Low-degree lower bounds via almost orthonormal bases

Simone Maria Giancola, Christophe Giraud, Nicolas Verzelen, Alexandra Carpentier

Abstract: Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a planted distribution P′ against a null distribution P with independent components -- the standard approach is to bound the advantage using an L2(P)-orthonormal family of polynomials. However, this method breaks down for estimation tasks or more complex testing problems where P has some planted structures, so that no simple L2(P)-orthogonal polynomial family is available. To address this challenge, several technical workarounds have been proposed [SW22,SW25], though their implementation can be delicate. In this work, we propose a more direct proof strategy. Focusing on random graph models, we construct a basis of polynomials that is almost orthonormal under P, in precisely those regimes where statistical-computational gaps arise. This almost orthonormal basis not only yields a direct route to establishing low-degree lower bounds, but also allows us to explicitly identify the polynomials that optimize the low-degree criterion. This, in turn, provides insights into the design of optimal polynomial-time algorithms. We illustrate the effectiveness of our approach by recovering known low-degree lower bounds, and establishing new ones for problems such as hidden subcliques, stochastic block models, and seriation models.