Do log factors matter? On optimal wavelet approximation and the foundations of compressed sensing
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A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements m for the approximation of piecewise α-Hölder functions. In this setting, we show the suboptimality of random Gaussian sampling, exhibit a provably optimal sampling strategy and prove that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent α is large enough. This resolves the claimed paradox, and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.
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