SparseStack constant-row-sparsity OSI conjecture

Establish that the SparseStack test matrix—a random d×k matrix formed by horizontally stacking ζ independent CountSketch blocks of width b = k/ζ and scaling by 1/√ζ—serves as an (r, 1/2)-oblivious subspace injection for some embedding dimension k = Θ(r) while using constant row sparsity ζ = Θ(1).

Background

Sparse test matrices enable near input-sparsity-time sketching, and the paper recommends the SparseStack construction for its empirical performance and practical advantages. The authors prove that SparseStack is an (r, 1/2)-OSI with embedding dimension k = Θ(r) and row sparsity ζ = Θ(log r), improving on prior OSE-based bounds.

Extensive numerical evidence in the paper suggests that constant sparsity (e.g., ζ = 4) already yields reliable OSI behavior with k ≈ 2r, and experiments indicate injectivity stabilizes while dilation grows, consistent with an OSI but not an OSE. Moreover, known OSE lower bounds rule out constant sparsity for OSEs with k = Θ(r) but do not preclude OSIs, motivating this conjecture.