Rank Bounds for Design Matrices and Applications
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Transcript of Rank Bounds for Design Matrices and Applications
Rank Bounds for Design Matrices and Applications
Shubhangi SarafRutgers University
Based on joint works with Albert Ai, Zeev Dvir, Avi Wigderson
Sylvester-Gallai Theorem (1893)
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Suppose that every line through two points passes through a third
Sylvester Gallai Theorem
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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.
ℓ
Pm
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
Hesse ConfigurationKelly’s Theorem:For every pair of points in , 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
Quantitative SG
For every point there are at least points s.t there is a third point on the line
𝛿𝑛
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BDWY: dimension
This work: dimension
Stable Sylvester-Gallai Theorem
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Suppose that for every two points there is a third that is -collinear with them
Stable Sylvester Gallai Theorem
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Suppose that for every two points there is a third that is -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 MatricesAn 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
m
n
· t
· q
¸ k
(q,k,t)-design matrix
q = 3k = 5t = 2
An example
Thm: Let A be an m x n complex (q,k,t)-design matrix then:
Not true over fields of small characteristic!
Holds for any field of char=0 (or very large positive char)
Earlier Bounds (BDWY):
Main Theorem: Rank Bound
Thm: Let A be an m x n complex (q,k,t)-design matrix then:
Rank Bound: no dependence on q
Square Matrices
• 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?
Thm: Let A be an n x n complex -design matrix then:
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
At
A=
m
m
n
n n
n
Diagonal entries ¸ k
Off-diagonals · t
“diagonal-dominant matrix”
Idea (BDWY) : reduce to easy case using matrix-scaling:
r1
r2
.
.
.
.
.
.rm
c1 c2 … cn
Replace Aij with ri¢cj¢Aij
ri, cj positive reals
Same rank, support.
Has ‘balanced’ coefficients:
General Case: Matrix scaling
Matrix scaling theoremSinkhorn (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
am + b
n · 1
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
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Suppose that for every two points there is a third that is -collinear with them
Stable Sylvester Gallai Theorem
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Suppose that for every two points there is a third that is -collinear with them
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 thatfor 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
Encoding
Corrupted Encodingfraction
Correction
Decoding
Local Correction & Decoding
Message
Encoding
Corrupted Encodingfraction
Correction
Decoding
Local
Local
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!