{
 "artifacts": null,
 "category": "geometry",
 "description": "Replace ordinary per-coordinate activation normalization or covariance whitening with a learned or frozen affine map based on a minimum-volume ellipsoid calibrated to the activation convex hull. Estimate the hull's Minkowski symmetry and use the bound \\(\\rho=\\sqrt{d/\\alpha}\\) to set a principled clipping radius, where \\(d\\) is feature dimension and \\(\\alpha\\) is the measured symmetry. This should reduce worst-case clipping of asymmetric activation distributions without requiring a much larger quantization range.",
 "formulas_latex": [
  "$$\\operatorname{sym}(S):=\\max\\{\\alpha\\geq 0:\\exists c\\in S\\text{ such that }-\\alpha(S-c)\\subseteq S-c\\}.$$",
  "$$E\\subseteq S-c\\subseteq\\sqrt{\\frac{d}{\\operatorname{sym}(S)}}\\,E.$$",
  "$$E=\\{Au:\\|u\\|_2\\leq1\\}=\\{z:z^TQ^{-1}z\\leq1\\},\\qquad Q=AA^T\\succ0.$$",
  "$$\\alpha^*=\\max_{\\alpha,c,\\lambda}\\ \\alpha\\quad\\text{s.t.}\\quad -\\alpha(x_j-c)=\\sum_{k=1}^{m}\\lambda_{jk}(x_k-c),\\ \\lambda_{jk}\\geq0,\\ \\sum_{k=1}^{m}\\lambda_{jk}=1\\quad(j=1,\\ldots,m).$$"
 ],
 "id": 2908,
 "implementation": "Integrate this at the output of a chosen MLP or attention block, before quantization, residual addition, or a sensitive nonlinear activation. Collect an activation calibration matrix X of shape m by d from 1,000-10,000 representative training examples; use a random subsample or k-means representatives when m is large. First estimate the Minkowski center and symmetry. For each candidate center c and scalar alpha, solve the finite convex-hull membership constraints in the displayed formula using simplex variables lambda[j,k]; in an MVP, replace the full pairwise formulation with a projected-gradient search over c and alpha, checking membership with a linear program. Set rho=sqrt(d/max(alpha,epsilon)). Next solve a minimum-volume or regularized ellipsoid fit for Q: minimize -logdet(Q) plus a small ridge penalty, subject to (x[j]-c)^T Q^{-1}(x[j]-c)\u003c=rho^2. Parameterize Q=LL^T and optimize the equivalent Cholesky variables if a generic SDP solver is unavailable. Store c and L^{-1}; at inference compute y=L^{-1}(x-c), clip y to the ball of radius rho (or coordinate-clip after rotation), then invert the transform if the following layer expects the original scale. The paper supplies the geometric guarantee; the engineer estimates alpha, c, Q, and the empirical coverage fraction from finite activations. Start with a 2-layer MLP on CIFAR-10 or a small Transformer on WikiText-2, comparing LayerNorm, diagonal min-max clipping, and ordinary covariance whitening. Measure validation accuracy or perplexity, fraction of clipped activations, INT8 calibration error, and added calibration/runtime cost. Success is lower clipping or quantization error at the same dynamic range, or equal error with a smaller range and at least 10% lower activation memory.",
 "math_summary": "For a full-dimensional convex body \\(S\\subset\\mathbb{R}^{d}\\), the paper defines the Minkowski symmetry \\(\\alpha=\\operatorname{sym}(S):=\\max\\{a\\geq0:\\exists c\\in S\\text{ such that }-a(S-c)\\subseteq S-c\\}\\); any maximizing \\(c\\) is a Minkowski center. Its main theorem guarantees an origin-centered ellipsoid \\(E\\) and center \\(c\\) satisfying \\(E\\subseteq S-c\\subseteq\\rho E\\), with \\(\\rho=\\sqrt{d/\\alpha}\\). Represent \\(E\\) as \\(E=\\{Au:\\|u\\|_2\\leq1\\}\\), equivalently \\(E=\\{z:z^TQ^{-1}z\\leq1\\}\\) for \\(Q=AA^T\\succ0\\). For a sampled activation set \\(X=\\{x_j\\}_{j=1}^m\\), use its convex hull \\(S=\\operatorname{conv}(X)\\) as a finite approximation. Estimate \\(\\alpha\\) and \\(c\\) by maximizing \\(a\\) subject to \\(-a(x_j-c)\\in\\operatorname{conv}(X)\\) for every sample \\(j\\); each membership condition can be represented with simplex variables \\(\\lambda_{jk}\\geq0\\), \\(\\sum_k\\lambda_{jk}=1\\), and \\(-a(x_j-c)=\\sum_k\\lambda_{jk}(x_k-c)\\). Then solve an ellipsoid program for \\(Q\\), using the paper's containment relation as the target: the practical outer constraints are \\((x_j-c)^TQ^{-1}(x_j-c)\\leq\\rho^2\\) for all samples, while inner containment can be enforced approximately by support-point or halfspace constraints. The affine normalization is \\(y=A^{-1}(x-c)\\), for which the calibrated region is contained in the Euclidean ball of radius \\(\\rho\\).",
 "math_tags": [
  "convex-analysis",
  "geometry",
  "optimization",
  "linear-algebra"
 ],
 "ml_areas": [
  "mlp",
  "transformer",
  "quantization",
  "initialization"
 ],
 "paper": {
  "arxiv_id": "2608.30876",
  "arxiv_url": "https://arxiv.org/abs/2608.30876",
  "summary_what_math_gives_to_ml": "The paper gives a constructive, symmetry-sensitive preconditioning principle: any full-dimensional convex data region can be enclosed between an ellipsoid and a dilation whose factor is only \\(\\sqrt{n/\\operatorname{sym}(S)}\\). The transferable asset is not merely ellipsoidal whitening, which is standard, but estimating the data region's Minkowski symmetry and using it to obtain a provably less conservative bound for anisotropic normalization, clipping, or robust optimization. A practical neural-network use is to compute this ellipsoid on activation clouds, transform activations into approximately isotropic coordinates, and use the symmetry-dependent radius for clipping or quantization. The main experiment should compare this adaptive ellipsoidal calibration against diagonal normalization and ordinary full-covariance whitening at equal calibration and inference cost.",
  "title": "Symmetry-dependence in Rounding of a Convex Body",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 4,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "accuracy",
  "memory"
 ],
 "title": "Minkowski-Symmetry Activation Ellipsoid",
 "url": "https://synthcore.org/idea/2908/minkowski-symmetry-activation-ellipsoid",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}