# Matrix Rigidity of Random Toeplitz Matrices

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2 Cotets 1 Itroductio Matrix Rigidity Goldreich-Wigderso s Project 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 Tests for AN Complexity ad AN2 Complexity Lower Bouds for the AN-Complexity of Small-Biased Matrices Explicit 4-Liear Fuctios with High AN-Complexity Lower Bouds for the AN2-Complexity of Radom Toeplitz Matrices Digest ad Ope Problems Digest Radomess-Rigidity Tradeoff Ope Problems Refereces 23 Appedices 24 A.1 Geeralizatio to Larger Fields A.2 The Structure of Matrices Associated with Geeral Biliear Circuits A.3 Characterizatio of AN-complexity for Biliear Forms

13 Recall that m = 2r ad k = /m to get which is o(1) for t 3 Pr [ i, j : A i,j is simple ] 2 2 log + 32 log t2 r/ + 16t 3 r 2 / 2 /16, 3 ad r. 1000r 2 Geeralizatio to Larger Fields. The choice of field F 2 was ot crucial i the proofs of Lemma 3.1, Theorem 3.2 ad Theorem 3.4. Oe ca sytactically replace the field size 2 by ay prime power q, keepig the proofs itact. Furthermore, i Theorem 3.2, we slightly beefit from takig a larger field. For details see Appedix A.1. 4 The Structure of Matrices of Small Biliear Circuits I this sectio we shall further refie the structure of matrices associated with small biliear circuits, beyod the structure captured by Defiitio 2.1 ad Theorem 2.4. We begi by explicitly statig structural results that are implicit i the proof of [GW13, Thm. 4.4]: Sectio 4.1 refers to the structure of biliear fuctios that are computed by depth-2 biliear circuits of small ANcomplexity, whereas Sectio 4.2 refers to geeral biliear circuits. These statemets ca be viewed as relatig AN-complexity to fier otios of structured rigidity (tha the oe of Defiitio 2.1). I Sectio 4.3 we go beyod [GW13], ad aalyze the structure of the submatrices of matrices associated with small biliear circuits, by startig with the foregoig structural results (of [GW13]) ad proceedig aalogously to the first part of the proof of Theorem 3.4. The results of Sectio 4.3 will play a pivotal role i the improved lower bouds proved i Sectio 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 [GW13, Thm. 4.4]). Let F be a biliear fuctio over x, y {0, 1} with C 2 (F ) m. The, F ca be expressed as (i,j) P L i (x)l j(y) + Q l (x, y) (10) where P is a subset of [m] [m], L 1,..., L m ad L 1,..., L m are m-sparse 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 P L row + S l (11) where L col is a m matrix with m-sparse colums, P is a geeral m 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. 11

14 Propositio 4.1 is proved explicitly i the warm-up part of the proof of [GW13, Thm. 4.4]. The followig propositio asserts that the coverse holds as well. This implies that the characterizatio of Prop. 4.1 captures C 2 completely. Propositio 4.2. Ay biliear form F that ca be writte as i Eq. (10), has C 2 (F ) = O(m). We defer the proof of this propositio to Appedix A The Structure of Matrices Associated with Geeral Biliear Circuits Propositio 4.3 (Structure of fuctios computed by geeral biliear circuits). Let F be a biliear fuctio over x, y {0, 1} with 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) (12) 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 (13) 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 4.3 is oly implicit i the proof of [GW13, Thm. 4.4], ad we iclude its proof i Appedix A.2. The followig propositio asserts that the coverse holds as well for m. This implies that the characterizatio of Prop. 4.3 captures C. Propositio 4.4. Ay biliear form F that ca be writte as i Eq. (12), has C(F ) = O(m + ). We defer the proof of this propositio to Appedix A Substructures I this subsectio, similarly to the first part of the proof of Theorem 3.4, we fid a structured submatrix of the matrix associated with ay biliear fuctio with low AN-complexity. I sectio 5, we prove that radom Toeplitz matrices ad small-biased matrices do ot have these structured submatrices, with high probability. This ultimately proves AN-complexity lower bouds for such radom matrices. Let F be a biliear fuctio over x, y {0, 1} with C(F ) m. Startig with Propositio 4.3, we write the matrix A associated with F as A = L col B + CL row + 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. T = m X l Y l, ad ote that T m 3. Deote by 12