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

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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)

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

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

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


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 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

(q,k,t)-design matrix

q = 3

k = 5

t = 2


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

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


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 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

m

n

n

Off-diagonals · t

At

A

=

n

n

m

Diagonal entries ¸ k

“diagonal-dominant matrix”


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

  • 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

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


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 that

for 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

Decoding

Encoding

Correction

Corrupted Encoding

fraction


Local Correction & Decoding

Message

Local

Decoding

Encoding

Local

Correction

Corrupted Encoding

fraction


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!


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