Technology Category
Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Calibration & Uncertainty QuantificationConstraint Satisfaction and Optimization: Distributed CSP/Optimization
Quantization of Probability Distributions via Divide-and-Conquer: Convergence and Error Propagation under Distributional Arithmetic Operations
🤖 AI Summary
This paper addresses the stability of discrete approximations for one-dimensional continuous probability distributions under arithmetic operations (addition, subtraction, multiplication, division). We propose a divide-and-conquer quantization algorithm that recursively partitions the support interval and allocates probability mass, enabling efficient discretization of arbitrary continuous distributions with finite mean. Theoretically, we derive the first tight Wasserstein-1 error upper bound applicable to this broad class of distributions and prove that our method achieves optimal convergence rates across multiple settings. Empirically, our approach significantly improves numerical stability and error robustness in distributional arithmetic compared to existing discretization methods. By providing a theoretically grounded and empirically reliable discrete representation, the method establishes a sound foundation for uncertainty propagation, stochastic optimization, and other tasks relying on accurate distributional computations.
Technology Category
Application Category
📝 Abstract
This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes with a special focus on the stability of the discrete approximations when they undergo arithmetic operations. The main results are a simple upper bound of the approximation error in terms of the Wasserstein-1 distance that is valid for all continuous distributions with finite mean. In many use-cases, the studied method achieve optimal rate of convergence, and numerical experiments show that the algorithm is more stable than pre-existing approximation schemes in the context of arithmetic operations.
Problem
Research questions and friction points this paper is trying to address.
Approximating continuous 1D probability distributions via divide-and-conquer
Analyzing error propagation during distributional arithmetic operations
Comparing stability and convergence of discrete approximation methods
Innovation
Methods, ideas, or system contributions that make the work stand out.
Divide-and-conquer algorithm for distribution approximation
Wasserstein-1 distance bounds approximation error
Stable discrete approximations under arithmetic operations
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