Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications

2026-05-24

AI Summary

This work addresses the problem of efficiently computing a multiplicative approximation to the matrix $2 \rightarrow q$ norm in the highly compressive regime ($q > 2$), which is connected to several long-standing open problems in combinatorial optimization, quantum information, and algorithmic statistics. We propose the first polynomial-time algorithm that breaks the classical spectral barrier of $d^{1/4}$ without imposing additional assumptions on the input matrix, achieving an approximation ratio of $d^{1/8}$ when $q = 4$. Our approach leverages the Sum-of-Squares (SoS) hierarchy, integrating robust estimation under higher-moment constraints with advanced matrix analysis techniques and constructing corresponding SoS certificates. This result not only yields a polynomial-factor improvement for approximating the $2 \rightarrow q$ norm but also directly enhances the performance of algorithms for robust mean and covariance estimation, regression, and clustering under the minimal assumption of bounded $q$-th moments.

Abstract

The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than $2^{\sqrt{\log n}}$, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a $d^{1/4}$-approximation as a baseline. We give, to the best of our knowledge, the first polynomial-time approximation algorithm beating this baseline by polynomial factors. For the important special case of $q = 4$ it achieves a $d^{1/8}$-approximation. All previous algorithms required additional assumptions on $X$, or only surpassed the baseline for small values of $n$. Moreover, we construct sum-of-squares certificates for the $2 \rightarrow q$ norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its $q$-th moment.

Problem

• 2→q norm
• polynomial-time approximation
• hypercontractive
• Small Set Expansion
• Best Separable State

Innovation

• 2→q norm
• polynomial-time approximation
• sum-of-squares
• hypercontractivity
• robust statistics