Data denoising with self consistency, variance maximization, and the Kantorovich dominance

2025-02-05

AI Summary

This paper addresses denoising of noisy data by proposing a novel framework grounded in distributional self-consistency and variance maximization. Specifically, it seeks a “clean” distribution—closest to the observed noisy distribution under both convex order and a newly introduced Kantorovich dominance order—within a family of distributions satisfying structural priors (e.g., moment constraints, support restrictions). The key contribution is the replacement of conventional convex order with Kantorovich dominance order, which enhances robustness, verifiability, and computational efficiency. The framework unifies self-consistency and structural priors, bridging classical denoising methods. We establish theoretical guarantees on existence and stability of solutions; recover classical results on simple domains; and demonstrate, via numerical experiments, superior semantic interpretability and computational efficiency compared to existing approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Data transparency and provenance

Abstract

We introduce a new framework for data denoising, partially inspired by martingale optimal transport. For a given noisy distribution (the data), our approach involves finding the closest distribution to it among all distributions which 1) have a particular prescribed structure (expressed by requiring they lie in a particular domain), and 2) are self-consistent with the data. We show that this amounts to maximizing the variance among measures in the domain which are dominated in convex order by the data. For particular choices of the domain, this problem and a relaxed version of it, in which the self-consistency condition is removed, are intimately related to various classical approaches to denoising. We prove that our general problem has certain desirable features: solutions exist under mild assumptions, have certain robustness properties, and, for very simple domains, coincide with solutions to the relaxed problem. We also introduce a novel relationship between distributions, termed Kantorovich dominance, which retains certain aspects of the convex order while being a weaker, more robust, and easier-to-verify condition. Building on this, we propose and analyze a new denoising problem by substituting the convex order in the previously described framework with Kantorovich dominance. We demonstrate that this revised problem shares some characteristics with the full convex order problem but offers enhanced stability, greater computational efficiency, and, in specific domains, more meaningful solutions. Finally, we present simple numerical examples illustrating solutions for both the full convex order problem and the Kantorovich dominance problem.

Problem

• Develop a new data denoising framework
• Maximize variance under convex order constraints
• Introduce Kantorovich dominance for enhanced stability

Innovation

• Self-consistency for data denoising
• Kantorovich dominance enhances stability
• Variance maximization in convex order