Adaptive Approximation Algorithms for Sparse Data Representation

在这个项目的第二阶段，我们专注于开发和数值分析用于高维信号数据处理的新型自适应近似方法，我们的联合研究将提供依赖于近似理论、谐波分析、微分几何和代数拓扑等现代工具的高效多尺度算法。特别强调的是(a)使用小波变换的分散数据去噪和(b)通过流形学习的非线性降维。在(a)中，我们将推广我们之前在简单路径小波变换（EPWT）上的结果，从图像数据到从高维信号中获取的有噪声的分散数据。为此，我们将开发新的基于扩散映射和小波变换的随机路径去噪方法。此外，我们将扩展我们之前关于渐近n项逼近的理论结果，通过EPWT获得分段H ø old光滑函数流形的最优稀疏数据表示。在(b)中，我们将继续就高效、稳健和可靠的非线性降维方法的设计和数值分析进行联合研究，其中自适应多尺度技术、持续同源方法和基于无网格核的近似方案将发挥关键作用。新方法将应用于高维信号的分离、分类和压缩等相关问题。

In the second period of this project we focus on the development and numerical analysis of noveladaptive approximation methods for high-dimensional signal data processing, where our joint researchwill provide efficient multiscale algorithms relying on modern tools from approximation theory, harmonical analysis, differential geometry, and algebraic topology. Special emphasis is placed on (a)scattered data denoising by using wavelet transforms and on (b) nonlinear dimensionality reductionby manifold learning. In (a), we will generalize our previous results on the Easy Path Wavelet Transform (EPWT), from image data to noisy scattered data taken from high-dimensional signals. To this end, we will develop new denoising methods based on diffusion maps and wavelet transforms along random paths. Moreover, we will extend our previous theoretical results on asymptotic N-term approximations to obtain optimally sparse data representations for piecewise H¨older smooth functionson manifolds by the EPWT. In (b), we will continue our joint research concerning the design andnumerical analysis of efficient, robust and reliable nonlinear dimensionality reduction methods, whereadaptive multiscale techniques, persistent homology methods, and meshfree kernel-based approximation schemes will play a key role. The new methods will be applied to relevant problems for the separation, classification, and compression of high-dimensional signals.