Adaptive Approximation Algorithms for Sparse Data Representation

点击翻译按钮获取中文摘要

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.