Adam W. Marcus - NCSU

Polynomial Techniques in Combinatorial Linear Algebra

Adam W. Marcus

Princeton [email protected]

Triangle LecturesMarch 30, 2019

Polynomial Techniques A. W. Marcus/PU

Frequent coauthors (MSS):

Dan SpielmanYale University

Nikhil SrivastavaUniversity of California, Berkeley

My involvement partially supported by:

NSF Postdoctoral Research Fellowship

NSF CAREER Grant DMS-1552520

2/33


Outline

1 Introduction

2 Method of Interlacing Polynomials

3 Applications

4 Recent Work

Introduction 3/33


What is this “Combinatorial Linear algebra?”

Combinatorics: What properties can collections of “things” have?

Linear algebra: “things” = vectors or matrices

Focus of this talk: eigenvalues of symmetric∗ matrices

Elements of

Random matrix theory

Geometric combinatorics

Real algebraic geometry∗∗

Free probability∗∗

Introduction 4/33


A fundamental question

Given (real) symmetric matrix

A = R

-2 · ·· 1 ·· · 7

RT

and a collection of symmetric matrices

Bk = Qk

1 · ·· 2 ·· · ·

QTk

(R,Qk orthogonal matrices).

Goal: minimize λmax(A + Bk)

Introduction 5/33


Example: Spectral Graph Theory

A is the Laplacian matrix of a graph, Bk Laplacian matrices of edges:

a

cd

be

1 · · · -1· 2 -1 · -1· -1 2 -1 ·· · -1 2 -1

-1 -1 · -1 3

a

cd

be⋃

+

1 -1 · · ·

-1 1 · · ·· · · · ·· · · · ·· · · · ·

· · · · ·· 1 · -1 ·· · · · ·· -1 · 1 ·· · · · ·

· · · · ·· · · · ·· · 1 · -1· · · · ·· · -1 · 1

Can we guarantee some edge is “good”?

Introduction 6/33


Probabilistic ansantzIf x1, . . . , xn are real numbers, then mink{xk} ≤ xk .

Compute an (easier?) expectation, get a bound.

We are interested in mink { λmax(A + Bk) } so look at...

1 λmax(A + B)?

2 λmax(A + B)?

For example:

EQ∈O(3)

QT

1 · ·· · ·· · ·

Q

=

1/3 · ·· 1/3 ·· · 1/3

Introduction 7/33


Probabilistic ansantz??

Let χA(x) = det(xI− A) be the characteristic polynomial of A, then

λmax(A) = maxroot {χA(x)} .

1 maxroot {χA+B(x)}?

2 maxroot{χA+B(x)

}?

3 maxroot{χA+B(x)

}(an average of polynomials!)?

Well defined?

Computable?

Relevant?

Introduction 8/33


Outline

1 Introduction


3 Applications

4 Recent Work

Method of Interlacing Polynomials 9/33


Well-defined?

In general, real-rootedness is not closed under addition.

What is maxroot {} of something with complex roots?



Example

For A in the Graph Laplacian example,

χA+B1 = x5 − 12x4 + 51x3 − 90x2 + 55x

χA+B2 = x5 − 12x4 + 50x3 − 82x2 + 40x

χA+B3 = x5 − 12x4 + 49x3 − 78x2 + 40x

so

1

3χA+B1 +

1

3χA+B2 +

1

3χA+B3 = x5 − 12x4 + 50x3 − 83

1

3x2 + 45x



Plotted



Lemma (Root Separation Lemma)

Let {pi} be degree d polynomials and let [s, t] ⊆ R an interval such that

pi (s) ≥ 0 for all i

pi (t) ≤ 0 for all i

Each pi has exactly one real root in [s, t].

Then every convex combination of {pi} has exactly one real root in [s, t]and it lies between the roots of some pa and pb.

Proof by picture:

st

Note: if there are d such intervals, then∑

i pi is real rooted.



Plotted, again

d separating intervals ⇐⇒ common interlacer



Corollary

If {pk} are degree d real-rooted polynomials with a common interlacer:

1 Every convex combination∑

k αkpk is real-rooted

2 There exist i , j such that kth-root{pi} ≤ kth-root{p} ≤ kth-root{pj}.

In particular, there exists an i such that maxroot {pi} ≤ maxroot {p}.

Theorem (Cauchy Interlacing)

If B1, . . . ,Bn satisfy rank(Bi − Bj) ≤ 2 for all i , j , then the polynomialspi = χA+Bi

have a common interlacer.

What about A + Bi + Cj?

Iterate?

Let Bi + Cj = Dij?



Think quantum (and iterate)

p11 p12 p21 p22

p1 p2

p∅

A rooted, connected tree of real-rooted polynomials where

1 Each parent is a convex combination of its children

2 Each set of children has a common interlacer

is called an interlacing family.



Putting it all together

Given symmetric matrices C1, . . . ,Ck

1 Form pi = χCi

2 (Try to) build an interlacing family on top of the pi

There exist i such that λmax(Ci ) ≤ maxroot {ptop} .

Theorem (MSS (2013) + Cohen(2015))

Let A be a symmetric matrix and let B1, . . . , Bk be independent “random”symmetric matrices. Then there exists a matrixY ∈ supp(A + B1 + · · ·+ Bk) such that

λmax(Y ) ≤ maxroot{E{χA+B1,...,Bk

(x)}}

(called the “mixed characteristic polynomial”).

Computing maxroot {}?Method of Interlacing Polynomials 17/33


Outline

1 Introduction


3 Applications

4 Recent Work

Applications 18/33


Picking vectors

Theorem (MSS (2013))

Let v1, . . . , vn ∈ Rd be a collection of vectors with∑viv

Ti = I

For all k ≤ d, there exists a set S ⊆ [n] with |S | = k such that

λmax

(∑i∈S

vivTi

)≤

(1 +

√k

d

)2(d

n

)

kd : fraction of dimensions you want to span

dn : average length of the vi

Applications 19/33


Partitioning vectors

Theorem (MSS (2013))


Ti = I and ‖vi‖2 ≤ ε.

For all r > 1, there exists a partition {S1, S2, . . . ,Sr} so that

λmax

∑i∈Sk

vivTi

≤ 1

r

(1 +√

rε)2

for all k.

Resolves (via a result of Weaver) question of Kadison and Singer.

Asymptotically tight (as d →∞), witnessed by Marchenko–Pastur.

Applications 20/33


Aside

Previous best used matrix concentration:

λmax

∑i∈Sj

vivTi

≤ f (r , δ)log d

log log d

Theorem (Matrix Chernoff/Bernstein/Hoeffding/etc)

If A1, . . . , An ∈ Rd×d are independent random self adjoint matrices then

P[λmax

(∑Ai > t

)]≤ d · e−f (t,A1,...,An).

Claim: any sufficiently generic concentration (“high” probability) boundmust depend on d .

Applications 21/33


The bad seed

Consider d copies of the diagonal matrices

1/d0

. . .

0

,

01/d

. . .

0

, . . . ,

00

. . .

1/d

Applications 22/33


Balls in bins

d identical balls

d different bins, chosen uniformly and independently

Xi = the number of balls in bin i

Well known fact:

E{

maxi

Xi

}= Θ

(log d

log log d

).

If we want asymptotic tightness:

P[

maxi

Xi = 1

]=

d!

dd≈ 1

ed.

Any successful method needs to find small probability events.

Applications 23/33


Fractional approximations of operators

Theorem (Akemann, Weaver (2013))


Ti ≤ I and ‖vi‖2 ≤ ε

and let 0 ≤ pi ≤ 1. There exist values η1, . . . , ηn ∈ {0, 1} such that∥∥∥∥∥n∑

i=1

ηivivTi −

n∑i=1

pivivTi

∥∥∥∥∥ = O(ε1/8)

Integrality gaps in semi-definite programming?

Applications 24/33


Biregular bipartite Ramanujan graphs

Theorem (Gribinski, M (2017))

For all integers n, k , d, there exists bipartite graph G = U ∪ V such that

|U| = n and |V | = kn

U is kd-regular, and V is d-regular

λ2(A) ≤√

d − 1 +√

kd − 1

where A is the adjacency matrix of G .

Furthermore, it is constructible in time poly(n) for fixed k, d.

Asymptotically optimal spectral expanders.

Applications 25/33


Partitioning into biregular Ramanujan graphs

Theorem (MSS (2013) + Boche–Wiese (2018))

For all integers n, d1, d2, there exists a coloring of the complete bipartitegraph K2nd1,2nd2 using 2n colors such that the adjacency matrix of everycolor class Ac satisfies

λ2(Ac) ≤√

d1 − 1 +√

d2 − 1.

Useful in building asymptotically optimal semantically secure codes.

All color classes need to act like hash functions.

Not constructive.

Applications 26/33


Outline

1 Introduction


3 Applications

4 Recent Work

Recent Work 27/33


Finite free probability

For d × d symmetric matrices A,B,

p(x) = det(xI−A) =∏i

(x − ai ) and q(x) = det(xI−B) =∏i

(x −bi )

Additive convolution:

[p � q](x) = EQ∈O(d)

{det(xI− A− QBQT )

}= Eπ∈S(d)

{∏i

(x − ai − bπ(i))

}

Linear in coefficients of p and q

Preserves real rootedness! (Borcea–Branden)

Gives unitarily invariant algebra of eigenvalues

Recent Work 28/33


Connection to Voiculescu Free Probability

Theorem (M, 2017)

Let A,B be symmetric matrices with (discrete) eigenvalue distributionsµA, µB . Set

p(x) = det(xI− A) and q(x) = det(xI− B)

and let µA+B be the free convolution of µ(A) and µ(B).

1 λ[pn � qn]D−→ µA+B

2 conv supp{λ[pn � qn]} ⊆ conv supp{µA+B}

Edge of spectrum of µA+B is an upper bound on maxroot {p � q}.

Explicit bounds (MSS, 2015).

Recent Work 29/33


Further work:

1 Multiplicative convolution

2 Additive Brownian motion

3 Central Limit Theorem

With A. Gribinski:

1 Rectangular matrices (singular values)

2 Entropy, Fisher information

With B. Mirabelli:

1 Multiplicative Brownian motion, CLT

2 Non-Hermitian square matrices

Recent Work 30/33


Hyperbolic manifolds

Conjecture (M, Sarnak)

There exists an infinite sequence of compact hyperbolic manifoldsM1,M2, . . . , for which the Laplacians L1, L2, . . . , satisfy

λ2(Li ) ≥1

4.

“Analogue” of Ramanujan graphs:

1 1/4 = spectral radius of the universal cover

2 λ1(L) = 0 for all compact manifolds.

Method of interlacing zeta functions?

Recent Work 31/33


Random Matrix Theory

Ansatz (M)

Let X (β) be a statement about random matrices for which

X (1) is true over RX (2) is true over CX (4) is true over H

Then limβ→∞ X (β) is true in finite free probability.

Known:

1 Additive, multiplicative Brownian motions (Tao)

2 β-corners process (with V. Gorin)

Jack polynomials?

Recent Work 32/33


Thank you for your attention!

Recent Work 33/33

Adam W. Marcus - NCSU

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