Algorithmic Contiguity from Low-Degree Heuristic II: Predicting Detection-Recovery Gaps

TL;DR

This paper introduces a general, simple, and flexible method to establish computational lower bounds for high-dimensional recovery problems by linking low-degree testing advantages to detection-recovery gaps.

Contribution

It develops a model-independent approach combining algorithmic contiguity and cross-validation to derive conditional lower bounds for recovery tasks from low-degree bounds.

Findings

Recovers existing low-degree bounds for planted submatrix, dense subgraph, and stochastic block model.

Provides new evidence for detection-recovery gaps in synchronization and multi-layer models.

Suggests mild low-degree advantage control explains computational barriers in high-dimensional inference.

Abstract

The low-degree polynomial framework has emerged as a powerful tool for providing evidence of statistical-computational gaps in high-dimensional inference. For detection problems, the standard approach bounds the low-degree advantage through an explicit orthonormal basis. However, this method does not extend naturally to estimation tasks, and thus fails to capture the \emph{detection-recovery gap phenomenon} that arises in many high-dimensional problems. Although several important advances have been made to overcome this limitation \cite{SW22, SW25, CGGV25+}, the existing approaches often rely on delicate, model-specific combinatorial arguments. In this work, we develop a general approach for obtaining \emph{conditional computational lower bounds} for recovery problems from mild bounds on low-degree testing advantage. Our method combines the notion of algorithmic contiguity in \cite{Li25} with a cross-validation reduction in \cite{DHSS25} that converts successful recovery into a hypothesis test with lopsided success probabilities. In contrast to prior unconditional lower bounds, our argument is conceptually simple, flexible, and largely model-independent. We apply this framework to several canonical inference problems, including planted submatrix, planted dense subgraph, stochastic block model, multi-frequency angular synchronization, orthogonal group synchronization, and multi-layer stochastic block model. In the first three settings, our method recovers existing low-degree lower bounds for recovery in \cite{SW22, SW25} via a substantially simpler argument. In the latter three, it gives new evidence for conjectured computational thresholds including the persistence of detection-recovery gaps. Together, these results suggest that mild control of low-degree advantage is often sufficient to explain computational barriers for recovery in high-dimensional statistical models.