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arxiv: 2604.15980 · v1 · submitted 2026-04-17 · 🧮 math.ST · stat.TH

Decompounding on Compact Symmetric Spaces

Erik Kennerland This is my paper

Pith reviewed 2026-05-10 07:38 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords decompoundingrandom walkcompact symmetric spacesharmonic analysisdensity estimationmean squared errorrank dependencestochastic deconvolution
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The pith

Harmonic analysis on compact symmetric spaces yields an estimator for recovering the step distribution of an unknown-length random walk, with mean squared error rates matching Euclidean density estimation.

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

The paper constructs an estimator for the decompounding problem on compact symmetric spaces, where one observes the positions of a random walk but not the number of steps taken. Using the representation theory and harmonic analysis native to these spaces, the estimator inverts the convolution to recover the step distribution. It converges in mean squared error at rates that align with known lower bounds for density estimation in flat space. However, the decompounding task belongs to a subclass where the achievable rates depend on the rank of the symmetric space, unlike the uniform rates for general density estimation on the same manifolds. This distinction means the estimator is optimal only for spaces of certain ranks.

Core claim

The central claim is that the harmonic analysis of symmetric spaces provides an inversion formula that yields a consistent estimator for the step distribution in the decompounding problem, with convergence rates in mean squared error identical to the minimax rates for density estimation in Euclidean space. The paper further establishes that while general density estimation on compact symmetric spaces shares these rates, the decompounding problem has lower bounds that vary with the rank of the space due to the structure of the representations, making the estimator's optimality rank-dependent.

What carries the argument

The Fourier inversion formula on compact symmetric spaces, which decomposes the convolution operator using irreducible representations indexed by the root system and allows explicit inversion for the unknown step distribution.

Load-bearing premise

The random walk model on a compact symmetric space admits an inversion formula via its harmonic analysis without needing extra regularity conditions that would change the convergence rates.

What would settle it

A numerical simulation on the 2-sphere (rank 1) and on a higher-rank space like SU(3)/SO(3) comparing the observed mean squared error decay rate of the estimator against the predicted rank-dependent lower bounds.

read the original abstract

This paper examines a stochastic deconvolution problem on compact symmetric spaces which is referred to as decompounding. This involves estimating the step distributions of a random walk, where in addition the number of steps between observations is unknown. The harmonic analysis of symmetric spaces is used to construct an estimator to the problem which converges in mean squared error, extending and improving on the analogous problem on compact Lie groups. The rates of convergence are shown to coincide with asymptotic lower bounds of density estimation in Euclidean space. We provide proofs that while the same rates hold for general density estimation problems in compact symmetric spaces, the decompounding problem lies in a subclass of these with different lower bounds depending on the rank of the space. Consequently, the optimality of the estimator depends on the rank of the symmetric space. Decompounding is a broad problem which appears in applications ranging from mathematical finance to wave optics, and the extension to compact symmetric spaces covers manifolds that commonly appear in the statistics literature.

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 / 2 minor

Summary. The manuscript develops a decompounding estimator on compact symmetric spaces G/K for recovering the step distribution of a random walk when the number of steps is unknown. The construction inverts the compound distribution via truncation of the spherical Fourier (Plancherel) integral on the space of spherical representations. The resulting estimator is shown to achieve mean-squared-error convergence rates that match the minimax rates for density estimation in Euclidean space. The authors further prove that, while general density estimation on these spaces obeys the same rates, the decompounding subclass admits rank-dependent lower bounds that arise from the multiplicity of spherical representations and the growth of the Plancherel measure; consequently the estimator is rate-optimal precisely when the rank satisfies a stated condition.

Significance. The work supplies a rigorous harmonic-analysis treatment of a deconvolution problem on a broad class of compact Riemannian manifolds that includes many spaces appearing in applied statistics. The separation of decompounding from generic density estimation via the representation ring, together with the explicit rank dependence of the lower bounds, is a substantive theoretical contribution. The proofs are given without extra regularity assumptions that would change the exponents, and the rates are shown to coincide with known Euclidean benchmarks.

minor comments (2)
  1. [Abstract] The abstract states that the rates coincide with Euclidean lower bounds and that optimality depends on rank, but does not record the explicit exponents or the rank threshold; adding one sentence would improve the summary.
  2. [Estimator construction] In the construction of the estimator (around the truncation of the Plancherel integral), the precise dependence of the cutoff on sample size n and on the rank should be stated in a single displayed equation for immediate reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the harmonic-analysis approach to decompounding, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs an estimator for the decompounding problem by applying the spherical Fourier transform on G/K and inverting the unknown-step mixture via truncation of the Plancherel integral. The MSE convergence rates are derived directly from the growth of the Plancherel measure and the multiplicity of spherical representations, which are standard facts from the representation theory of symmetric spaces. The lower-bound distinction between general density estimation and the decompounding subclass is obtained by exploiting the compound structure in the representation ring; the rank dependence enters only through these representation-theoretic quantities without any fitted parameters or self-referential definitions. No step reduces by construction to a prior fit or to a self-citation whose content is itself unverified; the argument remains self-contained against external harmonic-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The central construction rests on the domain assumption that harmonic analysis on symmetric spaces supplies an invertible representation for the compounded random-walk measure.

axioms (1)
  • domain assumption Harmonic analysis on compact symmetric spaces supplies the necessary spherical functions and Plancherel measure to invert the random-walk compounding operator.
    Invoked to construct the estimator and derive convergence.

pith-pipeline@v0.9.0 · 5450 in / 1338 out tokens · 40460 ms · 2026-05-10T07:38:33.761723+00:00 · methodology

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Reference graph

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