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Rank Bounds for Design Matrices and Applications
Shubhangi SarafRutgers University
Based on joint works with Albert Ai, Zeev Dvir, Avi Wigderson
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Sylvester-Gallai Theorem (1893)
v v
v
v
Suppose that every line through two points passes through a third
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Sylvester Gallai Theorem
v
v
vv
Suppose that every line through two points passes through a third
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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!
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• 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
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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
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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
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Quantitative SG
For every point there are at least points s.t there is a third point on the line
𝛿𝑛
vi
BDWY: dimension
This work: dimension
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Stable Sylvester-Gallai Theorem
v v
v
v
Suppose that for every two points there is a third that is -collinear with them
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Stable Sylvester Gallai Theorem
v
v
vv
Suppose that for every two points there is a third that is -collinear with them
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Other extensions
• High dimensional Sylvester-Gallai Theorem
• Colorful Sylvester-Gallai Theorem
• Average Sylvester-Gallai Theorem
• Generalization of Freiman’s Lemma
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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
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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
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(q,k,t)-design matrix
q = 3k = 5t = 2
An example
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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
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Thm: Let A be an m x n complex (q,k,t)-design matrix then:
Rank Bound: no dependence on q
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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:
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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
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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.
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Rank Bounds to Incidence Theorems
• Want: Upper bound on rank of V
• How?: Lower bound on rank of A
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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
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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”
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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
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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!
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Scaling + design perturbed identity matrix
• Let A be an scaled ()design matrix. (Column norms = , row norms = 1)
• Let
•
• BDWY:
• This work:
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Bounding the rank of perturbed identity matrices M Hermitian matrix,
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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
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Stable Sylvester-Gallai Theorem
v v
v
v
Suppose that for every two points there is a third that is -collinear with them
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Stable Sylvester Gallai Theorem
v
v
vv
Suppose that for every two points there is a third that is -collinear with them
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Not true in general ..
points in dimensional space s.t for every two points there exists a third point that is
-collinear with them
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Bounded Distances
• Set of points is B-balanced if all distances are between 1 and B
• triple is -collinear so that and
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Theorem
Let be a set of B-balanced points in so thatfor each there is a point such that the triple is
- collinear. Then
(
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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)
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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)
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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
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Correcting from Errors
Message
Encoding
Corrupted Encodingfraction
Correction
Decoding
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Local Correction & Decoding
Message
Encoding
Corrupted Encodingfraction
Correction
Decoding
Local
Local
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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
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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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Thanks!