Minimum-rank parameters of complements of threshold Kneser graphs

What the math gives to ML

The paper gives an exact low-dimensional positive-semidefinite Gram representation for the threshold intersection relation on k-subsets: two sets have a nonzero inner product exactly when their intersection has size at least s. The transferable asset is the combination of exterior-power features, interpolation that annihilates unwanted intersection sizes, and PSD factorization with dimension \(\binom{n-2(k-s)}{s}\), often much smaller than the naive \(\binom{n}{s}\) feature space. This suggests compressed set-valued embeddings or structured attention kernels whose sparsity pattern is determined by overlap while remaining a valid Gram kernel. The first practical test should compare a learned low-rank PSD overlap kernel against the standard s-subset incidence kernel on set retrieval and set-transformer tasks.