Faster Macaulay solvers for larger sparse systems

Develop faster Macaulay solvers that make simultaneous recovery of all sparse solutions practical for larger sparse-approximation problems.

Background

The Macaulay construction is formally guaranteed to recover all solutions of the polynomial system simultaneously, but its matrix dimensions grow rapidly with the number of variables and the polynomial degree. The paper therefore finds the method tractable only for small problems.

Improving the computational efficiency of Macaulay-based solution methods is explicitly identified as necessary for applying simultaneous all-solution recovery to larger problems.

References

Several directions remain open. First, the connection with EVD and CPD makes it in principle possible to derive upper bounds on the estimation error in the noisy case. Second, our approach extends to constrained sparse approximation, since any constraint that can be written polynomially can be appended to the system. Examples of such constrained variants are: the unit-norm constraint, structured sparsity patterns, and nonnegativity. Third, the approach may be extended to parametrized dictionaries if the dependence on the unknown parameters is polynomial, as those parameters may be treated as additional variables. Fourth, faster Macaulay solvers would make the simultaneous recovery of all solutions practical for larger problems.

— Sparse Approximation via Polynomial Equations  (2609.11215 - Tomić et al., 10 Sep 2026) in Section Conclusion and Future Work