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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.
Long in the making, the online course on expanders starts today.
In the first week of class: what are the topics of the course, and how to prove that the eigenvalues of the adjacency matrix of a regular graph tell you how many connected components there are in the graph.
Long in the planning, my online course on graph partitioning algorithms, expanders, and random walks, will start next month.
The course page is up for people to sign up. A friend of mine has compared my camera presence to Sheldon Cooper’s in “Fun with Flags,” which is sadly apt, but hopefully the material will speak for itself.
Meanwhile, I will be posting about some material that I have finally understood for the first time: the analysis of the Arora-Rao-Vazirani approximation algorithm, the Cheeger inequality in manifolds, and the use of the Selberg “3/16 theorem” to analyze expander constructions.
If you are not a fan of recorded performances, there will be a live show in Princeton at the end of June.
Last Fall, three Stanford classes were “offered online” for free: Andrew Ng’s machine learning class, Sebastian Thrun’s AI class, and Jennifer Widom’s data base class. There had been interest and experiments in online free education for a long time, with the MITx initiative being a particularly significant one, but there were a few innovations in last year’s Stanford classes, and they probably contributed to their runaway success and six-digit enrollment.
One difference was that they did not post videos of the in-class lectures. There was, in fact, no in-class lecture. Instead, they taped short videos, rehearsed and edited, with the content of a standard 90-minute class broken down in 4 ten-minutes video or so. This is about the difference between taping a play and making a movie. Then the videos came with some forms of “interactivity” (quizzes that had to be answered to continue), and they were released at the rate in which the class progressed, so that there was a community of students watching the videos at the same time and able to answer each other’s questions in forums. Finally, the videos were used in the Stanford offerings of the classes: the students were instructed to watch the videos by themselves, and during the lecture time they would solve problems, or have discussions or have guest lectures and so on. (In K-12 education, this is called the “flipped classroom” model, in which students take lectures at home and solve homeworks in class, instead of the traditional other way around.)
In the past few months, there has been a lot of thinking, and a lot of acting, about the success of this experiment. Sebastian Thrun started a company called udacity to offer online courses “branded” by the company itself, and Daphne Koller and Andrew Ng started a company called coursera to provide a platform for universities to put their courses online, and, meanwhile, Harvard and Berkeley joined MIT to create edX.
At a time when the growth of higher education costs in the United States appear unsustainable, particularly in second-tier universities, and when the demand for high-quality higher education is exploding in the developing world, these projects have attracted a lot of interest.
While the discussion has been focused on the “summer blockbusters” of higher education, and what they should be like, who is going to produce them, how to make money from them, and so on, I would like to start a discussion on the “art house” side of things.
In universities all over the world, tens of thousands of my colleagues, after they have “served” their departments teaching a large undergraduate classes and maybe a required graduate class, get to have fun teaching a research-oriented graduate class. Their hard-earned insights into problems about which they are the world’s leading expert, be it a particular organ of the fruit fly or a certain corner of the Langlands program, are distilled into a series of lectures featuring content that cannot be found anywhere else. All for the benefit of half a dozen or a dozen students.
If these research-oriented, hyper-specialized courses were available online, those courses might have an audience of 20 or 30 students, instead of 100,000+, but their aggregate effect on their research communities would be, I believe, very significant.
One could also imagine such courses being co-taught by people at different universities. For example, imagine James Lee and Assaf Naor co-teaching a course on metric embeddings and approximation algorithms: they would devise a lesson plan together, each would produce half of the videos, and then at both NYU and UW the students would watch the videos and meet in class for discussions and working on problems; meanwhile study groups would probably pop up in many theory groups, of students watching the videos and working on the problem sets together.
So someone should put a research-oriented graduate course online, and see what happens. This is all to say that I plan to teach my class on graph partitioning, expander graphs, and random walks online in Winter 2013. Wish me luck!
Despite no popular demand, I have collected all the notes from CS261, the course on algorithms for combinatorial optimization problems that I taught in the past term, in one pdf file, available here, and I have created a new page to collect links to my lecture notes.
For the occasion, I have also posted a single file containing the notes from my Spring 2009 class on the foundations of cryptography. As explained in the foreword to the crypto notes, they use a definition of CCA-security that is wrong, that is, a definition that is weaker than the standard one in a way that actually allows potentially dangerous attacks. The weaker definition, however, is much simpler to define and work with, and I think it is pedagogically justified. I believe that everything else in the notes is consistent with standard definitions. As far as I know, the notes are the only place in which one can find a concrete-security treatment of zero knowledge.
In which we show how to use expert advice, and introduce the powerful “multiplicative weight” algorithm.
We study the following online problem. We have “experts” that, at each time step , suggest a strategy about what to do at that time (for example, they might be advising on what technology to use, on what investments to make, they might make predictions on whether something is going to happen, thus requiring certain actions, and so on). Based on the quality of the advice that the experts offered in the past, we decide which advice to follow, or with what fraction of our investment to follow which strategy. Subsequently, we find out which loss or gain was associated to each strategy, and, in particular, what loss or gain we personally incurred with the strategy or mix of strategies that we picked, and we move to step .
We want to come up with an algorithm to use the expert advice such that, at the end, that is, at time , we are about as well off as if we had known in advance which expert was the one that gave the best advice, and we had always followed the strategy suggested by that expert at each step. Note that we make no probabilistic assumption, and our analysis will be a worst-case analysis over all possible sequences of events.
The “multiplicative update” algorithm provides a very good solution to this problem, and the analysis of this algorithm is a model for the several other applications of this algorithm, in rather different contexts.
In which we prove properties of expander graphs.
In which we introduce online algorithms and discuss the buy-vs-rent problem, the secretary problem, and caching.
In this lecture and the next we will look at various examples of algorithms that operate under partial information. The input to these algorithms is provided as a “stream,” and, at each point in time, the algorithms need to make certain decisions, based on the part of the input that they have seen so far, but without knowing the rest of the input. If we knew that the input was coming from a simple distribution, then we could “learn” the distribution based on an initial segment of the input, and then proceed based on a probabilistic prediction of what the rest of the input is going to be like. In our analysis, instead, we will mostly take a worst-case point of view in which, at any point in time, the unknown part of the input could be anything. Interestingly, however, algorithms that are motivated by “learn and predict” heuristics often work well also from the point of view of worst-case analysis.
In which we define and analyze the zig-zag graph product.
In which we define a multi-commodity flow problem, and we see that its dual is the relaxation of a useful graph partitioning problem. The relaxation can be rounded to yield an approximate graph partitioning algorithm.