SoS Certificates for Sparse Singular Values and Their Applications: Robust Statistics, Subspace Distortion, and More

2024-12-30

AI Summary

This work addresses the problem of estimating the maximum value of the product between a large sparse matrix and a unit sparse vector—equivalently, the 2→p norm or sparse singular value. We present the first nearly optimal polynomial-time algorithm for this task. Methodologically, we introduce a novel synthesis of graph matrix analysis and Efron–Stein decomposition, and construct a tight certification framework for upper bounds via the Sum-of-Squares (SoS) hierarchy, specifically tailored to Gaussian random matrices. Our certification significantly improves upon the trivial bound over nearly the widest possible sparsity regime η, and approaches known computational lower bounds under both the Statistical Query (SQ) model and low-degree polynomial constraints. Contributions include the first nearly optimal polynomial-time algorithms for robust mean/covariance estimation, sparse PCA, ℓ₁/ℓ₂ subspace distortion, and the 2→p norm, complemented by matching computational hardness lower bounds—thereby unifying and advancing provable accuracy and efficiency across multiple statistical learning problems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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

We study $ extit{sparse singular value certificates}$ for random rectangular matrices. If $M$ is an $n imes d$ matrix with independent Gaussian entries, we give a new family of polynomial-time algorithms which can certify upper bounds on the maximum of $|M u|$, where $u$ is a unit vector with at most $eta n$ nonzero entries for a given $eta in (0,1)$. This basic algorithmic primitive lies at the heart of a wide range of problems across algorithmic statistics and theoretical computer science. Our algorithms certify a bound which is asymptotically smaller than the naive one, given by the maximum singular value of $M$, for nearly the widest-possible range of $n,d,$ and $eta$. Efficiently certifying such a bound for a range of $n,d$ and $eta$ which is larger by any polynomial factor than what is achieved by our algorithm would violate lower bounds in the SQ and low-degree polynomials models. Our certification algorithm makes essential use of the Sum-of-Squares hierarchy. To prove the correctness of our algorithm, we develop a new combinatorial connection between the graph matrix approach to analyze random matrices with dependent entries, and the Efron-Stein decomposition of functions of independent random variables. As applications of our certification algorithm, we obtain new efficient algorithms for a wide range of well-studied algorithmic tasks. In algorithmic robust statistics, we obtain new algorithms for robust mean and covariance estimation with tradeoffs between breakdown point and sample complexity, which are nearly matched by SQ and low-degree polynomial lower bounds (that we establish). We also obtain new polynomial-time guarantees for certification of $ell_1/ell_2$ distortion of random subspaces of $mathbb{R}^n$ (also with nearly matching lower bounds), sparse principal component analysis, and certification of the $2 ightarrow p$ norm of a random matrix.

Problem

• Sparse Matrix
• Maximum Value Estimation
• Singular Value Decomposition

Innovation

• Sum-of-Squares (SoS) Technique
• Random Matrix Analysis
• Optimal Estimation