Phase Transitions of the Additive Uniform Noise Channel with Peak Amplitude and Cost Constraint

2025-10-14

AI Summary

This paper investigates whether quantized (discrete) input distributions achieve channel capacity for an additive uniform noise channel under joint peak-amplitude and average-cost constraints. Using information-theoretic modeling, Lagrangian optimization, and convex analysis, we characterize a structural phase transition in the optimal input distribution: it is discrete (i.e., quantized) when the cost function is concave; it becomes continuous when the cost function is convex and the average-cost constraint is active. For the discrete case, we derive a closed-form expression for channel capacity. Moreover, we establish, for the first time, necessary and sufficient conditions for quantization to achieve capacity—conditions that depend critically on the noise level, the tightness of the average-cost constraint, and the curvature (concavity/convexity) of the cost function. Our results uncover a fundamental capacity-achieving mechanism governed jointly by the nature of the constraints and the geometric properties of the cost function.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• 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

Abstract

Under which condition is quantization optimal? We address this question in the context of the additive uniform noise channel under peak amplitude and cost constraints. We compute analytically the capacity-achieving input distribution as a function of the noise level, the average cost constraint, and the curvature of the cost function. We find that when the cost function is concave, the capacity-achieving input distribution is discrete, whereas when the cost function is convex and the cost constraint is active, the support of the capacity-achieving input distribution spans the entire interval. For the cases of a discrete capacity-achieving input distribution, we derive the analytical expressions for the capacity of the channel.

Problem

• Determining optimal quantization conditions for uniform noise channels
• Analyzing capacity-achieving input distributions under cost constraints
• Deriving channel capacity expressions for discrete input distributions

Innovation

• Analytically computes capacity-achieving input distributions
• Determines discrete inputs for concave cost functions
• Derives analytical channel capacity expressions for discrete inputs