Operator-norm characterization for strided and dilated convolution

Extend the operator-norm characterization of Muon-C to strided and dilated convolution.

Background

The paper develops an operator-aligned Fourier geometry and a critical-grid norm characterization for finite convolutional kernels, with exact oracle and approximation guarantees for the stated convolution setting. It notes that the same kernel-grid update can be applied to strided convolution because the stored kernel support is unchanged, but the theoretical operator interpretation does not yet incorporate subsampling.

The open problem is to derive an operator-norm formulation and corresponding characterization of the Muon-C geometry for strided and dilated convolution, accounting explicitly for the effects of subsampling and dilation rather than relying only on the finite kernel-support update.

References

The same kernel-grid update can also be applied to a strided convolution because its stored $k_h\times k_w$ kernel support is unchanged, although an operator-norm characterization that explicitly incorporates subsampling remains open.

— Muon-C: Operator-Aligned Muon for Convolutional Kernels  (2609.09676 - Qing et al., 9 Sep 2026) in Section 4.4, subsection Practical Muon-C Algorithm; Section Discussion