Wavelet and tent space characterizations of $h^p(\mathbb{R}^n)$, $0 < p \le 1$, with exact $L^2$ convergence, and the maximal class of admissible functions for Goldberg-type splittings

What the math gives to ML

The paper supplies a multiscale quasi-Banach sparsity structure for wavelet coefficient pairs, rather than an ordinary coefficient-wise l_p penalty. Its local tent-space square function groups coarse father coefficients with fine mother coefficients according to spatial dyadic regions, while the atomic theorem gives an equivalent decomposition into localized atoms with l_p-summable weights. This can be transferred into neural networks as a structured, spatially localized sparsifier for wavelet or multiresolution feature maps, encouraging entire localized feature packets to disappear. The main caveat is that p<1 is nonconvex, so an iteratively reweighted implementation should be compared with group-lasso and coefficient-wise l_1 baselines.