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Rank Bounds for Design Matrices and Applications

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Rank Bounds for Design Matrices and Applications

ShubhangiSaraf

Rutgers University

Based on joint works with

Albert Ai, ZeevDvir, AviWigderson

Suppose that every line through two points passes through a third

v

v

v

v

Suppose that every line through two points passes through a third

v

v

v

v

- 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.

P

m

Q

ℓ

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

- 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

Kelly’s Theorem:

For every pair of points in , the line through them contains a third, then all points contained in a complex plane

Hesse Configuration

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

This work:

New proof using basic linear algebra

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

vi

BDWY: dimension

This work:

dimension

v

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

v

v

v

v

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

v

v

v

- High dimensional Sylvester-Gallai Theorem
- Colorful Sylvester-Gallai Theorem
- Average Sylvester-Gallai Theorem
- Generalization of Freiman’s Lemma

- 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

n

An m x n matrix is a

(q,k,t)-design matrix if:

Each row has at most q non-zeros

Each column has at least k non-zeros

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

· q

¸ k

m

· t

(q,k,t)-design matrix

q = 3

k = 5

t = 2

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

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

Not true over fields of small characteristic!

Earlier Bounds (BDWY):

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

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

- 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?

- 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

- 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.

- Want: Upper bound on rank of V
- How?: Lower bound on rank of A

- 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

m

n

n

Off-diagonals · t

At

A

=

n

n

m

Diagonal entries ¸ k

“diagonal-dominant matrix”

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

c1c2 … cn

r1

r2

.

.

.

.

.

.

rm

Replace Aij with ri¢cj¢Aij

ri, cjpositive reals

Same rank, support.

- Has ‘balanced’ coefficients:

Matrix scaling theorem

Sinkhorn (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

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!

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

- Let
- BDWY:
- This work:

M Hermitian matrix,

- 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

v

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

v

v

v

v

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

v

v

v

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

-collinear with them

- Set of points is B-balanced if all distances are between 1 and B
- triple is -collinear
so that

and

Let be a set of B-balanced points in so that

for each there is a point such that the triple is - collinear.

Then

(

- 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)

- 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)

- 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

Message

Decoding

Encoding

Correction

Corrupted Encoding

fraction

Message

Local

Decoding

Encoding

Local

Correction

Corrupted Encoding

fraction

- 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

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!