Tag Archives: random matrices

Summer school on Free Probability, Random Matrices, and Applications from June 6th to June 10th 2022 at the University of Wyoming

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Zhuang Niu and Ping Zhong are organizing a summer school on Free Probability, Random Matrices, and Applications from June 6th to June 10th 2022 at the University of Wyoming. This event is planned to be held in person, but all lectures will have hybrid components for remote participation. 

The summer school will bring together leading experts, young researchers, and students working in free probability, operator algebras, random matrices, and related fields with the aim of fostering new communications and collaborations. Four mini-courses will be given by leading researchers in the field. In addition, there will be some survey talks or research talks. The mini-courses are designed for graduate students and young researchers. However, everyone is very welcome to participate in the meeting.  

Mini-courses: 

  • Hari Bercovici (Indiana), Complex analysis in free probability. 
  • Benoit Collins (Kyoto), Around the operator norm convergence of random matrices in free probability.  
  • Ken Dykema (College Station), On spectral distributions and decompositions of operators in finite von Neumann algebras 
  • Alexandru Nica (Waterloo), Recent developments related to the use of cumulants in free probability 

The conference website and registration form can be found at https://sites.google.com/view/rmmc2022. Please indicate in the registration form if you would like to contribute a research talk.  

Financial supports are available, and priority will be given to graduate students, junior researchers, and other participants without travel grants. The deadline of registration for full consideration of a financial aid is April 30th, 2022.  

Please share this information with anyone who might be interested in this meeting. 

Queen’s Seminar on Free Probability and Random Matrices

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One of the weekly highlights during my times at Queens was the joint seminar with Jamie Mingo on free probability and random matrices. This seminar was of course going on after I left Queen’s, and I also installed different versions of that seminar in Saarbrucken, but at least for me it did not feel the same anymore. So I am happy that we decided now to join forces again and revive our joint seminar in online form. For now it is running on Thursdays at 10 am Eastern time, which is 4 pm German time. You can find the program on the seminar page. If you are interested, write to Jamie to be put on the mailing list.

Proof of Harer-Zagier now online

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This blog started actually a couple of years ago as a (not very successful) discussion forum for the recordings of my lectures on Free Probability, Non-Commutative Distributions, Random Matrices, Mathematical Aspects of Quantum Mechanics. During the last year I have still been producing quite a few videos, but those are on mathematics for engineers (and also in German), so not of much interest in this context here. But the Christmas break gave me some motivation and energy to do something more elaborate. So I came back to the random matrix class, where one lecture was, by some technical reasons, not recorded during the original class two years ago. I did a retake of this missing class, and so finally the playlist for the lecture series on random matrices is now also complete. The topic of this new recording is the proof of the theorem of Harer-Zagier. It gives more or less the original proof from the paper by Harer and Zagier, which I find still fun and amazing – consisting of a mixture of analysis, combinatorics, and generatingfunctionology, which I am most fond of. If you want to have a look see here, here, here, and here. For videos without audience (as is now common during corona times) I have become used to splitting the whole lecture into smaller units — which also has the advantage that I can clean the black board in between.

Maybe I will also find time and energy in a (probably far far away) future to do a reshooting of the whole free probability course. This is of course closest to my heart, but as this was my first attempt on recording lectures I needed some time to become aware of the importance of the audio quality – which is so bad in those videos that they did not make it to youtoube …

PhD position for a project on “Free Probability Aspects of Neural Networks” at Saarland University

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Update: the position has been filled!

I have funding from the German Science Foundation DFG for a PhD position on the interrelations between free probability, random matrices, and neural networks. Below are more details. See also here for the announcement as a pdf.

Project: Neural networks are, roughly speaking, functions of many parameters in a high-dimensional space and the training of such a network consists in finding the parameters such that the function does what it is supposed to do on the “training inputs”, but also generalizing this in a meaningful way to “real test inputs”. Random matrix and free probability theory are mathematical theories which deal with typical behaviours of functions (which have an underlying matrix structure) in high dimensions and in the limit of large matrix size. Thus it is not surprising that those theories should have something to offer for describing and dealing with neural networks. Accordingly, there have been approaches relying on random matrix and/or free probability theory to investigate questions around deep learning. This interaction between free probability and neural networks is hoped to be bi-directional in the long run. However, the project will not address practical purposes of deep learning; instead we want to take the deep learning challenges as new questions around random matrices and free probability and we aim to develop those theories further on a mathematical level.

Prerequisites: Applicants should have an equivalent of a Master’s degree and a background in at least one of the subjects

  • free probability
  • random matrices
  • neural networks

and an interest in learning the remaining ones and, in particular, in working on their interrelations.

Application: Inquiries and applications should be addressed to Roland Speicher. Your application, in German or in English, should arrive before June 20, 2021. It should contain your curriculum vitae and an abstract of your Master’s thesis. Arrange also for at least one recommendation letter to be sent directly to Roland Speicher, preferably by email. State in your application the name of those you asked for such a letter .

Contact:
Prof. Dr. Roland Speicher
Saarland University
Department of Mathematics
Postfach 15 11 50
66041 Saarbrücken
Germany
speicher@math.uni-sb.de
https://www.uni-saarland.de/lehrstuhl/speicher/

Correction on my lecture notes on random matrices (weak convergence versus convergence of moments)

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I just noticed that I have a stupid mistake in my random matrix lecture notes (and also in the recording of the corresponding lecture). I am replacing the notes with a new version which corrects this.

In Theorem 4.16, I was claiming that weak convergence is equivalent to convergence of moments, in a setting where all moments exist and the limit is determined by its moments. Of course, this is a too optimistic statement. What is true is the direction that convergence of moments implies weak convergence. That’s the important direction. The other direction would be more of a relevance for combinatorial aficionados like me, as it would allow me to claim that the combinatorial and the analytical approach in such a setting are equivalent. However, the other direction is clearly wrong without some additional assumptions; and thus there are nice-looking situations where one cannot prove weak convergence by dealing with moments.

Of course this is not a new insight. In the context of proving the convergence to the semicircle for Wigner matrices with non-zero mean for the entries we know that we cannot do this with moments (see for example, Remark 11 in Chapter 4 of my book with Jamie).

To get a kind of positive spin out of this annoying mistake, I started to think about what kind of convergence we actually want in our theorems in free probability. Usually our convergence is in distribution, i.e., we are looking on moments – which seems to be the natural thing to do in the multivariate case of several non-commuting operators. However, we can also project things down to the classical world of one variable by taking functions in our operators and ask for the convergence of all such functions. And then there might be a difference whether we ask for weak convergence or for convergence in distribution (i.e., convergence of all moments).

This might become kind of relevant in the context of rational functions. Sheng Yin showed in Non-commutative rational functions in strong convergent random variables that convergence in distribution goes over from polynomials to rational functions (in the case where we assume that the rational function in the limit is a bounded operator) if we assume strong convergence on the level of polynomials (i.e., also convergence of the operator norms). Without the assumption of strong convergence it is easy to see that there are examples (see page 12 of the paper of Sheng) where one has convergence in distribution for the polynomials, but not for the rational functions. However, though one does not have convergence of the moments of the rational function, it is still true in this example that one has weak convergence of the (selfadjoint) rational function. So maybe it could still be the case that, even without strong convergence assumptions, convergence in distribution for polynomials (or maybe weak convergence for polynomials) implies weak convergence for rational functions. At least at the moment we do not know a counter example to this.

The saga ends …

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I have now finished my class on random matrices. The last lecture motivated the notion of (asymptotic) freeness from the point of view of looking on independent GUE random matrices. So you might think that there should now be continuations on free probability and alike coming soon. But actually this part of the story was already written and recorded and if you don’t want to spoil the tension you should watch the series not in its historical but in its logical order:

  1. Random Matrices (videos, homepage of class)
  2. Free Probability Theory (videos, homepage of class)
  3. Non-commutative Distributions (and Operator-Valued free Probability Theory) (videos, homepage of class)

More information, in particular the underlying script (sometimes in a handwritten version, sometimes in a more polished texed version), can be found on the corresponding home page of the lecture series.

May freeness be with you …

Welcome to the Non-Commutative World!

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About two weeks ago I posted with Tobias Mai on the archive the preprint “A Note on the Free and Cyclic Differential Calculus”. Here is what we say in the abstract:

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu’s fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples – such as a discrete version of the Ito stochastic integral – can be treated.

At the moment I am not in the mood to say more specifically about this preprint (maybe Tobias or I will do so later), but I want to take the opportunity — in particular as the first anniversary of this blog is also coming closer — to put this in a bigger context and mumble a bit about the bigger picture and our dreams … so actually about what this blog should be all about.

Free probability theory has come a long way. Whereas born in the subject of operator algebras, the realization that is also has to say quite a bit about random matrices paved the way to its use in many (and, in particular, also applied) subjects. Hence there are now also papers in statistics, like this one, or in deep learning, like this one or this one, which use tools from free probability for their problems. The last words on how far the use of free probability goes in those subjects are surely not yet spoken but I am looking forward to see more on this.

This is of course all great and nice for our subject, but on the other hand there is also a bigger picture in the background, where I would hope for some more fundamental uses of free probability.

This goes roughly like this. There is the classical world, where we are dealing with numbers and functions and everything commutes; then there is our non-commutative world, where we are dealing with operators and limits of random matrices and where on the basic level nothing commutes. That’s where quite a bit of maximal non-commutative mathematics has been (and is still being) developed from various points of views:

  • free probability deals with a non-commutative notion of independence for non-commuting random variables;
  • there is a version of a non-commutative differential calculus which allows to talk about derivatives in non-commutative variables; my paper with Tobias mentioned above is in this context and tries to formalize and put all this a bit further;
  • free analysis (or free/non-commutative function theory) aims at a non-commutative version of classical complex analysis, i.e., a theory of analytic functions in non-commuting variables;
  • free quantum groups provide the right kind of symmetries for such non-commuting variables.

The nice point is that all those subjects have their own source of motivation but it turns out that there are often relations between them which are non-commutative analogues of classical results.

So, again this is all great and nice, BUT apart from the commutative and our maximal non-commutative world there is actually the, maybe most important, quantum world. This is of course also non-commutative, but only up to some point. There operators don’t commute in general, but commutativity is replaced by some other relations, like the canonical commutation relations, and there are actually still a lot of operators which commute (for example, measurements which are at space-like positions are usually modeled by commuting operators). Because of this commutativity, basic concepts of free probability do not have a direct application there.

Here is a bit more concretely what I mean with that. In free probability we have free analogues of such basic concepts as entropy or Fisher information. There are a lot of nice statements and uses of those concepts and via random matrices they can also be seen as arising as a kind of large N limit of the corresponding classical concepts. However, in the classical world those concepts have usually also a kind of operational meaning by being the answer to fundamental questions. For example, the classical Shannon entropy is the answer to the question how much information one can transmit over classical channels. Now there are quantum channels and one can ask how much information one can transmit over them; again there are answers in terms of an entropy, but this is unfortunately not free entropy, but von Neumann entropy, a more commutative non-commutative cousin of classical entropy. There are just too many tensor products showing up in the quantum world which prevent a direct use of basic free probability concepts. But still, I am dreaming of finding some day operational meanings of free entropy and similar quantities.

Anyhow, I hope to continue to explain in this blog more of the concrete results and problems which we have in free probability and related subjects; but I just wanted to point out that there are also some bigger dreams in the background.

Class on “Random Matrices”, Winter Term 2019/20

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Our winter term has just started, running from mid October 2019 to mid February 2020, with a two-week break around Christmas. This term I am giving an introduction to random matrices. Again, the lectures will be recorded and put online. The lectures can be found on our video platform; more info on the lectures are also on the website of the class.

The lectures will follow roughly the material from the same class of summer term 2018, for which there exist also texed lecture notes. There will be a few reorganizations and shifts in the material, so there might emerge also a new version of the lectures notes sometimes in the future …

Focus Program on Applications of Noncommutative Functions; Featuring a Celebration Banquet for Dan Voiculescu’s 70th Birthday

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Today started the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto. There will be two workshops: this week on the “Developments and Technical Aspects of Free Noncommutative Functions” and next week on “Applications to Random Matrices and Free Probability of Free Noncommutative Functions”. Both workshops look interesting to me; unfortunately I will miss most of the first one as I will fly only on Wednesday to Toronto.

I will give a series of three talks on the relation between free probability and random matrices. The first talk will be quite general and is also intended for a public academic audience. Its main purpose is to celebrate the 70th birthday of Dan Voiculescu by giving an idea of Dan’s achievements and of free probability theory. The talk and the banquet will be on the very day of Dan’s birthday.