# Matrix Rigidity of Random Toeplitz Matrices

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2 Cotets 1 Itroductio Matrix Rigidity The Project of Goldreich-Wigderso Resolvig the Foregoig Ope Problems Overview of the Proof of Theorem Orgaizatio Prelimiaries 5 3 Mai Results 6 4 The Structure of Matrices of Small Biliear Circuits The Structure of Matrices Associated with Depth Two Biliear Circuits The Structure of Matrices Associated with Geeral Biliear Circuits Substructures Testig AN Complexity ad AN2 Complexity Lower Bouds for the AN-Complexity of Small-Biased Matrices Lower Bouds for the AN2-Complexity of Radom Toeplitz Matrices Explicit 4-Liear Fuctios with AN-Complexity Ω( 2/3 ) Digest ad Ope Problems Digest Ope Problems Refereces 21 Appedices 22 A.1 Geeralizatio to Larger Fields A.2 The Structure of Matrices Associated with Geeral Biliear Circuits

10 Before provig Theorem 3.2, we state a immediate corollary of it. Corollary 3.3. Let A F 2 be a radom Hakel matrix. The, there exists a uiversal costat c > 0 such that for every ε > 0 1. With probability 1 o(1), the matrix A is c 2 2ε / log rigid for rak 1/2+ε. 2. With probability 1 o(1), the matrix A is m 3 rigid for rak m = c 3/5 log 1/5. 3. With probability 1 o(1), the matrix A is c 1+2ε / log rigid for rak 1 ε. Proof of Theorem 3.2. Suppose towards cotradictio that A ca be represeted as a sum of a matrix R of rak at most r, ad a s-sparse matrix S, where s 3 /160r 2 log. Let m = 2r, ad assume for coveiece that k = /m is a iteger. Cosider the followig partitio of A s etries ito (/m) 2 submatrices, each of dimesio m m. For i [/m] ad j [/m], let I i = {i, i + k,..., i + (m 1)k}, J j = {(j 1)m + 1, (j 1)m + 2,..., jm}. (6) Deote by A i,j (R i,j, S i,j, resp.) the matrix A (R, S, resp.) restricted to rows I i ad colums J j. See Figure 1 for a example of such a submatrix. The mai observatio is that for each a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 15 a 9 a 10 a 11 a 12 a 13 a 14 a 15 a 16 Figure 1: A submatrix A 1,1 of the matrix A, for m = 4 ad k = 2. (i, j) [/m] 2, the matrix A i,j is of the form eeded by the mai lemma. Aother observatio is that sice the submatrices S i,j partitios the sparse matrix S, oe of them has sparsity at most s s m2. I additio, sice rak of a submatrix may oly decrease, for every i, j, it holds that 2 rak(r i,j ) rak(r) r. We say that A i,j is simple if it ca be represeted as a sum of a s -sparse matrix ad a matrix of rak at most r. By the above discussio, A ca be represeted as S + R where S is s-sparse ad R is of rak at most r, oly if there exists a submatrix A i,j that is simple. We shall show that the latter occurs with very low probability: Pr [ i, j : A i,j is simple ] i,j i,j Pr[A i,j is simple] S F m m 2 : wt(s) s Pr[rak(A i,j + S) m/2] (Uio Boud) (Uio Boud) ( ) ( ) 2 m 2 m s 2 mk/16 (Lemma 3.1) < 2 (2m 2 ) s 2 /16. (km = ) Fially, usig s 40 log, which follows from s 3 160r 2 log, we get that Pr[ i, j : Ai,j is simple] = o(1), which completes the proof. 8

13 4.1 The Structure of Matrices Associated with Depth Two Biliear Circuits We say a row/colum i a matrix is m-sparse if it cotais at most m o-zero etries. Likewise, a liear fuctio l(x) (resp. l (y)) is m-sparse if it depeds o at most m etries i x (resp. y). Lastly, recall that by Defiitio 2.3, C 2 (F ) is the miimal AN-complexity of a depth-two biliear circuit computig F. Propositio 4.1 (Structure of fuctios computed by depth two biliear circuits). If C 2 (F ) = m, the F ca be expressed as L i(x)l i (y) + Q l (x, y) where L 1,..., L m are m-sparse liear fuctios, L 1,..., L m are geeral liear fuctios, ad each Q l is a biliear fuctio of at most m variables from x ad at most m variables from y. The matrix associated with F has the form A = CL row + S l (9) where C is a geeral m matrix, L row is a m matrix with m-sparse rows, ad each S l is a matrix whose oes reside i a m m rectagle. Propositio 4.1 is proved explicitly i the warm-up part of the proof of [GW13, Thm. 4.4]. 4.2 The Structure of Matrices Associated with Geeral Biliear Circuits Propositio 4.2 (Structure of fuctios computed by geeral biliear circuits). If C(F ) = m, the F ca be expressed as L i (x)l i(y) + M i(x)m i (y) + Q l (x, y) where L 1,..., L m ad M 1,..., M m are m-sparse liear fuctios, L 1,..., L m ad M 1,..., M m are geeral liear fuctios, ad each Q l is a biliear fuctio of at most m variables from x ad at most m variables from y. The matrix associated with F has the form A = L col B + CL row + S l (10) where L col is a m matrix with m-sparse colums, B is a geeral m matrix, C is a geeral m matrix, L row is a m matrix with m-sparse rows, ad each S l is a matrix whose oes reside i a m m rectagle. Propositio 4.2 is oly implicit i the proof of [GW13, Thm. 4.4], ad we iclude its proof i Appedix A Substructures I this subsectio, similarly to the first part of the proof of Theorem 3.4, we fid a submatrix (of a matrix associated with a biliear circuit) that has average rigidity-like parameters. Startig with Propositio 4.2, for C(F ) = m, we write the matrix A associated with F as A = L col B + CL row + m S l such that the o-zero etries of S l are a subset of X l Y l, where X l, Y l m. Deote 11