Spectral Reconstruction Function: Spectral Complexity and Its Connection to Barron Norms

Abstract

We introduce the Spectral Reconstruction Function (SRF), a cumulative descriptor ofthe spectral content of a digital image. The construction is based on the discrete Fourierdecomposition of an image and on the radial frequency associated with each Fourier mode. TheSRF measures the fraction of the total spectral amplitude contained in modes whose normalizedradial frequency does not exceed a prescribed threshold. This provides a natural spectralreconstruction viewpoint: increasing the threshold progressively incorporates higher-frequencycomponents of the image.We further consider the first moment associated with the SRF and show that it has adirect relation, up to normalization constants and the total spectral mass, with a discreteFourier-based quantity of Barron type. This connection suggests a quantitative interpretation ofthe SRF as a measure of spectral complexity. Finally, we formulate a computational experimentin which correctly classified MNIST images are progressively shifted towards higher-frequencycontent using a discrete Laplacian. The critical point at which a convolutional neural networkloses the correct classification is then compared with the spectral complexity of the perturbedimage. The purpose of this preliminary note is to establish the mathematical framework and thecomputational methodology for studying spectral robustness in convolutional neural networks.Keywords: spectral reconstruction function; Fourier transform; spectral complexity; Barron norm;convolutional neural networks; MNIST; spectral robustness.