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Compressed Single-Tone Frequency Estimation With Unknown Complex Gain: Singular Rates and Global Identifiability

Armon Rasooli

arXiv:2608.30623v1eess.SP

TL;DR

The paper asks when fixed compression of a single complex tone preserves useful frequency information without losing identifiability or regularity under unknown gain. It analyzes dark-response contact locally and optimizes whitened row spaces globally, obtaining sharp singular laws and exact even-rank design results. The findings distinguish local Fisher preservation, singular recoverability, and global identifiability as separate compression properties.

  • Problem

    Fixed compression can preserve local Fisher information yet create global aliases, while dark responses can make local frequency estimation nonregular.

  • Method

    The paper combines analytic contact analysis near dark frequencies with worst-frequency efficient-information optimization over whitened row-space projectors.

  • Results

    Finite contact yields local quotient identifiability and consistency with a rate set by radial plus contact orders; even-rank designs are solved exactly, rank two has a positive identification price, and regular-GI suprema agree from rank three onward.

  • Takeaways & Limitations

    Local information preservation, singular recoverability, and global identifiability obey distinct compression laws within the same single-tone experiment.

Abstract

from arXiv · show

Fixed linear compression can preserve local Fisher information for a sinusoid yet destroy global frequency identification, while a dark response can make local estimation nonregular. We study both failures for a single complex tone observed through a fixed complex-linear sketch with unknown nonzero complex gain and pre-sketch white Gaussian noise. Near an isolated analytic dark frequency, we separate radial signal vanishing from optimized projective contact between the two signed frequency branches. When the gain magnitude is constrained to a fixed nondegenerate interval, finite contact is equivalent to local quotient identifiability and minimax consistency. The sharp mean-square-error rate is determined by the sum of the radial and contact orders; infinite contact produces exact local aliases. Globally, we formulate a worst-frequency efficientinformation objective on whitened row spaces. We solve it exactly for every even output rank, prove uniqueness and quantitative rigidity of the symmetric-edge projector, and solve the evenaperture co-rank-one case. The rank-two optimum is aliased, and a winding obstruction gives a positive lower bound on the information price of global identification. From three outputs onward, sketches that are globally identifying with an immersive projective response are open and dense and have zero price at the level of suprema; for every even rank of at least four, the exact optimizer has this property. Thus local information preservation, singular recoverability, and global identifiability obey distinct compression laws within one estimation model.

I. INTRODUCTION

The paper studies how fixed compression can preserve local Fisher information while failing at global frequency identification or becoming nonregular near dark responses. It develops a unified quotient-geometry framework and quantitative design results under unknown complex gain.

  • Motivation: Fixed compression has two distinct failures: dark responses can cause nonregular local rates, while proportional responses at separated frequencies create global aliases.Both failures arise because the unknown gain makes only the compressed complex line physically observable.
  • Motivation: Fisher preservation is insufficient for global identification, because compressed likelihoods may retain degeneracies despite matching local information.Prior score-based and projector-based results motivate the paper’s quantitative harmonic design problem.
  • Contributions: The paper proves finite/infinite analytic-contact laws, exact even-rank information designs, a positive rank-two identification price floor, and regular-GI attainment for proved even-rank families.It also characterizes the zero set of the uniform information functional and lossless ranks.
  • Experiment and geometry: The experiment uses a fixed complex-linear sketch with pre-sketch white Gaussian noise and arbitrary invertible output calibration, reducing row-space behavior to a rank-m orthoprojector P.The resulting theorem does not cover passive hardware or post-combiner noise.
  • Experiment and geometry: After profiling the unknown gain, nonzero responses are represented projectively, while projectivization fails exactly when the compressed response is dark.The efficient-score interpretation applies only away from darkness and after fixing the probability model and covariance.

III. DARK RESPONSES: CONTACT AND MINIMAX RATE

Near an isolated dark frequency, the paper separates radial signal vanishing from projective contact between the signed frequency branches. Their interaction determines local identifiability, consistency, and the singular estimation rate.

  • Dark-response factorization: The whitened analytic response factors into a radial term and a nonvanishing analytic factor, with radial order r measuring how rapidly signal magnitude disappears.The order is invariant under local analytic source reparameterization and constant invertible output maps.
  • Projective contact: The contact order c measures how rapidly positive and negative frequency branches become indistinguishable as complex lines after gain profiling.Finite contact and eventual-zero contact are the two analytic alternatives.
  • Identifiability and rate: Finite opposite-projective contact is equivalent to local quotient identifiability and consistency, while infinite contact creates exact local aliases.Neither radial order nor projective contact alone determines the local behavior.

A. Why the Two Orders Add

Near a dark frequency, radial signal loss and opposite-branch projective contact combine to determine distinguishability. Under a fixed strict gain annulus, finite contact yields local identifiability and consistency, while infinite contact yields exact aliases.

  • Order decomposition: The separation scale is ρ^(r+c): radial vanishing contributes r, while gain-profiled projective contact contributes c.Theorem 1 rules out faster decay from imbalanced, same-side, or gain-boundary pairs.
  • Order decomposition: The first visible projective jet does not alone determine the rate: q_j(t) has projective order two but contact c=2j+1.This family shows that higher opposite-branch contact can dominate the singular exponent.
  • Identifiability: Finite contact c<∞ is equivalent to local quotient identifiability and signed minimax consistency under the fixed strict gain annulus.The equivalence holds on every sufficiently small signed interval.
  • Aliases: Infinite contact c=∞ produces admissible exact local aliases and a minimax risk bounded away from zero on every fixed sufficiently small interval.The positive risk floor need not be uniform as the neighborhood shrinks.
  • Scope: The theorem’s uniformity is conditional: constants are not asserted uniform as the gain annulus collapses or the neighborhood changes.The annulus is load-bearing; fixed gain magnitude can produce a different exponent.
  • Rates: The sharp mean-square-error rate is R^−1/[2(r+c)] when contact is finite.Localization and root-mean-square error follow the same exponent, with p=r+c.

IV. UNIFORM INFORMATION AND THE LOCAL DISCRIMINANT

The paper replaces pointwise information at dark responses with a continuous worst-frequency design functional. This classifies sketches into local degeneracy, globally aliased designs, and regular globally identifying designs.

  • Local discriminant: At a non-dark response, profiled information is positive precisely when the projected curve is projectively regular.Darkness requires a separate limiting analysis and can leave the frequency infimum unattained.
  • Design classes: Every design belongs to exactly one class: F=0 local degeneracy, F>0 with a global alias, or regular-GI.The functional F is the global classification criterion.
  • Dark responses: At a dark germ, the residual tends to zero after profiling, so F=0 even when the algebraic pointwise information value is positive.For non-dark immersions, the profile remains continuously positive on the compact frequency circle.

V. EXACT SPECTRAL DESIGN

The worst-frequency efficient-information problem is solved through spectral extremality of whitened row-space projectors. Symmetric-edge projectors are uniquely optimal for every even rank, with explicit global-identification behavior at boundary ranks.

  • Even-rank extremality: For every even output rank 2≤m≤L, the unique maximizing row-space projector is the symmetric-edge projector Q_L,m.The proof uses a trace ceiling and Ky Fan’s principle to select the top symmetric spectral pairs.
  • Rigidity: For m<L, every competing rank-m projector obeys a quantitative rigidity bound relative to Q_L,m.Equality is unique at the row-space projector level, up to invertible output-coordinate changes.
  • Global identification: For every even m≥4, Q_L,m is regular-GI and uniquely attains the global-identification optimum.Coordinates 0 and 1 remain, whose ratio e^iω certifies non-darkness, injectivity, and immersion.
  • Co-rank one: If L is even and m=L−1, exactly two central-coordinate deletions maximize, and both are regular-GI.Every nontrivial mixture of the two central kernel vectors is strictly worse.
  • Lossless designs: The lossless condition is Γ_L,m=1 exactly when m=L or L is odd and m=L−1.For odd L≥5, the proper lossless row space is the unique central deletion; L=3 is the aliased exception.
  • Boundary cases: Rank one supplies neither immersion nor global identification, while full rank uniquely gives P=I and F=1.For even L, no proper lossless rank exists; the nearest rank is the solved co-rank-one case.

VI. GLOBAL-IDENTIFICATION PRICE AND OUTPUT RANK

The global analysis characterizes how output rank affects efficient information and global identification, including an aliased rank-two optimum and a positive identification price floor.

  • The endpoint plane is uniquely optimal but aliased for every L ≥3, whereas an adjacent-coordinate plane has F = 1/(2SL) > 0.
  • The global-identification price equality holds as an equality of suprema, with exact attainment established for several even-rank and co-rank-one families.
  • Rank-two designs contain a winding-protected non-GI chamber, yielding a positive lower bound on the information price of global identification.
  • For m ≥3, compactifying physical frequency pairs into secants and tangents reduces bad designs to Schubert-incidence conditions.
  • At m = 3, the projected bad set has empty interior, while m = 2 admits an open winding-protected bad chamber.

VII. DISCUSSION AND LIMITS

The discussion separates output-rank landmarks and clarifies when globally identifying designs can approach or attain the information optimum, while identifying unresolved optimization problems.

  • Two outputs can identify frequency, yet a neighborhood of the information optimum cannot; from three outputs, the regular-GI constraint is free at the level of suprema.
  • Within the even-rank family, adding the next symmetric output pair has an exact marginal value.
  • For every η > 0, some nonempty open set of regular-GI designs satisfies F > ΓL,m −η, although a maximizer need not lie in that set.
  • Open problems include the exact rank-two GI-constrained optimum, proper residual odd ranks, singular local limits, exact minimax constants, and collapsing gain annuli.

VIII. CONCLUSION

The paper concludes that local information preservation, singular recoverability, and global identifiability follow distinct compression laws in the same unknown-gain estimation model.

  • Unknown-gain compression separates local singular collision, governed by radial vanishing and projective contact, from global aliasing, governed by optimizer geometry and projective incidence.

SUPPLEMENTARY MATERIAL

The supplement supplies analytic, subanalytic, geometric, and spectral tools supporting the local phase law and full-pair exponent results.

  • The supplement states that proper subanalytic minimization, Puiseux expansions, definable selection, Ky Fan’s principle, Rouché’s theorem, and semialgebraic dimension are the imported tools.
  • Infinite contact produces admissible exact opposite-branch mean collisions near the dark frequency.
  • Finite contact exponents are at least k0; odd k0 gives exponent k0 on every comparison cone, while even k0 permits higher-order contact only near s/t →1.
  • Under finite contact and a strict gain annulus, the full-pair exponent is p = r + c, with ∆A(ρ) = CAρp+o(ρp).The fixed-magnitude witness instead has exponent r + 3 rather than r + 5.
  • Uniformity follows by partitioning local pairs into exhaustive cases and taking the minimum of finitely many positive constants.
  • The example h(t) = tr(1, t2 + t3, t5)T attains strict-annulus order r + 5, while fixed magnitude attains r + 3.

SII. LOCAL SIGNED MINIMAX CONVERSION

Finite contact yields the sharp local signed minimax risk rate Θ(R^-1/p), while infinite contact prevents consistency. The conversion is established for all sufficiently large sample sizes through matching lower and upper bounds.

  • Risk conversion: Θ(R^-1/p) is the local signed minimax mean-square risk when finite p = r + c.The rate holds for every sufficiently large integer R under the fixed-neighborhood whitened experiment.
  • Risk conversion: Infinite contact prevents consistent frequency estimation.Exact local aliases produce parameter points with the same law but positive signed separation.
  • Lower bound: A two-point construction bounds the R-sample divergence and converts total variation into a squared-error lower bound.The midpoint estimator attains equality for the corresponding two-point subproblem.
  • Upper bound: The sufficient sample mean supplies the upper-bound estimator through a Borel-selected profile minimizer.The all-scale profile makes the resulting statement valid for all sufficiently large R, not merely a subsequence.

SIV. WINDING, INCIDENCE, AND CONTINUITY

The global analysis combines a winding obstruction, compactified incidence geometry, and continuity of the design objective. It shows a positive rank-two identification price, generic regular global identifiability from three outputs, and equality of constrained and unconstrained suprema.

  • Winding obstruction: Winding L−1 ≥ 2 rules out an injective projective loop in C×, creating a non-GI rank-two chamber and a positive information-price floor.The obstruction applies inside a strict principal-angle chamber; the resulting bound is a sufficient price floor, not the exact constrained optimum.
  • Incidence geometry: For 3 ≤ m < L, regular-GI projectors form an open dense subset of GrC(m, L).The incidence analysis uses the physical two-real-dimensional frequency-pair parameterization and compactified bad sets.
  • Incidence geometry: Regular-GI is exactly the complement of the compact projected bad set defined by darkness, projective criticality, and global aliases.The compactified wedge formulation identifies the relevant joint design–pair closure.
  • Continuity and attainment: Continuity of F on the Grassmannian makes regular-GI density imply equality of constrained and unconstrained suprema for m ≥ 3.A maximizer can be approximated by regular-GI projectors without assuming that every rank has an attaining regular-GI optimizer.
  • Continuity and attainment: Exact attainment is established for the proved design families, while no general odd-rank attaining projector is inferred.The limitation remains for unresolved proper odd ranks.
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