基于伪逆向的多尺度分析及其在流形值序列中的应用
Multiscale analysis via pseudo-reversing and applications to manifold-valued sequences
AI总结：
本文针对流形值序列的多尺度表示问题，提出基于伪逆向的多尺度变换框架，解决了部分上采样算子缺乏实用逆向的难题，并展示了其在流形数据处理中的应用。
摘要：
使用流形值对数据进行建模是一个强大的概念，具有众多优势，尤其是在处理非线性现象方面。这种方法能够捕捉数据内在的几何结构，从而得到更准确的描述符和更高效的计算过程。然而，在流形环境下，即便是压缩和数据增强这样的基本任务也面临着有意义的挑战。本文引入了一种多尺度变换，旨在以不同尺度表示流形值序列，从而为各种应用提供新颖的数据处理工具。与传统方法类似，我们的构造基于一个细化算子（其作用相当于上采样算子）以及一个相应的下采样算子。受Wiener引理的启发，我们将后者称为前者的逆向。结果表明，某些上采样算子（例如基于最小二乘的细化）并不具有实用的逆向。因此，我们引入了伪逆向的概念，并探讨了其分析性质和渐近行为。我们推导了所诱导的多尺度变换的分析性质，并在论文结尾给出了数值示例，展示了伪逆向的不同方面以及两个涉及流形的数据处理应用。
英文摘要：
Modeling data using manifold values is a powerful concept with numerous advantages, particularly in addressing nonlinear phenomena. This approach captures the intrinsic geometric structure of the data, leading to more accurate descriptors and more efficient computational processes. However, even fundamental tasks like compression and data enhancement present meaningful challenges in the manifold setting. This paper introduces a multiscale transform that aims to represent manifold-valued sequences at different scales, enabling novel data processing tools for various applications. Similar to traditional methods, our construction is based on a refinement operator that acts as an upsampling operator and a corresponding downsampling operator. Inspired by Wiener's lemma, we term the latter as the reverse of the former. It turns out that some upsampling operators, for example, least-squares-based refinement, do not have a practical reverse. Therefore, we introduce the notion of pseudo-reversing and explore its analytical properties and asymptotic behavior. We derive analytical properties of the induced multiscale transform and conclude the paper with numerical illustrations showcasing different aspects of the pseudo-reversing and two data processing applications involving manifolds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
