Recovery of Functions of Many Variables via Compressive Sensing

Recovery of functions of many variables from sample values usually suffers the curse of dimensionality: The number of required samples scales exponentially with the spatial dimension. In order to avoid this severe bottleneck, one needs to impose further structural properties of the function to be recovered apart from smoothness. Here, we build on ideas from compressive sensing and introduce a function model that involves “sparsity with respect to dimensions” in the Fourier domain. Using recent estimates on the restricted isometry constants of measurement matrices associated to randomly sampled trigonometric systems, we show that the number of required samples scales only logarithmically in the spatial dimension provided the function to be recovered follows the newly introduced highdimensional function model. Keywords— Functions in high dimensions, compressive sensing, sparse Fourier expansions, Fourier algebra, restricted isometry property.