The Fast Tree Transform is a locally-aggregated transform defined for arbitrary graph geometries that generalizes both the Hadamard frequency transform and the Haar wavelet transform on the circle. The pure frequency part of the transform approximately diagonalizes the graph Laplacian; this fact makes it useful as a preconditioner in geometric optimization problems. The full transform provides an only slightly-redundant representation of signals on the graph that allows sparse representation of signals that include both edges and textures.

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