Rank deficiency of Bernoulli random matrices for growing corank

What the math gives to ML

The paper identifies an explicit exponential law for large corank in i.i.d. Bernoulli matrices: for k=o(\sqrt{\log n}), rank deficiency is asymptotically dominated by sparse structural events such as zero columns, with probability approximately (1-p)^{kn}. This gives a quantitative design rule for binary or highly sparse neural linear layers, where dead input channels and duplicated linear dependencies can destroy information flow even when individual weights appear random. The most direct transfer is a rank-safe initialization and layer-width/sparsity controller that uses the bound to set p, resamples pathological matrices, and empirically verifies numerical rank rather than relying on dense Gaussian initialization.