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  1. Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theoremScott SchubrEdition 3 · 2026-10-27 Public
  2. Finite positivity and correction inertia for the Riemann theta formScott SchubrEdition 2 · 2026-10-27 Public
  3. Minimum-Degree Obstructions in the Riemann Theta Product: Classification, transition bounds and robust Gram certificatesScott SchubrEdition 3 · 2026-10-27 Public
  4. Collision and cancellation spectra for concentrated energy formsScott SchubrEdition 2 · 2026-10-27 Public
  5. Fekete convergence from balanced integer predecessors in shrinking corridorsScott SchubrEdition 2 · 2026-10-27 Public
  6. Sharp trace-degree certificates for rank-two permanent-derivative compensationScott SchubrEdition 2 · 2026-10-27 Public
  7. Residual Certificates and Empirical Sharpness for Entropic Self-Transport ScoresScott SchubrEdition 2 · 2026-10-27 Public
  8. Observation norms under double-exponential confinementScott SchubrEdition 2 · 2026-10-27 Public
  9. Residual Stopping and Pointwise Inference for Nonuniform Entropic Self-TransportScott SchubrEdition 2 · 2026-10-27 Public
  10. Finite-spectral recovery of Weyl-blind twisting modesScott SchubrEdition 2 · 2026-10-27 Public
  11. Local comparison counting for linear extensions and a bounded-range balance refinementScott SchubrEdition 2 · 2026-10-27 Public
  12. An explicit hyperbolicity threshold for xi Jensen polynomialsScott SchubrEdition 2 · 2026-10-27 Public
  13. A quantitative Chollet inequality for rank-four matrices of order sevenScott SchubrEdition 2 · 2026-10-27 Public
  14. Single-copy learning of stabilizers with correlated diagonal noiseScott SchubrEdition 2 · 2026-10-27 Public
  15. Relative finite optimization of arithmetic approximation distancesScott SchubrEdition 2 · 2026-10-27 Public
  16. Growing null subtraction and weighted Weil zero detectionScott SchubrEdition 2 · 2026-10-27 Public
  17. Frequency Localization for Weil Operator CertificatesScott SchubrEdition 3 · 2026-10-27 Public
  18. Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step spaceScott SchubrEdition 2 · 2026-10-27 Public
  19. Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil OperatorsScott SchubrEdition 3 · 2026-10-27 Public
  20. Sharp convergence of heat-kernel Sinkhorn metricsScott SchubrEdition 2 · 2026-10-27 Public
  21. Positive heat-flow observations on spherical shells: cutoffs, optimal preparation and concentrationScott SchubrEdition 2 · 2026-10-27 Public
  22. Curvature classification and certified Sinkhorn metrics for Fourier density modelsScott SchubrEdition 2 · 2026-10-27 Public
  23. Mean ergodic convergence from near-diagonal subadditivityScott SchubrEdition 2 · 2026-10-27 Public
  24. Quantitative Null-Space Compression for a Weighted Weil OperatorScott SchubrEdition 2 · 2026-10-27 Public
PublicEdition 2

Exact minimax estimation of homogeneous complex Gaussian energies

Scott Schubr

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Abstract

Let f be a fixed nonzero homogeneous holomorphic polynomial of degree n on ℂ^d. We estimate its Gaussian energy F_f(A)=E|f(X)|^2 from N independent centered proper complex Gaussian observations with unknown positive semidefinite covariance A. Singular covariances are included and F_f(A)>0. Writing W=Σ_{s=1}^N X_sX_s^*, a=(N rising n) and b=(N rising2n), we prove that the exact minimax risk under relative squared error is 1-a^2/b, attained by the estimator (a/b)F_f(W). The unbiased estimator F_f(W)/a has exact worst-case relative variance b/a^2-1, with equality at rank-one covariances. A classical complex Wishart moment operator supplies the uniform upper bound, while a scalar scale subexperiment and proper inverse-Gamma priors give the matching lower bound against all estimators. Permanent estimation is a special case. Uniform relative-error estimation has observation complexity of order n^2ε^(-2)log(1/δ), with computation unrestricted.

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