Sharp Phase Transitions in Estimation with Low-Degree Polynomials

TL;DR

This paper develops new techniques to establish lower bounds on low-degree polynomial algorithms, revealing sharp phase transitions in high-dimensional estimation problems like planted subgraphs and spiked models.

Contribution

It introduces novel methods for proving lower bounds on polynomial algorithms, capturing precise phase transitions in estimation tasks across multiple models.

Findings

Addresses estimation phase transitions such as BBP and Kesten-Stigum thresholds.

Rules out low-degree polynomial algorithms of degree n^δ for some δ>0.

Resolves open problems and supports conjectures in high-dimensional inference.

Abstract

High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it is conjectured to be computationally intractable in certain parameter regimes. A powerful approach to understanding this hardness involves proving lower bounds on the efficacy of low-degree polynomial algorithms. We introduce new techniques for establishing such lower bounds, leading to novel results across diverse settings: planted submatrix, planted dense subgraph, the spiked Wigner model, and the stochastic block model. Notably, our results address the estimation task -- whereas most prior work is limited to hypothesis testing -- and capture sharp phase transitions such as the "BBP" transition in the spiked Wigner model (named for Baik, Ben Arous, and P\'{e}ch\'{e}) and the Kesten-Stigum threshold in the stochastic block model. Existing work on estimation either falls short of achieving these sharp thresholds or is limited to polynomials of very low (constant or logarithmic) degree. In contrast, our results rule out estimation with polynomials of degree $n^{\delta}$ where $n$ is the dimension and $\delta > 0$ is a constant, and in some cases we pin down the optimal constant $\delta$. Our work resolves open problems posed by Hopkins & Steurer (2017) and Schramm & Wein (2022), and provides rigorous support within the low-degree framework for conjectures by Abbe & Sandon (2018) and Lelarge & Miolane (2019).