Low-degree lower bounds via almost orthonormal bases

TL;DR

The paper develops a direct LD-lower-bound framework for high-dimensional graph models with latent structure by constructing an invariant, almost-orthonormal basis of low-degree polynomials. It centers the adjacency data, enforces permutation invariance, and uses corrected, renormalized templates to form the polynomials $\Psi_G$, achieving near-orthogonality under the null in weak-signal regimes. This enables explicit LD upper and lower bounds for both estimation and complex testing across six models (HS-I, HS-P, SBM-I, SBM-P, TS-I, TS-P), and yields new bounds as well as recovery of known ones. The approach provides actionable insights into which polynomials drive LD performance and thus informs the design of optimal polynomial-time algorithms, with broad applicability to latent-structure inference problems. Overall, the work offers a simpler, constructive route to LD hardness results and highlights the role of symmetry and motif-based aggregations in computational-statistical gaps.

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 $\mathbb{P}'$ against a null distribution $\mathbb{P}$ with independent components -- the standard approach is to bound the advantage using an $\mathbb{L}^2(\mathbb{P})$-orthonormal family of polynomials. However, this method breaks down for estimation tasks or more complex testing problems where $\mathbb{P}$ has some planted structures, so that no simple $\mathbb{L}^2(\mathbb{P})$-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 $\mathbb{P}$, 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.

Figures (2)

• Figure 1: Illustration of two templates $G^{(1)}$ and $G^{(2)}$, a matching $\mathbf{M}$ and the symmetric difference graph $G_{\Delta}$.
• Figure 2: Illustration of the shadow of a graph. The information contained in the shadow are all labels of the nodes which are colored. The labels of the black part is not registered in the shadow --- but we know the "shape" and that all nodes in the black part are perfectly matched.

Theorems & Definitions (53)

• Lemma 3.1
• Definition 1: Collection $\mathcal{G}_{\leq D}$
• Lemma 3.2
• Remark
• Lemma 3.3
• Theorem 3.4
• Lemma 3.5
• Theorem 3.6
• Theorem 4.1
• Theorem 4.2