Sample Abundance for Signal Processing: A Brief Introduction

2025-07-25

AI Summary

This paper addresses high computational complexity in constrained optimization for one-bit/few-bit signal processing—particularly involving semidefinite and low-rank constraints. We introduce the novel concept of “sample-abundant singularity”: when the number of measurements vastly exceeds classical requirements, high-dimensional nonconvex/nonlinear constraints degenerate into an overdetermined linear feasibility problem. Leveraging the finite-volume property, we theoretically establish that this phenomenon guarantees exact signal recovery. Methodologically, we integrate low-precision quantization models with efficient linear feasibility solvers, eliminating iterative optimization and matrix decomposition. Experiments demonstrate that our framework reduces computational cost by several orders of magnitude while maintaining high accuracy in phase retrieval and covariance estimation. Moreover, it exhibits superior practicality in hardware-constrained settings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

This paper reports, by way of introduction, on the advances made by our group and the broader signal processing community on the concept of sample abundance; a phenomenon that naturally arises in one-bit and few-bit signal processing frameworks. By leveraging large volumes of low-precision measurements, we show how traditionally costly constraints, such as matrix semi-definiteness and rank conditions, become redundant, yielding simple overdetermined linear feasibility problems. We illustrate key algorithms, theoretical guarantees via the Finite Volume Property, and the sample abundance singularity phenomenon, where computational complexity sharply drops.

Problem

• Explores sample abundance in low-precision signal processing frameworks
• Simplifies costly constraints via overdetermined linear feasibility problems
• Analyzes computational complexity drop via sample abundance singularity

Innovation

• Leveraging large volumes of low-precision measurements
• Simplifying overdetermined linear feasibility problems
• Reducing computational complexity via sample abundance