Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs

摘要: Recent work by some of the authors presented a novel construction multiresolution analysis on manifolds and graphs, acted upon given symmetric Markov semigroup {T t } t≥0 , for which T has low rank large t.1 This includes important classes diffusion-like operators, in any dimension, manifolds, nonhomogeneous media. The dyadic powers an operator are used to induce analysis, analogous classical Littlewood-Paley 14 wavelet theory, while associated packets can also be constructed. 2 extends multiscale function signal processing class spaces, such as with efficient algorithms. Powers functions (notably its Green's function) efficiently computed, represented compressed. is related generalizes certain Fast Multipole Methods, 3 representation Calderon-Zygmund pseudo-differential 4 relates algebraic multigrid techniques. 5 original diffusion yields orthonormal bases spaces {V j }. orthogonality requirement advantages from numerical perspective, but several drawbacks terms space frequency localization basis functions. Here we show how relax this order construct biorthogonal scaling wavelets. more compact representations operator, better localized new applies non self-adjoint semigroups, arising many applications.