Analysis of the higher-order spectral algorithm

Analyze the spectral regularization algorithm obtained by iterating the operators $H_{varepsilon_1,varepsilon_2,dots,varepsilon_k}$ to obtain a spectral approach to $k$-th order Fourier analysis.

Background

The paper develops a spectral regularization algorithm in the quadratic setting and observes that the same order-increment construction can be iterated to produce operators approximating structured components at arbitrary higher orders. Although the construction is formally described, the authors do not analyze its accuracy, stability, or algorithmic performance beyond the quadratic case.

References

By further iterating this process, we obtain a spectral approach to $k$-th order Fourier analysis based on a regularization operator $f\mapsto H_{\varepsilon_1,\varepsilon_2,\dots,\varepsilon_k}(f)$ with $k$ parameters. We leave the analysis of this algorithm to a subsequent paper.

— Spectral algorithms in higher-order Fourier analysis  (2501.12287 - Candela et al., 21 Jan 2025) in Remark "Higher-order versions" following Algorithm 1, Section 1.2