Rank bounds for design matrices and applications
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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!


Rank bounds for design matrices and applications

  • 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


Rank bounds for design matrices and applications

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:


Rank bounds for design matrices and applications

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:


Bounding the rank of perturbed identity matrices

Bounding the rank of perturbed identity matrices

M Hermitian matrix,


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


Rank bounds for design matrices and applications

Thanks!


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