The Grassmannian of indefinite subspaces

What the math gives to ML

The paper supplies a concrete manifold of subspaces whose intrinsic bilinear form has a prescribed mixed signature, represented using ordinary Euclidean projectors and a fixed indefinite metric. This gives neural modules a controlled way to separate positive and negative latent directions while retaining orthonormal, low-dimensional coordinates. The most direct transfer is an indefinite low-rank embedding or adapter: learn an orthonormal basis U, constrain the compressed metric U^*I_{m,n}U to have p positive and q negative eigenvalues, and use the resulting signed inner product in attention, retrieval, or residual modulation. The construction is differentiable and straightforward to compare against ordinary low-rank parameterizations, although the paper itself does not establish task-level machine-learning benefits.