Well-invertible column subsets of sparse matrices are rare

What the math gives to ML

The paper identifies a deterministic failure mode for sparse matrices: tree-like support patterns create explicit near-null vectors, so many column subsets have very small minimum singular value even when the matrix is globally isotropic. The transferable asset is the support-graph certificate, not merely the asymptotic impossibility theorem: a local motif can be converted into a concrete vector witness and a cheap conditioning diagnostic. This suggests designing sparse neural linear layers and routing matrices by suppressing these motifs or increasing row degree when they appear. The most practical first test is a mask regularizer or rewiring rule for sparse MLP or MoE expert matrices, evaluated through Jacobian conditioning and downstream accuracy.