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Two Proofs of the Existance of Ramanujan Graphs Spectral Two proofs of the existance of Ramanujan graphs Spectral Algorithms Workshop, Banff Adam W. Marcus Princeton University [email protected] August 1, 2016 Two proofs of Ramanujan graphs A. W. Marcus/Princeton Acknowledgements: Joint work with: Dan Spielman Yale University Nikhil Srivastava University of California, Berkeley 2/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton Outline Motivation and the Fundamental Lemma Exploiting Separation: Interlacing Families Mixed Characteristic Polynomials Ramanujan Graphs Proof 1 Proof 2 Further directions Motivation and the Fundamental Lemma 3/51 Typically, we can figure out how the eigenvalues change when adding a fixed smaller matrix, but many times one might want to add edges to a graph randomly. In this talk, I will discuss the idea of adding randomly chosen matrices to other matrices and then show how the idea can be used to show existance of Ramanujan graphs in two ways. Two proofs of Ramanujan graphs A. W. Marcus/Princeton Motivation In spectral graph theory, we are interested in eigenvalues of some matrix (Laplacian, adjacency, etc). Since graphs are a union of edges, there is a natural way to think of these matrices as a sum of smaller matrices. Motivation and the Fundamental Lemma 4/51 In this talk, I will discuss the idea of adding randomly chosen matrices to other matrices and then show how the idea can be used to show existance of Ramanujan graphs in two ways. Two proofs of Ramanujan graphs A. W. Marcus/Princeton Motivation In spectral graph theory, we are interested in eigenvalues of some matrix (Laplacian, adjacency, etc). Since graphs are a union of edges, there is a natural way to think of these matrices as a sum of smaller matrices. Typically, we can figure out how the eigenvalues change when adding a fixed smaller matrix, but many times one might want to add edges to a graph randomly. Motivation and the Fundamental Lemma 4/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton Motivation In spectral graph theory, we are interested in eigenvalues of some matrix (Laplacian, adjacency, etc). Since graphs are a union of edges, there is a natural way to think of these matrices as a sum of smaller matrices. Typically, we can figure out how the eigenvalues change when adding a fixed smaller matrix, but many times one might want to add edges to a graph randomly. In this talk, I will discuss the idea of adding randomly chosen matrices to other matrices and then show how the idea can be used to show existance of Ramanujan graphs in two ways. Motivation and the Fundamental Lemma 4/51 All such inequalities have two things in common: 1 They give results with high probability 2 The bounds depend on the dimension This will always be true | tight concentration (in this respect) depends on the dimension (consider n=d copies of basis vectors). Two proofs of Ramanujan graphs A. W. Marcus/Princeton Well-known techniques exist for bounding the eigenvalues of random sums of matrices. Theorem (Matrix Chernoff, for example) Let vb1;:::; vbn be independent random vectors with kvbi k ≤ 1 and P T i vbi vbi = Vb. Then h i −nD(θkλmax (EVb)) P λmax (Vb) ≤ θ ≥ 1 − d · e Similar inequalities by Rudelson (1999), Ahlswede{Winter (2002). Motivation and the Fundamental Lemma 5/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton Well-known techniques exist for bounding the eigenvalues of random sums of matrices. Theorem (Matrix Chernoff, for example) Let vb1;:::; vbn be independent random vectors with kvbi k ≤ 1 and P T i vbi vbi = Vb. Then h i −nD(θkλmax (EVb)) P λmax (Vb) ≤ θ ≥ 1 − d · e Similar inequalities by Rudelson (1999), Ahlswede{Winter (2002). All such inequalities have two things in common: 1 They give results with high probability 2 The bounds depend on the dimension This will always be true | tight concentration (in this respect) depends on the dimension (consider n=d copies of basis vectors). Motivation and the Fundamental Lemma 5/51 Furthemore, I want to keep the \probabilistic" nature: Theorem If θb is a random variable with finite support, then h i h i P θb ≥ Eθb > 0 and P θb ≤ Eθb > 0 In other words, I want to study one object (here Eθb) and then be able to assert the existence of something at least as good (in both directions). Two proofs of Ramanujan graphs A. W. Marcus/Princeton The goal I want to find a bound on the eigenvalues that is independent of dimension. Motivation and the Fundamental Lemma 6/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton The goal I want to find a bound on the eigenvalues that is independent of dimension. Furthemore, I want to keep the \probabilistic" nature: Theorem If θb is a random variable with finite support, then h i h i P θb ≥ Eθb > 0 and P θb ≤ Eθb > 0 In other words, I want to study one object (here Eθb) and then be able to assert the existence of something at least as good (in both directions). Motivation and the Fundamental Lemma 6/51 But this isn't true (pick just vb as (0; 1) or (1; 0) uniformly). So instead, we make an observation: Observation The eigenvalues of matrix are the roots of its characteristic polynomial. That is, if A is a d × d real, symmetric matrix with eigenvalues λ1; : : : ; λd , then d Y χA(x) := det [xI − A] = (x − λi ): i=1 Two proofs of Ramanujan graphs A. W. Marcus/Princeton In fairy-tale land P T So given a random frame Vb = i vbi vbi , I would like to say: h i P λmax (Vb) ≥ λmax (EVb) > 0 and h i P λmax (Vb) ≤ λmax (EVb) > 0 Motivation and the Fundamental Lemma 7/51 So instead, we make an observation: Observation The eigenvalues of matrix are the roots of its characteristic polynomial. That is, if A is a d × d real, symmetric matrix with eigenvalues λ1; : : : ; λd , then d Y χA(x) := det [xI − A] = (x − λi ): i=1 Two proofs of Ramanujan graphs A. W. Marcus/Princeton In fairy-tale land P T So given a random frame Vb = i vbi vbi , I would like to say: h i P λmax (Vb) ≥ λmax (EVb) > 0 and h i P λmax (Vb) ≤ λmax (EVb) > 0 But this isn't true (pick just vb as (0; 1) or (1; 0) uniformly). Motivation and the Fundamental Lemma 7/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton In fairy-tale land P T So given a random frame Vb = i vbi vbi , I would like to say: h i P λmax (Vb) ≥ λmax (EVb) > 0 and h i P λmax (Vb) ≤ λmax (EVb) > 0 But this isn't true (pick just vb as (0; 1) or (1; 0) uniformly). So instead, we make an observation: Observation The eigenvalues of matrix are the roots of its characteristic polynomial. That is, if A is a d × d real, symmetric matrix with eigenvalues λ1; : : : ; λd , then d Y χA(x) := det [xI − A] = (x − λi ): i=1 Motivation and the Fundamental Lemma 7/51 P T That is, given a random frame Vb = i vbi vbi , maybe we can say: maxroot χ ≥ maxroot χ > 0 P Vb E Vb and maxroot χ ≤ maxroot χ > 0 P Vb E Vb Certainly this is nonsense, but let's play along with a toy problem: Let A be a matrix and wb a random vector (taking values u or v uniformly). T What can we say about the eigenvalues of A + wbwb ? Two proofs of Ramanujan graphs A. W. Marcus/Princeton REAL fairy-tale land So now, maybe we can do what we want in terms of polynomials! Motivation and the Fundamental Lemma 8/51 Certainly this is nonsense, but let's play along with a toy problem: Let A be a matrix and wb a random vector (taking values u or v uniformly). T What can we say about the eigenvalues of A + wbwb ? Two proofs of Ramanujan graphs A. W. Marcus/Princeton REAL fairy-tale land So now, maybe we can do what we want in terms of polynomials! P T That is, given a random frame Vb = i vbi vbi , maybe we can say: maxroot χ ≥ maxroot χ > 0 P Vb E Vb and maxroot χ ≤ maxroot χ > 0 P Vb E Vb Motivation and the Fundamental Lemma 8/51 Two proofs of Ramanujan graphs A. W. Marcus/Princeton REAL fairy-tale land So now, maybe we can do what we want in terms of polynomials! P T That is, given a random frame Vb = i vbi vbi , maybe we can say: maxroot χ ≥ maxroot χ > 0 P Vb E Vb and maxroot χ ≤ maxroot χ > 0 P Vb E Vb Certainly this is nonsense, but let's play along with a toy problem: Let A be a matrix and wb a random vector (taking values u or v uniformly). T What can we say about the eigenvalues of A + wbwb ? Motivation and the Fundamental Lemma 8/51 Adding polynomials is a function of the coefficients and we are interested in the roots. In general, it is easy to get the coefficients from the roots but hard to get the roots from the coefficients. Example: p(x) = (x − 2)2 − 1 (has double root at1) and q(x) = (x + 2)2 − 1 (has double root at −1). p(x) + q(x) = x2 + 6 p does not have any real roots (roots are ± −6). Two proofs of Ramanujan graphs A. W. Marcus/Princeton Still playing along We would (naively) start by looking at the expected polynomial 1 1 p(x) = χ T (x) + χ T (x) 2 A+uu 2 A+vv Why is this naive? Motivation and the Fundamental Lemma 9/51 Example: p(x) = (x − 2)2 − 1 (has double root at1) and q(x) = (x + 2)2 − 1 (has double root at −1).
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