Approximation in shift-invariant spaces with deep ReLU neural networks
This work addresses approximation challenges in signal and image processing by providing theoretical guarantees for neural networks, though it is incremental as it builds on existing methods for known bottlenecks.
The paper tackles the problem of approximating functions in dilated shift-invariant spaces using deep ReLU neural networks, deriving error bounds based on network width and depth, and shows that their construction is asymptotically optimal for Sobolev spaces up to a logarithmic factor.
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the $L^p (1\le p \le \infty)$ approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.