small cycle with 31 vertices. It goes faster on the hypercube, which has a much larger .
A better way to apply the power method to find small eigenvalues of the Laplacian is to apply the power method to the pseudoinverse of the Laplacian. If the Laplacian of a connected graph has eigenvalues , then the pseudoinverse has eigenvalues with the same eigenvectors, so approximately finding the largest eigenvalue of is the same problem as approximately finding the second smallest eigenvalue of .
Although we do not have fast algorithms to compute , what we need to run the power method is, for a given , to find the such that , that is, to solve the linear system in given and .
For this problem, Spielman and Teng gave an algorithm nearly linear in the number of nonzero of , and new algorithms have been developed more recently (and with some promise of being practical) by Koutis, Miller and Peng and by Kelner, Orecchia, Sidford and Zhu.
Coincidentally, just this week, Nisheeth Vishnoi has completed his monograph Lx=b on algorithms to solve such linear systems and their applications. It’s going to be great summer reading for those long days at the beach.