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# Rank Bounds for Design Matrices and Applications - PowerPoint PPT Presentation

Rank Bounds for Design Matrices and Applications. Shubhangi Saraf Rutgers University Based on joint works with Albert Ai, Zeev Dvir , Avi Wigderson. Sylvester- Gallai Theorem (1893). Suppose that every line through two points passes through a third. v. v. v. v.

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### Rank Bounds for Design Matrices and Applications

ShubhangiSaraf

Rutgers University

Based on joint works with

Albert Ai, ZeevDvir, AviWigderson

Sylvester-Gallai Theorem (1893)

Suppose that every line through two points passes through a third

v

v

v

v

Sylvester Gallai Theorem

Suppose that every line through two points passes through a third

v

v

v

v

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.

P

m

Q

dist(Q, m) < dist(P, ℓ)

• 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

Stable Sylvester-Gallai Theorem

v

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

v

v

v

Stable Sylvester Gallai Theorem

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

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:

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!

Scaling + design perturbed identity matrix

• 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

Stable Sylvester-Gallai Theorem

v

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

v

v

v

Stable Sylvester Gallai Theorem

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