Smaller Low-Depth Circuits for Kronecker Powers

A linear circuit for computing an N × N matrix M is a circuit with N inputs corresponding to the entries of a vector x and N outputs corresponding to the entries of the transformed vector Mx, and where each gate computes a linear combination of its inputs. Each gate may have unbounded fan-in, and the size of the circuit is the number of wires. This model captures most known algorithms for computing linear transforms, and (in the constant-depth or 'synchronous' settings) is equivalent to factoring M as the product of sparse matrices. We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization of the Fast Walsh-Hadamard transform) shows that any such N × N matrix has a depth-2 circuit of size O(N1.5). We improve on this for all such matrices, and especially for some such matrices of particular interest: • For any integer q > 1 and any matrix which is the Kronecker power of a fixed q × q matrix, we construct a depth-2 circuit of size O(N1.5-aq), where aq > 0 is a positive constant depending only on q. No bound beating size O(N1.5) was previously known for any q > 2. • For the case q = 2, i.e., for any matrix which is the Kronecker power of a fixed 2 × 2 matrix, we construct a depth-2 circuit of size O(N1.446), improving the prior best size O(N1.493) [Alman, 2021]. • For the Walsh-Hadamard transform, we construct a depth-2 circuit of size O(N1.443), improving the prior best size O(N1.476) [Alman, 2021]. • For the disjointness matrix (the communication matrix of set disjointness, or equivalently, the matrix for the linear transform that evaluates a multilinear polynomial on all 0/1 inputs), we construct a depth-2 circuit of size O(N1.258), improving the prior best size O(N1.272) [Jukna and Sergeev, 2013]. Our constructions also generalize to improving the standard construction for any depth ≤ O (log N). Our main technical tool is an improved way to convert a nontrivial circuit for any matrix into a circuit for its Kronecker powers. Our new bounds provably could not be achieved using the approaches of prior work.