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的noveladaptive逼近方法的高维信号数据处理，我们的联合researchwill提供有效的多尺度算法依赖于现代工具，从近似理论，谐波分析，微分几何，代数拓扑。特别强调的是（a）分散的数据去噪使用小波变换和（B）非线性降维流形学习。在（a）中，我们将推广我们以前的易路径小波变换（EPWT）的结果，从图像数据到从高维信号中提取的噪声散射数据。为此，我们将开发新的去噪方法的基础上扩散地图和小波变换沿着随机路径。此外，我们将推广我们以前的理论结果的渐近N项逼近获得最佳稀疏数据表示的分段H？older光滑函数在流形上的EPWT。在（B）中，我们将继续我们的联合研究，涉及有效的，鲁棒的和可靠的非线性降维方法的设计和数值分析，其中自适应多尺度技术，持久同源方法和无网格核基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.