Parametrized dictionaries with polynomial parameter dependence

Extend the polynomial-equation sparse-approximation approach to parametrized dictionaries whose dependence on unknown parameters is polynomial by treating those parameters as additional variables.

Background

The paper considers a fixed dictionary matrix A in the sparse approximation problem. It observes that dictionaries depending on unknown parameters could be incorporated into the polynomial formulation when that dependence is polynomial.

A method for solving and analyzing this augmented system, including the interaction between sparse coefficients and dictionary parameters, is not provided and remains an open direction.

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.

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