2018/01/01 by Miguel del Álamo, del Álamo, Miguel, Housen Li +3 · 2 citations
Engineering · Medicine · #62G05 #62G20 #62M40 #FOS: Mathematics #Medical Imaging Techniques and Applications #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1807.02038
openalex publication_date 2018/07/05 · openalex created_date 2022/08/04 · openalex updated_date 2026/04/28
Despite the popularity and practical success of total variation (TV)\nregularization for function estimation, surprisingly little is known about its\ntheoretical performance in a statistical setting. While TV regularization has\nbeen known for quite some time to be minimax optimal for denoising\none-dimensional signals, for higher dimensions this remains elusive until\ntoday. In this paper we consider frame-constrained TV estimators including many\nwell-known (overcomplete) frames in a white noise regression model, and prove\ntheir minimax optimality w.r.t. Lq-risk (1\≤ q<\∞) up to a\nlogarithmic factor in any dimension d\≥ 1. Overcomplete frames are an\nestablished tool in mathematical imaging and signal recovery, and their\ncombination with TV regularization has been shown to give excellent results in\npractice, which our theory now confirms. Our results rely on a novel connection\nbetween frame-constraints and certain Besov norms, and on an interpolation\ninequality to relate them to the risk functional.\n