Complex generalised weighing matrices in centraliser algebras of monomial representations

What the math gives to ML

The paper studies sparse complex matrices whose nonzero entries are roots of unity and whose rows are exactly orthogonal. After scaling, these matrices are isometries, so they can mix channels without changing activation or gradient norms while using only nw nonzero operations instead of n^2 dense operations. The representation-theoretic centralizer construction is valuable because it can generate structured families of such mixers with compact orbital descriptions rather than storing arbitrary sparse weights. The most direct neural-network test is to replace selected dense channel-mixing layers in an MLP or Transformer with fixed, normalized sparse mixers and measure whether the exact isometry offsets the optimization damage normally caused by unstructured sparsity.