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 vv 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 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 Hesse Configuration [Elkies, Pretorius, Swanpoel 2006]: First elementary proof This work: New proof using basic linear algebra

Quantitative SG vivi

Stable Sylvester-Gallai Theorem v v v v

Stable Sylvester Gallai Theorem v v vv

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 m n · t · q ¸ k

(q,k,t)-design matrix q = 3 k = 5 t = 2 An example

Not true over fields of small characteristic! Holds for any field of char=0 (or very large positive char) Main Theorem: Rank Bound

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?

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

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 AtAt 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: r1r rmr1r rm c 1 c 2 … c n Replace A ij with r i ¢ c j ¢ A ij r i, c j positive reals Same rank, support. Has ‘balanced’ coefficients: General Case: Matrix scaling

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!

Bounding the rank of perturbed identity matrices

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

Stable Sylvester Gallai Theorem v v vv

Not true in general..

Bounded Distances

Theorem

Incidence theorems to design matrices

proof

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 Encoding Correction Decoding

Local Correction & Decoding Message Encoding Corrupted Encoding Correction Decoding Local

Stable Codes over the Reals

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!