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Title: Multiscale splines and local polynomials Authors:  Maarten Jansen - Université libre de Bruxelles (Belgium) [presenting]
Abstract: The benefits of non-linear estimation of piecewise smooth data through a sparse multiresolution decomposition, such as offered by a wavelet transform, can be incorporated into well-established methods, such as splines and kernel-based approaches, by upgrading the unit scale approach into a multiscale version, thus adding locality in scale (or frequency) to the already existing spatial locality. The combination of splines or kernel-based methods with a multiresolution analysis extends the scope of the latter beyond the dyadic, equispaced setting of most wavelet methods. The construction of the multiscale spline, kernel or local polynomial methods proceeds through the so-called lifting scheme. The understanding of this scheme is crucial for successful applications in nonparametric statistical estimation. In particular, two issues play an important role. On one hand, the smoothness of the reconstruction through the process of multiscale refinement should be monitored. On the other hand, the variance propagation throughout the scheme should be controlled, by looking at the singular value decomposition of the underlying projection matrix.