Core-KAN: Continuous Vision Kernels with Kolmogorov-Arnold Networks

Conventional convolutional kernels are typically defined on fixed discrete grids, limiting their ability to accommodate heterogeneous local structures. Existing adaptive operators improve flexibility but often couple geometric scale variation with content-dependent filtering, while incurring high computational cost from per-location kernel generation. To decouple geometric scale adaptation from content-dependent filtering while avoiding expensive per-location kernel generation, we propose Continuous Relative-scale KAN (Core-KAN), a relative-scale-conditioned continuous convolution operator. Core-KAN maps input features into a compact latent basis space and uses a lightweight scale controller to predict local scales relative to an exponential moving average reference. A KAN-based generator represents depth-wise kernel bases as continuous coordinate functions, allowing the operator to synthesize spatial filters at arbitrary resolutions rather than being confined to a fixed lattice. Instead of synthesizing independent kernels at every location, it constructs a compact bank of scale-conditioned kernel responses and interpolates them according to the predicted local scale map. An independent mixing controller further combines the interpolated basis responses based on local content, explicitly decoupling geometric scale adaptation from content-dependent filtering. Together with lightweight pointwise projections, this design forms a low-rank dynamic convolution that scales efficiently with kernel size and integrates readily into hierarchical vision backbones. Experiments across three representative vision tasks show Core-KAN consistently outperforms strong convolutional and dynamic-kernel baselines with only marginal parameter and computational overhead, offering an efficient, general framework for continuous, scale-adaptive convolution.