Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target functions. We analyze their learnability using an explicit three-layer fitting model trained via layer-wise spectral estimators. Although the target is globally a high-degree polynomial, its compositional structure allows learning to proceed in stages: an intermediate representation reveals structure that is inaccessible at the input level. This reduces learning to simpler spectral estimation problems, well studied in the context of multi-index models, whereas any shallow estimator must resolve all components simultaneously. Our analysis relies on Gaussian universality, leading to sharp separations in sample complexity between two and three-layer learning strategies.
@article{arxiv.2602.10867,
title = {Deep Learning of Compositional Targets with Hierarchical Spectral Methods},
author = {Hugo Tabanelli and Yatin Dandi and Luca Pesce and Florent Krzakala},
journal= {arXiv preprint arXiv:2602.10867},
year = {2026}
}