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) 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 ℓ 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

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?

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

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,

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)

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!

Rank Bounds for Design Matrices and Applications