Data denoising with self consistency, variance maximization, and the Kantorovich dominance
Joshua Zoen-Git Hiew, Tongseok Lim, Brendan Pass, Marcelo Cruz de Souza
TL;DR
The paper develops a novel data denoising framework that seeks a signal distribution $\mu$ within a structure-defining domain ${\cal D}$ that is self-consistent with observed data $\nu$, linking this to a variance-maximization problem under convex order, $\mu \preceq_{\rm C} \nu$. To address computational and stability challenges, it introduces Kantorovich dominance, a weaker coupling-based condition $\mu \preceq_{\rm K} \nu$, which preserves key optimality features and yields a robust, more efficient denoising method. The authors establish existence and stability results, show equivalence to relaxed Wasserstein-projection problems for conic domains, and connect the framework to principal component analysis and principal curves. Numerical experiments on curves with various geometric constraints demonstrate the method’s scalability and comparative performance against traditional PCA/principal-curve approaches, highlighting enhanced stability and practical utility in high-dimensional denoising tasks.
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.