Matrix and tensor rigidity and Lp-approximation

Abstract

We apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good ℓp-approximation for p<2. Hence the first n functions of Walsh system can be approximated by a low-dimensional linear space:dn1−δ({w1,…,wn},Lp[0,1])⩽n−δ,1≤p<2, where δ=δ(p)>0. We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon–Frankl–Rödl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be ℓ1-approximated by low-rank tensors. This implies lower bounds for Θm — the error of m-term approximation of multivariate functions by sums of tensor products u1(x1)⋯ud(xd). For the set of trigonometric polynomials with spectrum in ∏j=1d[−nj,nj] and of norm ‖t‖∞⩽1, and m⩽c(d)∏nj/max⁡{nj}, we haveΘm(T(n1,…,nd)∞,L1[−π,π]d)⩾c1(d)>0. Sharp bounds follow for classes of dominated mixed smoothness in the case 2⩽p⩽∞,1⩽q⩽2:Θm(Wp(r,r,…,r),Lq[0,1]d)≍m−rdd−1.