2 Stochastic Perturbation Theory Allow us to sincerely flatter G. W.(Pete)Stewart by including his words on this subject verbatim from his survey paper on Stochastic Perturbation theory [1 which has been included in the course reader(Random Matrices-Il) Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, presumed small 1. Approximate F(A+E) 2. Bound‖F(A+E)-F(A)‖ in terms of‖E‖ Here‖l.‖ is some norm of interest The first problem is usually, but not always, solved by assuming that F is differentiable at A with derivative F(A).Then F(A+E)=F(A)+ FA(E)+O(E D, so that for E sufficiently small FA(E) is the required approximation. The problem then reduces to finding tractable expressions for FA(E)which in itself is often a nontrivial task. The second problem may be treated a variety of ways; the basic idea being that the size of the perturbation is used as a bound even thought it is likely to be an overestimate. Stochastic perturbation theory is different from this approach because it is, in some sense, intermediate of the other two. We let e be a stochastic matrix and compute expectations of quantities derived from the perturbation expansion. roject Idea Pete Stewart's paper was published in 1990. The work of Marcenko and Pastur was ediscovered around that time; hence you will notice that there are no references to their work. An interesting project would be to condense what Pete Stewart has to say and to determine if what we learned in class can strengthen any of the ideas presented in his paper. Thinking of applications of this would be a bont 3 Other ideas These some other ideas based on the survey papers on the website. If you want any clarification feel free to contact us What is the replica method? What is the connection between statistical physics, spin glasses and random matrices? What can random matrix theory tell us about real world graphs Suppose we were given the moments of the semi-circle(numerically ) Could you compute its density using techniques described by what is known as the "Classical Moment Problem"? Construct a bijection between the MANOVA matrix and McKays theorem(ask us for the exact relationship) Derive Jonsson's result in a more direct manner using the bipartite graph bijection(possibly) What is the connection between Jack Polynomials and Free probability. What aspects of random matrix theory can be useful for principal component analysis? References 1 G. W. Stewart. Stochastic perturbation theory. SIAM Rev, 32(4): 579-610, 19902 Stochastic Perturbation Theory Allow us to sincerely flatter G. W. (Pete) Stewart by including his words on this subject verbatim from his survey paper on Stochastic Perturbation theory [1] which has been included in the course reader (Random Matrices - II). Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, presumed small, 1. Approximate F(A + E), 2. Bound � F(A + E) − F(A) � in terms of � E �. Here � . � is some norm of interest. The first problem is usually, but not always, solved by assuming that F is differentiable at A with ′ derivative F (A). Then ′ F(A + E) = F(A) + FA(E) + o(� E �), (5) ′ so that for E sufficiently small FA(E) is the required approximation. The problem then reduces to finding ′ tractable expressions for FA(E) which in itself is often a nontrivial task. The second problem may be treated a variety of ways; the basic idea being that the size of the perturbation is used as a bound even thought it is likely to be an overestimate. Stochastic perturbation theory is different from this approach because it is, in some sense, intermediate of the other two. We let E be a stochastic matrix and compute expectations of quantities derived from the perturbation expansion. Project Idea: Pete Stewart’s paper was published in 1990. The work of Marˇcenko and Pastur was rediscovered around that time; hence you will notice that there are no references to their work. An interesting project would be to condense what Pete Stewart has to say and to determine if what we learned in class can strengthen any of the ideas presented in his paper. Thinking of applications of this would be a bonus. 3 Other ideas These some other ideas based on the survey papers on the website. If you want any clarification feel free to contact us. • What is the replica method? • What is the connection between statistical physics, spin glasses and random matrices? • What can random matrix theory tell us about real world graphs? • Suppose we were given the moments of the semi-circle (numerically). Could you compute its density using techniques described by what is known as the “Classical Moment Problem”? • Construct a bijection between the MANOVA matrix and McKay’s theorem (ask us for the exact relationship). • Derive Jonsson’s result in a more direct manner using the bipartite graph bijection (possibly). • What is the connection between Jack Polynomials and Free probability. • What aspects of random matrix theory can be useful for principal component analysis? References [1] G. W. Stewart. Stochastic perturbation theory. SIAM Rev., 32(4):579–610, 1990