A spectral inverse theorem and a spectral regularity theorem show that leading eigenvectors of Fourier-denoised matrices recover quadratic Fourier structure, giving new algorithms for quadratic denoising and character decomposition.
Higher order Fourier analysis as an algebraic theory I
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abstract
Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\'edi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between these approaches. Our main method is to take the ultra product of abelian groups and to develop a precise algebraic theory of higher order characters on it. These results then can be turned back into approximative statements about finite Abelian groups.
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Spectral algorithms in higher-order Fourier analysis
A spectral inverse theorem and a spectral regularity theorem show that leading eigenvectors of Fourier-denoised matrices recover quadratic Fourier structure, giving new algorithms for quadratic denoising and character decomposition.