Stanford University and Bala Rajaratnam...

Critical exponents of graphs

Apoorva Khare

Stanford University

Joint work with Dominique Guillot (U. Delaware)and Bala Rajaratnam (Stanford)

AMS Sectional Meeting, LoyolaOctober 3, 2015

Working example

Question. Suppose

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

Raise each entry to the αth power for some α > 0.

When is the resulting matrix positive semide�nite?

Working example

Question. Suppose

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

Raise each entry to the αth power for some α > 0.

When is the resulting matrix positive semide�nite?

Motivation from high-dimensional statistics

Graphical models: Connections between statistics andcombinatorics.

Let X1, . . . , Xp be a collection of random variables.

In a very large vector, it is rare that all the variables dependstrongly on each other.

Many variables are independent or conditionally independent.

Variables independent of Xi are of no use to predict Xi.

The covariance matrix Σ of the vector (X1, . . . , Xp) captureslinear relationships between variables:

Σ = (σjk)pj,k=1 = (Cov(Xj , Xk))

pj,k=1,

Important problem: Estimate Σ given data x1, . . . , xn ∈ Rp of(X1, . . . , Xp).

pj,k=1,

Covariance estimation

Classical estimator (sample covariance matrix):

n− 1

n∑j=1

(xj − x)(xj − x)T .

S is positive semide�nite.

S has rank at most n.

Dense matrix (no graphical structure).

In modern �large p, small n� problems, S is known to be a poor

estimator of Σ.

Modern approach: Convex optimization: obtain sparse estimateof Σ (e.g., penalized likelihood methods)

Works well for dimensions up to a few thousands.

Does not scale to modern problems with 100, 000+ variables.

n− 1

n∑j=1

(xj − x)(xj − x)T .

estimator of Σ.

n− 1

n∑j=1

(xj − x)(xj − x)T .

estimator of Σ.

n− 1

n∑j=1

(xj − x)(xj − x)T .

estimator of Σ.

n− 1

n∑j=1

(xj − x)(xj − x)T .

estimator of Σ.

Covariance estimation (cont.)

Alternate approach: Thresholding covariance matrices

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

Natural to threshold small entries (thinking the variables areindependent):

0.95 0.18 00.18 0.96 0.470 0.47 0.98

Can be signi�cant if p = 1, 000, 000 and only, say, ∼ 1% of theentries of the true Σ are nonzero.

Highly scalable. Analysis on the cone - no optimization.

Question: When does this procedure preserve positive(semi)de�niteness?

Critical for applications, since covariance matrices are positivesemide�nite.

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

0.95 0.18 00.18 0.96 0.470 0.47 0.98

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

0.95 0.18 00.18 0.96 0.470 0.47 0.98

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

0.95 0.18 00.18 0.96 0.470 0.47 0.98

Highly scalable.

Analysis on the cone - no optimization.

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

0.95 0.18 00.18 0.96 0.470 0.47 0.98

True Σ =

1 0.2 00.2 1 0.50 0.5 1

0.95 0.18 0.020.18 0.96 0.470.02 0.47 0.98

0.95 0.18 00.18 0.96 0.470 0.47 0.98

Entrywise functions preserving positivity

More generally: Given a function f : I → R, when is it true that

f [A] := (f(ajk)) ∈ Pn for all A ∈ Pn(I)?

Problem: What kind of functions have this property?

Entrywise functions preserving positivity on Pn for all n:Schoenberg (Duke, 1942), Rudin (Duke, 1959), and others.Analytic, with nonnegative Taylor coe�cients.

Preserving positivity for �xed n: Hard problem, open forn ≥ 3. Recent characterization for polynomials:[Belton-Guillot-Khare-Putinar, 2015].

Focus on distinguished families to get insights into generalcase. Well-studied family in theory and applications:power functions xα where α > 0.(Applications use functions such as hard- and soft-thresholding, and powers, to regularize covariance matrices.)

Question: Which power functions applied entrywise preservepositivity on Pn (for �xed n)?

Preserving positivity for �xed n: Hard problem, open forn ≥ 3.

Recent characterization for polynomials:[Belton-Guillot-Khare-Putinar, 2015].

Focus on distinguished families to get insights into generalcase. Well-studied family in theory and applications:power functions xα where α > 0.

(Applications use functions such as hard- and soft-thresholding, and powers, to regularize covariance matrices.)

Powers preserving positivity

Theorem (FitzGerald and Horn, J. Math. Anal. Appl. 1977)

Let n ≥ 2. Then:

1 f(x) = xα preserves positivity on Pn((0,∞)) if α ≥ n− 2.

2 If α < n− 2 is not an integer, there is a matrix

A = (ajk) ∈ Pn such that A◦α := (aαjk) 6∈ Pn.In other words, f(x) = xα preserves positivity on Pn((0,∞)) if and

only if α ∈ N ∪ [n− 2,∞).

Critical exponent: n− 2 = smallest α0 such that α ≥ α0

preserves positivity.

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

, all powers α ∈ N∪ [3,∞) work.

Can we do better?

Let n ≥ 2. Then:

A = (ajk) ∈ Pn such that A◦α := (aαjk) 6∈ Pn.

In other words, f(x) = xα preserves positivity on Pn((0,∞)) if and

only if α ∈ N ∪ [n− 2,∞).

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

Can we do better?

Let n ≥ 2. Then:

only if α ∈ N ∪ [n− 2,∞).

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

Can we do better?

Let n ≥ 2. Then:

only if α ∈ N ∪ [n− 2,∞).

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

Can we do better?

Matrices with structures of zeros

Re�ne the FitzGerald�Horn problem for matrices with zeros.

A graph G = (V,E) is a set of vertices V and edges E ⊂ V × V :

The pattern of zeros of a symmetric n× n matrix is naturallyencoded by a graph on V = {1, 2, . . . , n}:

Edge between j and k ⇐⇒ ajk 6= 0.

1 3 0 23 1 5 00 5 1 42 0 4 1

←→

1 3 0 23 1 5 00 5 1 42 0 4 1

←→

1 3 0 23 1 5 00 5 1 42 0 4 1

←→

The cone PGGiven a graph G = (V,E) with V = {1, . . . , n} we de�ne a subsetof Pn by

PG := {A ∈ Pn : ajk = 0 if (j, k) 6∈ E and j 6= k}.

Note: ajk can be zero if (j, k) ∈ E.Example:

∗ ∗ 0 ∗∗ ∗ ∗ 00 ∗ ∗ ∗∗ 0 ∗ ∗

De�ne the set of powers preserving positivity for G:

HG := {α ≥ 0 : A◦α ∈ PG for all A ∈ PG([0,∞))}CE(G) := smallest α0 s.t. xα preserves positivity on PG,∀α ≥ α0.

Problem: How does the structure of G relate to the set of powerspreserving positivity? (FitzGerald-Horn studied the case G = Kn.)

PG := {A ∈ Pn : ajk = 0 if (j, k) 6∈ E and j 6= k}.Note: ajk can be zero if (j, k) ∈ E.Example:

∗ ∗ 0 ∗∗ ∗ ∗ 00 ∗ ∗ ∗∗ 0 ∗ ∗

HG := {α ≥ 0 : A◦α ∈ PG for all A ∈ PG([0,∞))}CE(G) := smallest α0 s.t. xα preserves positivity on PG,∀α ≥ α0.

∗ ∗ 0 ∗∗ ∗ ∗ 00 ∗ ∗ ∗∗ 0 ∗ ∗

HG := {α ≥ 0 : A◦α ∈ PG for all A ∈ PG([0,∞))}CE(G) := smallest α0 s.t. xα preserves positivity on PG, ∀α ≥ α0.

∗ ∗ 0 ∗∗ ∗ ∗ 00 ∗ ∗ ∗∗ 0 ∗ ∗

HG := {α ≥ 0 : A◦α ∈ PG for all A ∈ PG([0,∞))}CE(G) := smallest α0 s.t. xα preserves positivity on PG, ∀α ≥ α0.

Problem: How does the structure of G relate to the set of powerspreserving positivity? (FitzGerald-Horn studied the case G = Kn.) 9

A �rst example: trees

De�nition: A tree is a connected graph containing no cycles.

Theorem (Guillot, Khare, Rajaratnam, TAMS-1, 2015)

Let T be a tree with at least 3 vertices. Then HT = [1,∞).

The proof uses induction on n, and Schur complements:

SM◦α − (SM )◦α ∈ Pn−1.

General graphs

CE(T ) = 1 for all trees T , and CE(Kn) = n− 2.

What is CE(G) in general? Some preliminary observations:

1 If G has n vertices then α ≥ n− 2 preserves positivity.

2 If G contains Km as an induced subgraph, then α < m− 2does not preserve positivity (α 6∈ N).

Consequence: m− 2 ≤ CE(G) ≤ n− 2.

Question: Is the critical exponent of G equal to the clique numberminus 2?

Answer: No. Counterexample: G = K(1)4 (K4 minus a chord).

Clearly, the maximal clique is K3. However,we can show that H

= {1} ∪ [2,∞).

General graphs

What is CE(G) in general?

Some preliminary observations:

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

General graphs

= {1} ∪ [2,∞).

Chordal graphs

Trees are graphs with no cycles of length n ≥ 3.

De�nition: A graph is chordal if it does not contain an inducedcycle of length n ≥ 4.

Chordal Not Chordal

Names: Triangulated, decomposable, rigid circuit graphs. . .

Examples: Trees, Complete graphs, Triangulation of anygraph, Apollonian graphs, Band graphs, Split graphs, etc.

Occur in many applications: positive de�nite completionproblems, maximum likelihood estimation in graphical models,Gaussian elimination, etc.

Chordal graphs

Chordal Not Chordal

Chordal graphs

Chordal Not Chordal

Chordal graphs

Chordal Not Chordal

Chordal graphs

Chordal Not Chordal

Chordal graphs

Chordal Not Chordal

Preserving positivity for chordal graphs

Theorem (Guillot, Khare, Rajaratnam, JCT-A 2015)

Let G be any chordal graph with at least 2 vertices and let r be the

largest integer such that either Kr or K(1)r is an induced subgraph

of G. Then

HG = N ∪ [r − 2,∞).

In particular, CE(G) = r − 2.

E.g., for band graphs with bandwidth d, CE(G) = min(d, n− 2).

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

, all powers ≥ 2 = d work.

Preserving positivity for chordal graphs

Let G be any chordal graph with at least 2 vertices and let r be the

largest integer such that either Kr or K(1)r is an induced subgraph

of G. Then

HG = N ∪ [r − 2,∞).

In particular, CE(G) = r − 2.

E.g., for band graphs with bandwidth d, CE(G) = min(d, n− 2).

So for A =

1 0.6 0.5 0 0

0.6 1 0.6 0.5 00.5 0.6 1 0.6 0.50 0.5 0.6 1 0.60 0 0.5 0.6 1

, all powers ≥ 2 = d work.

Preserving positivity for chordal graphs (cont.)

Some key ideas for the proof:

1 Matrix decompositions: If (A,S,B) is a decomposition of G,every M ∈ PG decomposes as M = M1 +M2 withM1 ∈ PA∪S and M2 ∈ PB∪S :MAA MAS 0

MTAS MSS MSB

0 MTSB MBB

MAA MAS 0

MTAS MT

ASM−1AAMAS 0

0 MSS −MTASM

−1AAMAS MSB

0 MTSB MBB

2 Loewner super-additive functions:

f [A+B]− (f [A] + f [B]) ∈ Pn ∀A,B ∈ Pn.

Loewner super-additive powers classi�ed in[Guillot, Khare, Rajaratnam], J. Math. Anal. Appl., 2015.

3 Induction and properties of chordal graphs (decomposition,ordering of cliques, etc.).

1 Matrix decompositions: If (A,S,B) is a decomposition of G,every M ∈ PG decomposes as M = M1 +M2 withM1 ∈ PA∪S and M2 ∈ PB∪S :

MAA MAS 0MTAS MSS MSB

0 MTSB MBB

MAA MAS 0

MTAS MT

ASM−1AAMAS 0

0 MSS −MTASM

−1AAMAS MSB

0 MTSB MBB

f [A+B]− (f [A] + f [B]) ∈ Pn ∀A,B ∈ Pn.

MTAS MSS MSB

0 MTSB MBB

MAA MAS 0

MTAS MT

ASM−1AAMAS 0

0 MSS −MTASM

−1AAMAS MSB

0 MTSB MBB

f [A+B]− (f [A] + f [B]) ∈ Pn ∀A,B ∈ Pn.

MTAS MSS MSB

0 MTSB MBB

MAA MAS 0

MTAS MT

ASM−1AAMAS 0

0 MSS −MTASM

−1AAMAS MSB

0 MTSB MBB

f [A+B]− (f [A] + f [B]) ∈ Pn ∀A,B ∈ Pn.

MTAS MSS MSB

0 MTSB MBB

MAA MAS 0

MTAS MT

ASM−1AAMAS 0

0 MSS −MTASM

−1AAMAS MSB

0 MTSB MBB

f [A+B]− (f [A] + f [B]) ∈ Pn ∀A,B ∈ Pn.

Non-chordal graphs

For all n ≥ 3, HCn = [1,∞).

Remark: 1 is the biggest r − 2 such that Kr or K(1)r ⊂ Cn.

Suppose G is a connected bipartite graph with at least 3 vertices.

Then HG = [1,∞).

Proof uses a completely di�erent approach based on the fact that,

ρ(A◦α) ≤ ρ(A)α for A ∈ Pn, α ≥ 1,

where ρ(M) = spectral radius of M .

Remark: 1 is the biggest r − 2 such that Kr or K(1)r ⊂ G.

Non-chordal graphs

For all n ≥ 3, HCn = [1,∞).

Then HG = [1,∞).

ρ(A◦α) ≤ ρ(A)α for A ∈ Pn, α ≥ 1,

Non-chordal graphs

For all n ≥ 3, HCn = [1,∞).

Then HG = [1,∞).

ρ(A◦α) ≤ ρ(A)α for A ∈ Pn, α ≥ 1,

Non-chordal graphs

For all n ≥ 3, HCn = [1,∞).

Then HG = [1,∞).

ρ(A◦α) ≤ ρ(A)α for A ∈ Pn, α ≥ 1,

Non-chordal graphs

For all n ≥ 3, HCn = [1,∞).

Then HG = [1,∞).

ρ(A◦α) ≤ ρ(A)α for A ∈ Pn, α ≥ 1,

Remark: 1 is the biggest r − 2 such that Kr or K(1)r ⊂ G. 15

Open problems

1 What is the critical exponent of a given graph?

2 For any graph G, is HG = N ∪ [r − 2,∞), where r is the

biggest integer such that Kr or K(1)r ⊂ G?

3 The critical exponent of a graph always appears to be aninteger. Can this be proved directly (without computing thecritical exponent explicitly)?

4 Variants for matrices with negative entries.Need to work with these, even for above questions.

5 Connections to other (purely combinatorial) graph invariants?

Open problems

Bibliography

[1] D. Guillot, A. Khare, B. Rajaratnam, Critical exponents of graphs, ac-cepted in J. Combin. Theory, Ser. A, 2015.

[2] D. Guillot, A. Khare, and B. Rajaratnam, Complete characterization of

Hadamard powers preserving Loewner positivity, monotonicity, and convex-

ity, J. Math. Anal. Appl. 425(1):489-507, 2015.

[3] D. Guillot, A. Khare, B. Rajaratnam, Preserving positivity for matrices

with sparsity constraints, Trans. Amer. Math. Soc., in press, 2015.

[4] D. Guillot, A. Khare, B. Rajaratnam, Preserving positivity for rank-

constrained matrices, Trans. Amer. Math. Soc., in press, 2015.

[5] A. Belton, D. Guillot, A. Khare, and M. Putinar, Matrix positivity pre-

servers in �xed dimension. I, Adv. Math., under review, 2015.

Papers available at:http://web.stanford.edu/∼khare/

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