2017/05/09 by Desai Cheng, Kasso A. Okoudjou, Cheng, Desai +1
Computer Science · Mathematics · #42C15 #60D05 #94A12 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Information Theory (cs.IT) #Mathematical Analysis and Transform Methods #cs.IT #math.CA #math.IT #msc:42C15 #msc:60D05 #msc:94A12
paper · pdf · doi:10.48550/arxiv.1705.03437
25 pages, 3 figures
arxiv created 2017/05/09 · openalex publication_date 2017/05/09 · arxiv updated 2017/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A probabilistic frame is a Borel probability measure with finite second moment whose support spans ℝd. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in ℝd. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the 2-Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabilistic frame can be approximated in the 2-Wasserstein metric with (compactly supported) finite frames whose bounds can be controlled. On the other hand, we establish some fine continuity properties of the function that maps a probabilistic frame to its canonical Parseval probabilistic frame. Our results generalize similar ones for finite frames and their associated Parseval frames.