# Rank Bounds for Design Matrices and Applications

```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)
v
v
v
v
Suppose that
every line
through two
points passes
through a
third
Sylvester Gallai Theorem
v
v
v
v
Suppose that
every line
through two
points passes
through a
third
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
β
dist(Q, m) &lt; dist(P, β)
• 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:
Hesse Configuration
For every pair of points in π π ,
the line through them contains
a third, then all points
contained in a complex plane
[Elkies, Pretorius, Swanpoel 2006]:
First elementary proof
This work:
New proof using basic
linear algebra
πΏπ
Quantitative SG
For every point there are at
least πΉπ points s.t there is a
third point on the line
1
BDWY: dimension ≤ π(πΏ 2 )
This work:
dimension ≤ πΆ
π
πΉ
vi
Stable Sylvester-Gallai Theorem
v
v
v
v
Suppose that
for every two
points there
is a third that
is π -collinear
with them
Stable Sylvester Gallai Theorem
v
v
v
v
Suppose that
for every two
points there
is a third that
is π -collinear
with them
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
An m x n matrix is a
(q,k,t)-design matrix if:
1. Each row has at most q
non-zeros
2. Each column has at
least k non-zeros
3. The supports of every
two columns intersect
in at most t rows
n
&middot;q
&cedil;k
m
&middot;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:
ππππ
ππππ ≥ π −
π
Holds for any field of char=0 (or very large positive char)
Not true over fields of small characteristic!
Earlier Bounds (BDWY): ππππ ≥ π −
ππ‘π 2
2π
Rank Bound: no dependence on q
Thm: Let A be an m x n complex
(q,k,t)-design matrix then:
πππ ππ
ππππ ≥ π −
ππ
Square Matrices
Thm: Let A be an n x n complex
≈ π, ≈ π, ≈ π -design matrix then:
ππππ ≥ π(π)
• 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 π£1 , π£2 , … , π£π ∈ π π
• For every collinear triple π£π , π£π , π£π , ∃ πΌπ , πΌπ , πΌπ so that πΌπ π£π + πΌπ π£π
+ πΌπ π£π = 0
• Construct π &times; π matrix V s.t π π‘β row is π£π
• Construct π &times; π matrix π¨ s.t for each collinear triple π£π , π£π , π£π
there is a row with πΌπ , πΌπ , πΌπ in positions π, π, π resp.
• π¨⋅π½=π
Rank Bounds to Incidence Theorems
• π΄⋅π =0
• 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
n
=n
At
A
Off-diagonals
&middot;t
m
Diagonal entries &cedil; k
“diagonal-dominant matrix”
General Case: Matrix scaling
Idea (BDWY) : reduce to easy case using matrixscaling:
c c …
c
1
Replace Aij with ri&cent;cj&cent;Aij
ri, cj positive reals
Same rank, support.
Has ‘balanced’ coefficients:
r1
r2
.
.
.
.
.
.
rm
2
n
Matrix scaling theorem
Sinkhorn (1964) / Rothblum and Schneider (1989)
Thm: Let A be a real m x n matrix with nonnegative entries. Suppose every zero minor of A
of size a x b satisfies
a
b
+ &middot; 1
m n
Then for every &sup2; there exists a scaling of A with
row sums 1 &plusmn; &sup2; and column sums (m/n) &plusmn; &sup2;
Can be applied also to squares of entries!
Scaling + design →
perturbed identity matrix
• Let A be an π &times; π scaled (π, π, π‘)design matrix.
π
(Column norms = , row norms = 1)
π
• Let π = π΄∗ π΄
•
πππ =
π
π
• πππ ?
• BDWY:
• This work:
Mij ≤ π π‘
π≠π
2
πππ
≤ π‘π 1 −
1
π
Bounding the rank of perturbed
identity matrices
M Hermitian matrix, πππ ≥ πΏ
ππππ π ≥
ππΏ
ππΏ2 +
2
π≠π
2
πππ
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
v
v
v
Suppose that
for every two
points there
is a third that
is π -collinear
with them
Stable Sylvester Gallai Theorem
v
v
v
v
Suppose that
for every two
points there
is a third that
is π -collinear
with them
Not true in general ..
∃π points in ≈ π dimensional space s.t for
every two points there exists a third point that
is
π-collinear with them
Bounded Distances
• Set of points is B-balanced if all distances are
between 1 and B
• triple π£π , π£π , π£π is π-collinear
∃ πΌπ , πΌπ , πΌπ so that
πΌπ π£π + πΌπ π£π + πΌπ π£π ≤ π
and
1
4B
≤ πΌ1 , πΌ2 , πΌ3 ≤ 1
Theorem
Let π£1 , π£2 , … , π£π be a set of B-balanced points
in πΆ π so that
for each π£π , π£π there is a point π£π such that
the triple is π- collinear.
Then
π·πππ′ (π£1 , π£2 , … , π£π ) ≤ π π΅6
Incidence theorems to design matrices
• Given B-balanced π£1 , π£2 , … , π£π ∈ πΆ π
• For every almost collinear triple π£π , π£π , π£π , ∃ πΌπ , πΌπ , πΌπ &gt;
|πΌπ π£π + πΌπ π£π + πΌπ π£π | ≤ π
1
4π΅
so that
• Construct π &times; π matrix V s.t π π‘β row is π£π
• Construct π &times; π matrix π¨ s.t for each almost collinear triple
π£π , π£π , π£π there is a row with πΌπ , πΌπ , πΌπ in positions π, π, π resp.
• π¨ ⋅ π½ = π¬ (Each row has small norm)
proof
• π¨⋅π½=π¬
• 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
Encoding
Correction
≤ πΉfraction
Corrupted Encoding
Local Correction &amp; Decoding
Message
Encoding
Local
Correction
≤ πΉfraction
Corrupted Encoding
Stable Codes over the Reals
• Linear Codes
πΈ: ππ → ππ
• Corruptions:
– arbitrarily corrupt πΏπ locations
– small β∞ 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!
```