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[2410.03041] Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula
Summary
The paper introduces Minmax Trend Filtering (MTF), a novel approach to Total Variation Denoising (TVD) that utilizes a local minmax/maxmin formula for improved nonparametric regression methods.
Why It Matters
This research advances the field of statistical theory and machine learning by providing a new framework for denoising and smoothing techniques. The MTF method offers a simpler local analysis of estimation errors, enhancing the understanding of local adaptivity in TVD, which is crucial for various applications in data science and signal processing.
Key Takeaways
- Introduces Minmax Trend Filtering (MTF) as a new estimator for Total Variation Denoising.
- MTF is based on a local minmax/maxmin formula, providing pointwise estimation error bounds.
- Offers higher order polynomial versions of TVD, expanding the nonparametric regression toolbox.
- Demonstrates local adaptivity of TVD/MTF through new pointwise convergence rates.
- Simplifies the analysis of TVD compared to traditional methods.
Mathematics > Statistics Theory arXiv:2410.03041 (math) [Submitted on 3 Oct 2024 (v1), last revised 12 Feb 2026 (this version, v4)] Title:Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula Authors:Sabyasachi Chatterjee View a PDF of the paper titled Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula, by Sabyasachi Chatterjee View PDF HTML (experimental) Abstract:Total Variation Denoising (TVD) is a fundamental denoising and smoothing method. In this article, we identify a new local minmax/maxmin formula producing two estimators which sandwich the univariate TVD estimator at every point. Operationally, this formula gives a local definition of TVD as a minmax/maxmin of a simple function of local averages. Moreover we find that this minmax/maxmin formula is generalizeable and can be used to define other TVD like estimators. In this article we propose and study higher order polynomial versions of TVD which are defined pointwise lying between minmax and maxmin optimizations of penalized local polynomial regressions over intervals of different scales. These appear to be new nonparametric regression methods, different from usual Trend Filtering and any other existing method in the nonparametric regression toolbox. We call these estimators Minmax Trend Filtering (MTF). We show how the proposed local definition of TVD/MTF estimator makes it tractable to bound pointwise estimation...