Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at Irregularly Spaced Data
This provides fundamental theoretical limits on interpolation capabilities of deep ReLU networks for irregular data, with implications for approximation theory and neural network design.
The paper establishes that deep ReLU neural networks require at least Ω(N) parameters to interpolate N irregularly spaced datapoints when the separation distance δ is exponentially small in N, matching the known upper bound of O(N) parameters. This result also shows that bit-extraction techniques for VC dimension bounds fail for irregular data and yields a lower bound on approximation rates for Sobolev spaces.
We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at $N$ datapoints in the unit ball which are separated by a distance $δ$. We show that $Ω(N)$ parameters are required in the regime where $δ$ is exponentially small in $N$, which gives the sharp result in this regime since $O(N)$ parameters are always sufficient. This also shows that the bit-extraction technique used to prove lower bounds on the VC dimension cannot be applied to irregularly spaced datapoints. Finally, as an application we give a lower bound on the approximation rates that deep ReLU neural networks can achieve for Sobolev spaces at the embedding endpoint.