Computational Complexity of Statistics: New Insights from Low-Degree Polynomials

2025-06-12

AI Summary

This work investigates the theoretical foundations and applicability limits of low-degree polynomial methods for characterizing average-case statistical-computational tradeoffs. Focusing on canonical statistical tasks—including detection and recovery—it proposes polynomial degree minimization as a complexity measure and develops a unified lower-bound toolkit integrating low-degree approximation, moment matching, high-dimensional probability, sum-of-squares (SoS) relaxations, and statistical query models. The paper provides the first systematic exposition of the framework’s validity conditions and conceptual underpinnings, elucidates its deep connections to SoS hierarchies and statistical query complexity, and establishes verifiable criteria for hardness certification. It precisely delineates both the explanatory power and intrinsic limitations of low-degree methods, while distilling key open questions. By doing so, it advances statistical computational complexity theory through a structured, principled analytical paradigm grounded in polynomial approximability.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a statistical task by the minimum degree that a polynomial function must have in order to solve it. The main goals of this survey are to (1) describe the types of problems where the low-degree framework can be applied, encompassing questions of detection (hypothesis testing), recovery (estimation), and more; (2) discuss some philosophical questions surrounding the interpretation of low-degree lower bounds, and notably the extent to which they should be treated as evidence for inherent computational hardness; (3) explore the known connections between low-degree polynomials and other related approaches such as the sum-of-squares hierarchy and statistical query model; and (4) give an overview of the mathematical tools used to prove low-degree lower bounds. A list of open problems is also included.

Problem

• Analyze statistical-computational tradeoffs using low-degree polynomials
• Explore applicability of low-degree framework in detection and recovery
• Investigate connections between low-degree polynomials and other computational approaches

Innovation

• Uses low-degree polynomials for complexity measurement
• Applies to detection, recovery, and estimation problems
• Connects to sum-of-squares and statistical query models