A characterization of idempotent Schur multipliers

What the math gives to ML

The paper gives a structural decomposition of Boolean interaction patterns: any Boolean matrix with bounded Schur-multiplier, equivalently factorization, norm can be expressed as a signed sum of a bounded number of blocky masks. A blocky mask consists of disjoint complete bipartite rectangles, with each row and column participating in at most one rectangle, so multiplying by it can be implemented through grouped reductions rather than individual pairwise interactions. The transferable opportunity is a structured sparse or grouped-attention operator whose affinity matrix is represented by a small signed collection of such masks. The main practical risk is that the theorem only guarantees an exponential bound in the norm and the supplied material does not provide a decomposition algorithm, so experiments should first target small masks and measure whether useful low-complexity decompositions occur in learned or data-derived patterns.