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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) < dist(P, β) 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: 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 πΏπ Quantitative SG For every point there are at least πΉπ points s.t there is a third point on the line 1 BDWY: dimension ≤ π(πΏ 2 ) This work: dimension ≤ πΆ π πΉ 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 ·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: ππππ ππππ ≥ π − π Holds for any field of char=0 (or very large positive char) Not true over fields of small characteristic! Earlier Bounds (BDWY): ππππ ≥ π − ππ‘π 2 2π Rank Bound: no dependence on q Thm: Let A be an m x n complex (q,k,t)-design matrix then: πππ ππ ππππ ≥ π − ππ Square Matrices Thm: Let A be an n x n complex ≈ π, ≈ π, ≈ π -design matrix then: ππππ ≥ 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 , ∃ πΌπ , πΌπ , πΌπ so that πΌπ π£π + πΌπ π£π + πΌπ π£π = 0 • Construct π × π matrix V s.t π π‘β row is π£π • Construct π × π matrix π¨ s.t for each collinear triple π£π , π£π , π£π there is a row with πΌπ , πΌπ , πΌπ in positions π, π, π resp. • π¨⋅π½=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 ·t m Diagonal entries ¸ k “diagonal-dominant matrix” General Case: Matrix scaling Idea (BDWY) : reduce to easy case using matrixscaling: c c … c 1 Replace Aij with ri¢cj¢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 + · 1 m n 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 π × π scaled (π, π, π‘)design matrix. π (Column norms = , row norms = 1) π • Let π = π΄∗ π΄ • πππ = π π • πππ ? • BDWY: • This work: Mij ≤ π π‘ π≠π 2 πππ ≤ π‘π 1 − 1 π Bounding the rank of perturbed identity matrices M Hermitian matrix, πππ ≥ πΏ ππππ π ≥ ππΏ ππΏ2 + 2 π≠π 2 πππ 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 ∃ πΌπ , πΌπ , πΌπ so that πΌπ π£π + πΌπ π£π + πΌπ π£π ≤ π and 1 4B ≤ πΌ1 , πΌ2 , πΌ3 ≤ 1 Theorem Let π£1 , π£2 , … , π£π be a set of B-balanced points in πΆ π so that for each π£π , π£π there is a point π£π such that the triple is π- collinear. Then π·πππ′ (π£1 , π£2 , … , π£π ) ≤ π π΅6 Incidence theorems to design matrices • Given B-balanced π£1 , π£2 , … , π£π ∈ πΆ π • For every almost collinear triple π£π , π£π , π£π , ∃ πΌπ , πΌπ , πΌπ > |πΌπ π£π + πΌπ π£π + πΌπ π£π | ≤ π 1 4π΅ so that • Construct π × π matrix V s.t π π‘β row is π£π • Construct π × π matrix π¨ s.t for each almost collinear triple π£π , π£π , π£π there is a row with πΌπ , πΌπ , πΌπ in positions π, π, π resp. • π¨ ⋅ π½ = π¬ (Each row has small norm) proof • π¨⋅π½=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 ≤ πΉfraction Corrupted Encoding Local Correction & Decoding Message Encoding Local Correction ≤ πΉfraction Corrupted Encoding Stable Codes over the Reals • Linear Codes πΈ: ππ → ππ • Corruptions: – arbitrarily corrupt πΏπ 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!