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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

Rank Bounds for Design Matrices and Applications

ShubhangiSaraf

Rutgers University

Based on joint works with

Albert Ai, ZeevDvir, AviWigderson


Sylvester gallai theorem 1893
Sylvester-Gallai Theorem (1893)

Suppose that every line through two points passes through a third

v

v

v

v


Sylvester gallai theorem
Sylvester Gallai Theorem

Suppose that every line through two points passes through a third

v

v

v

v


Proof of sylvester gallai
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)

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
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
Points in Complex space

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


Quantitative sg
Quantitative SG

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
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 theorem1
Stable Sylvester Gallai Theorem

v

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

v

v

v


Other extensions
Other extensions

  • High dimensional Sylvester-Gallai Theorem

  • Colorful Sylvester-Gallai Theorem

  • Average Sylvester-Gallai Theorem

  • Generalization of Freiman’s Lemma


The plan1
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 matrices
Design Matrices

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


An example
An example

(q,k,t)-design matrix

q = 3

k = 5

t = 2


Main theorem rank bound
Main Theorem: Rank Bound

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):


Rank bound no dependence on q
Rank Bound: no dependence on q

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


Square matrices
Square Matrices

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?


The plan2
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
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 theorems1
Rank Bounds to Incidence Theorems

  • Want: Upper bound on rank of V

  • How?: Lower bound on rank of A


The plan3
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

m

n

n

Off-diagonals · t

At

A

=

n

n

m

Diagonal entries ¸ k

“diagonal-dominant matrix”


General case matrix scaling
General Case: Matrix scaling

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!


Scaling design perturbed identity matrix
Scaling + design perturbed identity matrix

  • Let A be an scaled ()design matrix.

    (Column norms = , row norms = 1)

  • Let

  • BDWY:

  • This work:



The plan4
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 theorem2
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 theorem3
Stable Sylvester Gallai Theorem

v

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

v

v

v


Not true in general
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
Bounded Distances

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

  • triple is -collinear

    so that

    and


Theorem
Theorem

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

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

Then

(


Incidence theorems to design matrices
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
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 plan5
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
Correcting from Errors

Message

Decoding

Encoding

Correction

Corrupted Encoding

fraction


Local correction decoding
Local Correction & Decoding

Message

Local

Decoding

Encoding

Local

Correction

Corrupted Encoding

fraction


Stable codes over the reals
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
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



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