Mohammad Sadegh Narimani

arXiv2026

Global minimax risk and acquisition laws for heterogeneous information fusion

We prove an all-allocation global target-risk theorem for independent Gaussian sources that share a scalar nuisance and an unknown contact coordinate. For a fixed immersed nuisance curve with finitely many multiple fibres and pairwise nonparallel branch tangents, squared target risk is comparable to a primary estimation floor plus a target-gap-weighted Gaussian discrimination profile. Independent localization, finite branch selection and target-class refitting give an estimator attaining this comparison across every nonnegative integer allocation. A full-box specialization has positive-definite primary Fisher information everywhere and globally identifies its target, yet models with identical derivatives of every order along a critical hypersurface have different polynomial risk exponents. We construct a finite certified estimator and quantify the numerical accuracy needed to preserve rare branch decisions and acquisition windows. A polynomial tangency family both demonstrates the geometric boundary and quantifies its repair: known auxiliary gain, source contact order and target vanishing order determine a risk law uniform through zero gain. For a shared spherical direction, a separately proved composite-testing result transfers through an unknown contact coordinate using only counted observations. These theorems distinguish the resources needed for local estimation, discrimination between parameter regions and nuisance alignment, and yield acquisition thresholds under explicit scalar-observation costs.

math.ST↗

arXiv2026

Sharp Minimax Limits and Compatibility Spectra for Critical Near-DFT Index-Only Frequency Estimation

An $M$-channel discrete Fourier transform (DFT) channelizer routes an on-grid sinusoid to a single output. When each frame reports only one energy-proportional channel index, signed sub-bin frequency estimation becomes nonregular: dark-channel probability is quadratic in the offset, whereas orientation enters cubically. We study $N$ independent labeled reports under a known uniform-replacement probability $ε_N$ and a single deterministic unitary shared by all frequencies and frames, constrained to routing defect $τ/N$. If $\sqrt{N}ε_N \to λ< \infty$, we establish an attained global-in-frequency minimax limit at the critical scales $N^{-1/4}$ for frequency offset, $N^{-1/2}$ for unitary perturbation and replacement, and $N^{-1}$ for routing defect. The effective unitary tangent is a symmetric complete-graph edge field modulo one centering nuisance, and its first variation is a radius-dependent weighted divergence. For $λ>0$, evaluation at $k$ distinct normalized offset magnitudes yields an exact Fourier-Cauchy nullity spectrum: $\lfloor(M-1)/2\rfloor$ evaluations are necessary and sufficient, for every choice of distinct magnitudes, to certify neutrality at all magnitudes. The terminal nullspace has a greatest-common-divisor dimension formula and positive-definite aggregate curvature. Consequently, exact DFT routing is uniquely minimax within the complete critical tangent class for $M=3$, whereas every positive critical defect budget strictly improves the minimax constant for $M\geq4$. The analysis also yields a smallest-prime curvature-visibility law and, for $M\geq5$, discontinuous compatibility geometry but a continuous minimax value at the zero critical replacement floor $λ=0$.

eess.SP↗