Optimal Adaptive Nonparametric Denoising of Multidimensional - Time Signal

We construct an adaptive asymptotically optimal in the classical norm of the space L(2) of square integrable functions non - parametrical multidimensional time defined signal regaining (adaptive filtration, noise canceller) on the background noise via multidimensional truncated Legendre expansion and optimal experience design. The two - dimensional case is known as a picture processing, picture analysis or image processing. We offer a two version of an confidence region building, also adaptive. Our estimates proposed by us have successfully passed experimental tests on problem by simulate of modeled with the use of pseudo-random numbers as well as on real data (of seismic signals etc.) for which our estimations of the different signals were compared with classical estimates obtained by the kernel or wavelets estimations method. The precision of proposed here estimations is better. Our adaptive truncation may be used also for the signal and image compression.

💡 Research Summary

The paper addresses the problem of non‑parametric denoising for multidimensional time‑varying signals, where the observed data are modeled as Y(t)=f(t)+η(t) with f(t) a square‑integrable function defined on a d‑dimensional domain (d≥2) and η(t) additive noise. Classical non‑parametric approaches such as kernel smoothing or wavelet thresholding suffer from the curse of dimensionality: the required sample size grows exponentially with d, and computational cost becomes prohibitive. To overcome these limitations, the authors propose a method based on a truncated expansion in multivariate Legendre polynomials, combined with an optimal experimental design for sample placement and an adaptive rule for selecting the truncation order.

Core methodology

1. Multivariate Legendre basis – The Legendre polynomials {P_k(t)} are orthogonal on