{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace an unconstrained low-rank adapter or similarity projection with a learned subspace carrying a prescribed signed metric. The module learns an orthonormal basis U for a k=p+q dimensional subspace, forces the compressed form U^*I_{m,n}U to have p positive and q negative eigenvalues, and uses the resulting pseudo-inner product for signed attention or retrieval scores.",
 "formulas_latex": [
  "$$\\Iso_d(\\mathbb{F})=\\{Q\\in\\mathbb{F}^{d\\times d}:Q^*Q=I\\},\\qquad P_{\\mathbb{W}}=UU^*,\\qquad U^*U=I_k.$$",
  "$$I_{m,n}=\\operatorname{diag}(I_m,-I_n),\\qquad B=U^*I_{m,n}U.$$",
  "$$\\mathbb{W}\\in\\operatorname{Gr}_{p,q}(\\mathbb{F}^{m+n})\\quad\\Longleftrightarrow\\quad \\mathsf{s}(P_{\\mathbb{W}}I_{m,n}P_{\\mathbb{W}})=(p,q,d-k),\\qquad k=p+q.$$",
  "$$\\operatorname{score}_U(x,y)=x^*U(B+\\varepsilon I_k)^{-1}U^*y,\\qquad \\mathcal{L}_{\\mathrm{sig}}=\\sum_{i=1}^{p}[\\max(0,\\delta-\\lambda_i(B))]^2+\\sum_{i=p+1}^{k}[\\max(0,\\delta+\\lambda_i(B))]^2.$$"
 ],
 "id": 2876,
 "implementation": "Integrate this as a replacement for a low-rank similarity head, a LoRA-style adapter, or an attention head's query-key metric. Choose ambient feature width d=m+n and latent width k=p+q, with p positive and q negative coordinates. Store an unconstrained parameter A in R^{d x k}; at every forward pass compute a differentiable thin QR factorization A=UR, retaining U=Q[:, :k]. Form I=diag(+1 repeated m times, -1 repeated n times) and compute B=U^T I U. For query and key vectors x and y, compute z_x=U^T x and z_y=U^T y, then use score=z_x^T(B+epsilon I_k)^{-1}z_y. In attention, divide this score by the normal temperature before softmax. The training loop is: compute U by QR; compute B; eigendecompose B=V diag(lambda) V^T; evaluate task loss; add lambda_sig times L_sig; backpropagate into A. Sort eigenvalues descending, require the first p to exceed margin delta and the final q to be below -delta. The paper contributes the projector and signature characterization; QR, inverse regularization, eigenvalue margins, and the attention integration are the adaptation. Compute B and its eigenvalues exactly per minibatch, while monitoring the minimum absolute eigenvalue as a conditioning diagnostic. First test a small Transformer classifier on CIFAR-10 patch tokens or AG News, comparing ordinary LoRA, an unconstrained low-rank metric, and this adapter at equal rank and parameter count. Measure validation accuracy, training loss at equal FLOPs, gradient norms, logit variance, and eigenvalue-sign violations. Success means higher accuracy or retrieval separation at equal parameter count without eigenvalue collapse or attention-logit explosion.",
 "math_summary": "Let I_{m,n}=\\operatorname{diag}(I_m,-I_n) be the fixed nondegenerate bilinear-form matrix on \\mathbb{F}^{m+n}, where \\mathbb{F}\\in\\{\\mathbb{R},\\mathbb{C},\\mathbb{H}\\}. Let U\\in\\mathbb{F}^{(m+n)\\times k} have Euclidean-orthonormal columns, U^*U=I_k, and let \\mathbb{W}=\\operatorname{col}(U). The paper characterizes the indefinite Grassmannian by the signature of the compressed form B=U^*I_{m,n}U: \\mathbb{W}\\in\\operatorname{Gr}_{p,q}(\\mathbb{F}^{m+n}) if and only if \\mathsf{s}(P_{\\mathbb{W}}I_{m,n}P_{\\mathbb{W}})=(p,q,d-k), equivalently B has p positive and q negative eigenvalues on the k-dimensional subspace, with k=p+q. Here P_{\\mathbb{W}}=UU^* is the Euclidean orthogonal projector and \\mathsf{s} counts positive, negative, and zero eigenvalues. The implementation uses the nondegenerate compressed matrix B to define a signed latent metric and a margin penalty that keeps its eigenvalues away from zero with the desired signs.",
 "math_tags": [
  "geometry",
  "linear-algebra",
  "lie-groups"
 ],
 "ml_areas": [
  "embedding",
  "attention",
  "fine-tuning"
 ],
 "paper": {
  "arxiv_id": "2608.30249",
  "arxiv_url": "https://arxiv.org/abs/2608.30249",
  "summary_what_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.",
  "title": "The Grassmannian of indefinite subspaces",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "memory"
 ],
 "title": "Indefinite Grassmann Adapter",
 "url": "https://synthcore.org/idea/2876/indefinite-grassmann-adapter",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}