Modulation Spaces and the Curse of Dimensionality

Rahul Parhi, Michael Unser

Abstract: We investigate the $L^2$-error of approximating functions in the modulation spaces $M^s_{1,1}(\mathbb{R}^d)$, $s \geq 0$, by linear combinations of Wilson bases elements. We analyze a nonlinear method for approximating functions in $M^s_{1,1}(\mathbb{R}^d)$ with $N$-terms from a Wilson basis. Its $L^2$-approximation error decays at a rate of $N^{-\frac{1}{2} - \frac{s}{2d}}$. 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 $L^2$-approximation error cannot decay faster than $N^{-\frac{s}{2d}}$. This shows that linear methods, contrary to the nonlinear ones, necessarily suffer the curse of dimensionality in these spaces.

Submission Type: Full Paper