Morphological Representation Theory in the Fourier Inf-Semilattice: Universal Decomposition of Frequency-Domain Deep Learning Operators
arXiv:2608.20399
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper supplies a constructive representation of increasing, upper-semicontinuous, positively homogeneous operators on nonnegative Fourier magnitudes as suprema of max-times erosions. The transferable asset is not the Fourier interpretation alone, but the exact algebraic constraints: monotonicity, scale equivariance, and adjoint erosion-dilation pairs that produce idempotent openings. A practical neural implementation is a spectral morphological mixer whose learned positive kernels are combined by a pointwise maximum of quotient responses, optionally followed by an adjoint opening for stable band-limited feature processing. This gives a falsifiable alternative to unconstrained spectral convolutions or MLPs, with exact positive homogeneity and monotonicity rather than merely encouraging them through regularization.
Ideas from this paper
Unverified
2026
Replace an unconstrained Fourier-domain linear mixer with a bank of positive spectral kernels and a max-times erosion aggregator. For a nonnegative Fourier magnitude f, each kernel produces a quotient response f/psi_k and the layer takes the pointwise supremum over kernels, giving exact positive homogeneity and monotonicity. This is most suitable as a drop-in spectral mixing block in a CNN, vision transformer, or state-space model.
Useful6/10
Difficulty5/10
Novelty7/10
Unverified
2026
Construct a learnable spectral pooling block as an erosion followed by its adjoint dilation, making the resulting opening idempotent, increasing, and anti-extensive. The block can suppress frequencies outside a learned passband while guaranteeing that applying it twice does not continue changing the representation, which is useful in multi-stage CNN pyramids and U-Net skip paths.
Useful5/10
Difficulty6/10
Novelty6/10