DAREK -- Distance Aware Error for Kolmogorov Networks
This work addresses the need for tighter, distance-aware error bounds in KANs for applications like shape estimation from sparse data, representing an incremental improvement over existing methods.
The paper tackles the problem of providing distance-aware error bounds for Kolmogorov Arnold Networks (KANs) by introducing DAREK, which generalizes error bounds from Newton's polynomial to splines and nested compositions. The result shows that DAREK is faster than Monte Carlo approaches and reliably encloses the true obstacle shape when estimating an object's shape from sparse laser scan points.
In this paper, we provide distance-aware error bounds for Kolmogorov Arnold Networks (KANs). We call our new error bounds estimator DAREK -- Distance Aware Error for Kolmogorov networks. Z. Liu et al. provide error bounds, which may be loose, lack distance-awareness, and are defined only up to an unknown constant of proportionality. We review the error bounds for Newton's polynomial, which is then generalized to an arbitrary spline, under Lipschitz continuity assumptions. We then extend these bounds to nested compositions of splines, arriving at error bounds for KANs. We evaluate our method by estimating an object's shape from sparse laser scan points. We use KAN to fit a smooth function to the scans and provide error bounds for the fit. We find that our method is faster than Monte Carlo approaches, and that our error bounds enclose the true obstacle shape reliably.