Kronecker Products, Low-Depth Circuits, and Matrix Rigidity

For a matrix M and a positive integer r, the rank r rigidity of M is the smallest number of entries of M which one must change to make its rank at most r. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: • For any $d> 1$, and over any field F, the N × N Walsh-Hadamard transform has a depth-d linear circuit of size O(d · N1 + 0.96/d). This circumvents a known lower bound of Ω(d · N1 + 1/d) for circuits with bounded coefficients over C by Pudlak (2000), by using coefficients of magnitude polynomial in N. Our construction also generalizes to linear transformations given by a Kronecker power of any fixed 2 × 2 matrix. • The N × N Walsh-Hadamard transform has a linear circuit of size ≤ (1.81 + o(1)) N log₂ N, improving on the bound of ≈ 1.88 N log₂ N which one obtains from the standard fast Walsh-Hadamard transform. • A new rigidity upper bound, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant's approach: $-$ for any field F and any function f : \0,1\ⁿ → F, the matrix Vf ∈ F2ⁿ × 2ⁿ given by, for any x,y ∈ \0,1\ⁿ, Vf[x,y] = f(x y), and $-$ for any field F and any fixed-size matrices M₁, …, Mₙ ∈ Fq × q, the Kronecker product M₁ ⊗ M₂ ⊗ ⋯ ⊗ Mₙ. This generalizes recent results on non-rigidity, using a simpler approach which avoids needing the polynomial method.