🤖 AI Summary
In high-dimensional statistical inference, a computational-statistical gap often separates detection and recovery tasks, yet existing low-degree polynomial methods struggle to directly characterize computational lower bounds for recovery. This work proposes a general framework that leverages algorithmic continuity and cross-validation reductions to translate weak low-degree detection advantages into conditional computational lower bounds for recovery. The approach provides, for the first time, a model-agnostic and conceptually simple unification of the detection-recovery gap across multiple canonical models. Applying this framework to six problems—including planted matrix, dense subgraph detection, and the stochastic block model—it either reproduces known computational thresholds or offers new evidence for them, thereby demonstrating the framework’s broad applicability and effectiveness.
📝 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.