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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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Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-26 · public

Research details

Subjects
Statistics and Data Science · Statistical Theory · Statistical Inference
Keywords
minimax estimation, Gaussian energy, complex Wishart, relative squared error, covariance functionals, permanent
Manuscript date
2026-10-26
Reader reuse
No additional permission granted

Scope

For each fixed nonzero degree-n homogeneous holomorphic energy of a proper complex Gaussian with unknown PSD covariance and positive energy, exact minimax relative squared risk is 1−(N rising n)²/(N rising 2n), including singular covariances.

Disclosures

OpenAI ChatGPT and Codex tools assisted mathematical exploration and derivation, literature discovery, manuscript drafting, verification-code development and checking. The author is responsible for the statements, references, code and conclusions. This research received no external funding. The author declares no competing interests.

Reproduction

The 8-page manuscript and six-member source package contain the complete proofs, editable TeX, identical PDF, citation metadata, README and SHA-256 manifest. Compile main.tex twice with PDFLaTeX. Run the standard-library program Gaussian moment controls.py; it writes its exact finite diagnostic results beside the script. The 21 permanent-coordinate controls check normalization, second moments, rank-one equality and shrinkage risk. The manuscript provides the general tensor and decision-theoretic proofs.

References and related resources

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