BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Modulation Spaces and the Curse of Dimensionality

R. Parhi, M. Unser

Proceedings of the Fourteenth International Workshop on Sampling Theory and Applications (SampTA'23), Yale NH, USA, July 10-14, 2023.

We investigate the L2-error of approximating functions in the modulation spaces Ms1,1(ℝd), s ≥ 0, by linear combinations of Wilson bases elements. We analyze a nonlinear method for approximating functions in Ms1,1(ℝd) with N-terms from a Wilson basis. Its L2-approximation error decays at a rate of N− ½ − s ∕ 2d. We show that this rate is optimal by proving a matching lower bound. Remarkably, these rates do not grow with the input dimension d. Finally, we show that the best linear L2-approximation error cannot decay faster than N−s ∕ 2d. This shows that linear methods, contrary to the nonlinear ones, necessarily suffer the curse of dimensionality in these spaces. 

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AUTHOR="Parhi, R. and Unser, M.",
TITLE="Modulation Spaces and the Curse of Dimensionality",
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