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A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new "local compression-alignment-modified- prediction (L-CAMP)" methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score < 29. The localness score of the 9/7 is, in that same dimension, > 575 000 000 000.