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arxiv: 2604.08949 · v3 · submitted 2026-04-10 · 💻 cs.IT · math.IT· math.PR

From Distance to Angle: One-Shot Detection Under Isotropic Multivariate Cauchy Noise

Yen-Chi Lee This is my paper

Pith reviewed 2026-05-10 17:49 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PR
keywords Cauchy noisesymbol error probabilityVoronoi regionsasymptotic analysisconstellation geometryheavy-tailed noiseone-shot detectionrecession cone
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The pith

Under isotropic Cauchy noise, symbol error bounds depend on reciprocal distances with long-range geometry in small noise but converge to angular measures of Voronoi cones in large noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes one-shot detection of finite constellations when additive noise is isotropic multivariate Cauchy. Maximum-likelihood decisions produce the same Euclidean Voronoi regions as under Gaussian noise, so the difference appears only in how probability mass falls inside those fixed regions. In the small-noise regime an upper bound on symbol error probability is obtained from a reciprocal distance spectrum; this bound continues to sense distant constellation points, unlike the nearest-neighbor dominance seen under Gaussian noise. In the large-noise regime the probability of correct detection approaches a limit fixed solely by the solid angle of the recession cone belonging to each Voronoi cell. The work therefore exhibits a clean regime-dependent switch from distance-based to angle-based reliability descriptors.

Core claim

For finite constellations observed in isotropic multivariate Cauchy noise the symbol error probability admits a reciprocal distance-spectrum upper bound in the small-noise regime that retains dependence on the global geometry, while the correct-decision probability converges in the large-noise regime to a quantity determined only by the angular measure of the associated Voronoi recession cone.

What carries the argument

The reciprocal distance-spectrum upper bound on symbol error probability together with the angular measure of each Voronoi recession cone.

If this is right

  • Constellation design under average power must penalize both nearest-neighbor distances and farther points when noise is light.
  • Under heavy noise only the angular width of each decision cone survives as a reliability metric, so scaling the entire constellation has no effect.
  • The descriptors supply a lightweight screening rule that flags geometric collapse in planar constellations before full simulation.
  • The transition between distance and angle regimes occurs at a finite noise level that can be located numerically for any given constellation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Heavy-tailed noise may make error floors in communication systems more sensitive to the outer geometry of the constellation than Gaussian analyses predict.
  • The same angular-cone limit could be tested for other spherically symmetric heavy-tailed distributions whose characteristic functions decay slowly.
  • Practical receivers operating in mixed noise environments could switch between distance-spectrum and angular-cone metrics according to measured noise strength.

Load-bearing premise

The noise must be exactly isotropic multivariate Cauchy and the analysis applies only to the asymptotic small-noise and large-noise regimes for finite constellations.

What would settle it

Direct Monte-Carlo measurement of symbol error rates for any fixed finite constellation at noise variances that are either very small or very large, compared against the explicit reciprocal-distance bound or the explicit angular-cone limit, would confirm or refute the predicted asymptotics.

Figures

Figures reproduced from arXiv: 2604.08949 by Yen-Chi Lee.

Figure 1
Figure 1. Figure 1: Finite-γ validation of the baseline asymptotic descriptors for the asymmetric four-point constellation in Appendix A. Panels (a)–(b) illustrate the small-noise upper-bound descriptor in Theorem 1, while panels (c)–(d) illustrate convergence toward the angular limits in Theorem 2, including the collapse of P1. For this constellation, Appendix A gives the conditional large-noise limits lim γ→∞ Pc(P1; γ) = 0,… view at source ↗
Figure 2
Figure 2. Figure 2: Descriptor-guided comparison for the hull-only pri [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Descriptor-guided comparison for the reciprocal-d [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Geometric and quantitative analysis for the illustr [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We study one-shot detection under isotropic multivariate Cauchy noise using finite constellations, with emphasis on the geometric mechanisms governing symbol-level reliability. Under isotropic Cauchy noise, the maximum-likelihood rule induces the same Euclidean Voronoi decision regions as in the Gaussian case, so the distinction lies not in the decision geometry itself but in how probability mass is distributed over these fixed regions. In the small-noise regime, we derive a reciprocal distance-spectrum upper bound for the symbol error probability (SEP), showing that this bound, and the associated reliability descriptor, retain a longer-range dependence on the global constellation geometry than under additive white Gaussian noise. In the large-noise regime, we prove that the correct-decision probability converges to a limit determined solely by the angular measure of the associated Voronoi recession cone. These results formalize a regime-dependent transition from bound-based distance descriptors to angle-based reliability descriptors under heavy-tailed noise. Beyond asymptotic characterization, we show that these descriptors also admit a lightweight design interpretation for planar constellations under a common average power budget. The theory is further illustrated through an asymmetric four-point example exhibiting geometric collapse, a standard four-point Quadrature Amplitude Modulation (4QAM) sanity check, and finite-$\gamma$ numerical validation for both asymptotic regimes, together with descriptor-guided design comparisons that reveal collapse avoidance and reciprocal-distance burden as practically meaningful screening criteria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper studies one-shot detection of finite constellations under isotropic multivariate Cauchy noise. It shows that the ML detector induces the same Euclidean Voronoi decision regions as under AWGN. In the small-noise regime, a reciprocal distance-spectrum upper bound on the symbol error probability is derived that retains longer-range dependence on global constellation geometry than the corresponding AWGN bound. In the large-noise regime, the correct-decision probability is proved to converge to a limit fixed solely by the angular measure of the Voronoi recession cone. The results are illustrated via an asymmetric four-point example, a 4QAM sanity check, finite-scale numerical validation, and descriptor-guided design comparisons under average power constraint.

Significance. If the derivations hold, the work supplies a clean geometric characterization of reliability under heavy-tailed noise, documenting an explicit transition from distance-based bounds at low noise to angle-based descriptors at high noise. This supplies both theoretical insight and lightweight design criteria for planar constellations. The explicit asymptotic statements, the coincidence of ML regions with Voronoi cells, and the numerical checks for both regimes are concrete strengths that make the contribution falsifiable and usable.

minor comments (4)
  1. The precise definition of the 'reciprocal distance-spectrum' and the associated reliability descriptor should be stated explicitly in the main text (rather than only in the abstract) before the small-noise bound is derived.
  2. In the large-noise section, the normalization of the angular measure (solid angle versus normalized solid angle) should be clarified with an explicit formula or reference to the surface measure on the unit sphere.
  3. The numerical validation plots would benefit from error bars or multiple Monte-Carlo runs to confirm that the observed convergence matches the predicted angular limit within sampling variability.
  4. A short remark on the relation to existing literature on Cauchy noise in communications (e.g., works on impulsive noise channels) would help situate the geometric results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring rebuttal or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from Cauchy density and geometry

full rationale

The paper's small-noise upper bound follows directly from the known univariate Cauchy tail P(|X|>t) ~ 1/t combined with a union bound over Euclidean distances to constellation points; the large-noise limit follows from the fact that the direction of an isotropic Cauchy vector becomes uniform on the sphere as the scale parameter tends to infinity, yielding the normalized solid angle of the recession cone of the Voronoi cell. Both steps use only the explicit form of the multivariate Cauchy pdf, the observation that ML regions coincide with Euclidean Voronoi cells (because the density depends solely on Euclidean norm), and standard solid-angle geometry. No fitted parameters are renamed as predictions, no self-citations supply load-bearing uniqueness theorems, and the results are not obtained by re-labeling known empirical patterns. The analysis is restricted to the two asymptotic regimes and finite constellations, rendering the derivation chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are explicitly introduced beyond the noise model itself.

axioms (2)
  • domain assumption Isotropic multivariate Cauchy distribution properties govern the noise
    Used to derive the probability mass over fixed Voronoi regions.
  • standard math Maximum-likelihood decision regions coincide with Euclidean Voronoi partition
    Stated as identical to the Gaussian case and used as the fixed geometry for both regimes.

pith-pipeline@v0.9.0 · 5534 in / 1454 out tokens · 52133 ms · 2026-05-10T17:49:12.695828+00:00 · methodology

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