Polynomially constrained sparse approximation

Extend the polynomial-equation sparse-approximation framework to constrained sparse approximation by appending polynomially expressible constraints, including unit-norm constraints, structured sparsity patterns, and nonnegativity constraints.

Background

The proposed formulation represents sparsity and the linear data-fitting condition as polynomial equations, so additional constraints that admit polynomial representations could in principle be incorporated into the same system.

The paper does not develop or analyze these constrained variants; it specifically identifies unit-norm constraints, structured sparsity patterns, and nonnegativity as examples of unresolved extensions.

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.

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