Nested recovery for approximately sparse products

Determine whether the nested deterministic recovery method for recovering an unknown sparse matrix from two-sided measurements can be extended to approximately sparse matrix products, with the goal of building on robust reductions for approximate recovery.

Background

The paper introduces a nested deterministic recovery procedure for an unknown sparse matrix whose two-sided measurements can be computed efficiently, and applies it to obtain an optimal deterministic algorithm for fully sparse matrix multiplication over arbitrary associative rings. The recovery procedure uses inner and outer sparse-recovery layers, with the outer layer operating over a direct-product ring.

The conclusion identifies approximate sparsity as an unresolved direction and specifically asks whether nested recovery can function for approximately sparse products. Such an extension would connect the paper’s exact sparse-recovery framework with robust sparse-recovery reductions, potentially enabling guarantees when the product is only approximately sparse rather than exactly sparse.