Rank Bounds for Design Matrices and Applications

Shubhangi SarafRutgers University

Based on joint works with Albert Ai, Zeev Dvir, Avi Wigderson

Sylvester-Gallai Theorem (1893)

v v

v

v

Suppose that every line through two points passes through a third

Sylvester Gallai Theorem

v

v

vv

Suppose that every line through two points passes through a third

Proof of Sylvester-Gallai:• By contradiction. If possible, for every pair of points, the line through

them contains a third.• Consider the point-line pair with the smallest distance.

ℓ

Pm

Q

dist(Q, m) < dist(P, ℓ)

Contradiction!

• Several extensions and variations studied– Complexes, other fields, colorful, quantitative, high-dimensional

• Several recent connections to complexity theory– Structure of arithmetic circuits– Locally Correctable Codes

• BDWY:– Connections of Incidence theorems to rank bounds for design matrices– Lower bounds on the rank of design matrices– Strong quantitative bounds for incidence theorems– 2-query LCCs over the Reals do not exist

• This work: builds upon their approach– Improved and optimal rank bounds– Improved and often optimal incidence results– Stable incidence thms

• stable LCCs over R do not exist

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Points in Complex space

Hesse ConfigurationKelly’s Theorem:For every pair of points in , the line through them contains a third, then all points contained in a complex plane

[Elkies, Pretorius, Swanpoel 2006]: First elementary proof

This work: New proof using basic linear algebra

Quantitative SG

For every point there are at least points s.t there is a third point on the line

𝛿𝑛

vi

BDWY: dimension

This work: dimension

Stable Sylvester-Gallai Theorem

v v

v

v

Suppose that for every two points there is a third that is -collinear with them

Stable Sylvester Gallai Theorem

v

v

vv

Suppose that for every two points there is a third that is -collinear with them

Other extensions

• High dimensional Sylvester-Gallai Theorem

• Colorful Sylvester-Gallai Theorem

• Average Sylvester-Gallai Theorem

• Generalization of Freiman’s Lemma

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Design MatricesAn m x n matrix is a (q,k,t)-design matrix if:

1. Each row has at most q non-zeros

2. Each column has at least k non-zeros

3. The supports of every two columns intersect in at most t rows

m

n

· t

· q

¸ k

(q,k,t)-design matrix

q = 3k = 5t = 2

An example

Thm: Let A be an m x n complex (q,k,t)-design matrix then:

Not true over fields of small characteristic!

Holds for any field of char=0 (or very large positive char)

Earlier Bounds (BDWY):

Main Theorem: Rank Bound

Thm: Let A be an m x n complex (q,k,t)-design matrix then:

Rank Bound: no dependence on q

Square Matrices

• Any matrix over the Reals/complex numbers with same zero-nonzero pattern as incidence matrix of the projective plane has high rank– Not true over small fields!

• Rigidity?

Thm: Let A be an n x n complex -design matrix then:

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Rank Bounds to Incidence Theorems

• Given

• For every collinear triple , so that

• Construct matrix V s.t row is

• Construct matrix s.t for each collinear triple there is a row with in positions resp.

Rank Bounds to Incidence Theorems

• Want: Upper bound on rank of V

• How?: Lower bound on rank of A

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Proof

Easy case: All entries are either zero or one

At

A=

m

m

n

n n

n

Diagonal entries ¸ k

Off-diagonals · t

“diagonal-dominant matrix”

Idea (BDWY) : reduce to easy case using matrix-scaling:

r1

r2

.

.

.

.

.

.rm

c1 c2 … cn

Replace Aij with ri¢cj¢Aij

ri, cj positive reals

Same rank, support.

Has ‘balanced’ coefficients:

General Case: Matrix scaling

Matrix scaling theoremSinkhorn (1964) / Rothblum and Schneider (1989)

Thm: Let A be a real m x n matrix with non-negative entries. Suppose every zero minor of A of size a x b satisfies

am + b

n · 1

Then for every ² there exists a scaling of A with row sums 1 ± ² and column sums (m/n) ± ²

Can be applied also to squares of entries!

Scaling + design perturbed identity matrix

• Let A be an scaled ()design matrix. (Column norms = , row norms = 1)

• Let

•

• BDWY:

• This work:

Bounding the rank of perturbed identity matrices M Hermitian matrix,

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Stable Sylvester-Gallai Theorem

v v

v

v

Suppose that for every two points there is a third that is -collinear with them

Stable Sylvester Gallai Theorem

v

v

vv

Suppose that for every two points there is a third that is -collinear with them

Not true in general ..

points in dimensional space s.t for every two points there exists a third point that is

-collinear with them

Bounded Distances

• Set of points is B-balanced if all distances are between 1 and B

• triple is -collinear so that and

Theorem

Let be a set of B-balanced points in so thatfor each there is a point such that the triple is

- collinear. Then

(

Incidence theorems to design matrices

• Given B-balanced

• For every almost collinear triple , so that

• Construct matrix V s.t row is

• Construct matrix s.t for each almost collinear triple there is a row with in positions resp.

• (Each row has small norm)

proof

• Want to show rows of V are close to low dim space

• Suffices to show columns are close to low dim space

• Columns are close to the span of singular vectors of A with small singular value

• Structure of A implies A has few small singular values (Hoffman-Wielandt Inequality)

The Plan • Extensions of the SG Theorem

• Improved rank bounds for design matrices

• From rank bounds to incidence theorems

• Proof of rank bound

• Stable Sylvester-Gallai Theorems– Applications to LCCs

Correcting from Errors

Message

Encoding

Corrupted Encodingfraction

Correction

Decoding

Local Correction & Decoding

Message

Encoding

Corrupted Encodingfraction

Correction

Decoding

Local

Local

Stable Codes over the Reals

• Linear Codes

• Corruptions: – arbitrarily corrupt locations– small perturbations on rest of the coordinates

• Recover message up to small perturbations

• Widely studied in the compressed sensing literature

Our Results

Constant query stable LCCs over the Reals do not exist. (Was not known for 2-query LCCs)

There are no constant query LCCs over the Reals with decoding using bounded coefficients

Thanks!