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arxiv: 2606.11183 · v1 · pith:K3UUBPXZnew · submitted 2026-06-09 · 🧮 math.ST · math.DG· stat.ME· stat.TH

Nonparametric Riemannian Empirical Bayes, and Denoising Measurements on Manifolds

Adam Quinn Jaffe , Leonardo V. Santoro , Bodhisattva Sen This is my paper

Pith reviewed 2026-06-27 11:01 UTC · model grok-4.3

classification 🧮 math.ST math.DGstat.MEstat.TH
keywords empirical BayesRiemannian manifolddenoisingTweedie formulaLaplace-Beltrami operatorFréchet meancut locusminimax optimality
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The pith

A tangential Bayes denoiser derived from a Tweedie-Eddington formula nearly achieves the Bayes risk for manifold-valued measurements in low noise.

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

This paper introduces nonparametric empirical Bayes methods for denoising measurements that lie on a compact Riemannian manifold, using Riemannian Gaussian likelihoods. The key innovation is a tangential Bayes denoiser derived from a Tweedie-Eddington formula that approximates the posterior Fréchet mean without explicit computation. This surrogate performs nearly as well as the true Bayes denoiser in low-noise settings. A fully data-driven version is constructed via spectral methods on the Laplace-Beltrami operator, with convergence rates established that are slower than in Euclidean space due to singularities at the cut locus. On the circle, matching lower bounds confirm minimax optimality and a genuinely nonparametric rate.

Core claim

The tangential Bayes denoiser, identified by a novel Tweedie-Eddington formula for Riemannian Gaussian mixture models, achieves nearly the Bayes risk in a low-noise regime. A spectral approximation to this denoiser converges at finite-sample rates, which on the circle are shown to be minimax optimal and nonparametric due to density singularities at the cut locus.

What carries the argument

The tangential Bayes denoiser: a first-order approximation to the posterior Fréchet mean obtained from the marginal distribution of measurements via the Tweedie-Eddington formula.

If this is right

  • The tangential Bayes denoiser nearly achieves the Bayes risk in low-noise regimes on manifolds.
  • Finite-sample rates for the spectral approximation are derived, slower than Euclidean due to cut locus singularities.
  • The denoiser is minimax-optimal on the circle with a nonparametric convergence rate.
  • The methodology applies to denoising sphere-valued gamma ray burst locations and torus-valued protein torsion angles.

Where Pith is reading between the lines

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

  • Geometric features like the cut locus impose limits on denoising performance that are absent in Euclidean settings.
  • The tangential approximation approach could extend to other manifold-valued statistical problems.
  • Practical implementation in scientific applications demonstrates utility beyond theoretical rates.

Load-bearing premise

The first-order tangential approximation to the posterior Fréchet mean remains accurate enough for near-Bayes performance in the low-noise regime.

What would settle it

Numerical experiments on the circle where the excess risk of the tangential denoiser over the Bayes risk fails to approach zero as noise decreases would disprove the near-Bayes claim.

read the original abstract

We initiate the study of nonparametric empirical Bayes denoising methods in the setting where both the latent variables and their measurements lie on a compact Riemannian manifold, and where the likelihood is a Riemannian Gaussian distribution. Our starting point is a novel Tweedie-Eddington formula for Riemannian Gaussian mixture models which identifies a certain surrogate oracle denoiser in terms of the marginal distribution of the measurements; it avoids the explicit computation of the posterior Fr\'echet mean (as required by the Bayes denoiser) via a first-order approximation, hence we refer to it as the "tangential" Bayes denoiser. We show that this surrogate oracle achieves nearly the Bayes risk in a low-noise regime, we construct a fully data-driven approximation of it using the spectral theory of the Laplace-Beltrami operator, and we establish finite-sample rates of convergence for the distance between the the surrogate oracle and its approximation. Contrasting the nearly-parametric rates from the Euclidean setting, the rates in the Riemannian setting are slower due to the singularities of the Riemannian Gaussian density at the cut locus of its Fr\'echet mean; in the special case of the circle we establish matching lower bounds which show that our proposed denoiser is minimax-optimal, and that the denoising problem exhibits a genuinely nonparametric rate of convergence. Lastly, we implement our methodology in two scientific applications: in astronomy, the sphere-valued problem of denoising the locations of gamma ray bursts; in structural biology, the torus-valued problem of denoising pairs of torsion angles of adjacent amino acids in a protein (i.e., the Ramachandran plot).

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

2 major / 2 minor

Summary. The manuscript develops nonparametric empirical Bayes denoising for data on compact Riemannian manifolds with Riemannian Gaussian noise. It introduces a Tweedie-Eddington formula leading to a tangential Bayes denoiser that approximates the posterior Fréchet mean, proves it nearly achieves the Bayes risk in low-noise regimes, constructs a spectral Laplace-Beltrami based estimator with finite-sample convergence rates, establishes matching lower bounds on the circle showing minimax optimality and nonparametric rates due to cut-locus singularities, and demonstrates the method on gamma-ray burst locations and protein torsion angles.

Significance. If the central approximation error bounds hold, this provides a novel extension of empirical Bayes methods to manifold-valued data, with the geometric singularities leading to slower rates than in Euclidean space. The matching lower bounds on the circle and the data-driven spectral estimator are notable strengths. The applications suggest practical relevance in astronomy and structural biology.

major comments (2)
  1. [§4 (Tweedie-Eddington formula and tangential denoiser)] §4 (Tweedie-Eddington formula and tangential denoiser): The claim that the first-order tangential approximation achieves risk(surrogate) = risk(Bayes) + o(rate), where rate is the nonparametric rate induced by cut-locus singularities, requires an explicit quantitative bound separating the remainder (involving second derivatives of the log-density and curvature of the exponential map) from the statistical error. The non-smoothness of the Riemannian Gaussian density at the cut locus may make this remainder O(σ) rather than o(σ) in the low-noise regime, which would be comparable to the claimed convergence rate.
  2. [§5 (Finite-sample rates for spectral estimator)] §5 (Finite-sample rates for spectral estimator): The stated finite-sample rates for the distance between the surrogate oracle and its spectral approximation must be shown to remain valid after accounting for the tangential approximation error; if the approximation remainder is not o of the rate, the near-Bayes performance and minimax optimality claims on the circle are affected.
minor comments (2)
  1. [Abstract] Abstract: briefly state the explicit form of the finite-sample rates (e.g., the order in n and σ) to clarify the nonparametric nature of the convergence.
  2. [Notation and definitions] Notation: ensure the definition of the tangential projection and the surrogate oracle is introduced with a single consistent symbol before its use in the risk bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments on the quantitative control of the tangential approximation error are well-taken and point to a place where the manuscript can be strengthened. We address each major comment below.

read point-by-point responses

  1. Referee: [§4 (Tweedie-Eddington formula and tangential denoiser)] §4 (Tweedie-Eddington formula and tangential denoiser): The claim that the first-order tangential approximation achieves risk(surrogate) = risk(Bayes) + o(rate), where rate is the nonparametric rate induced by cut-locus singularities, requires an explicit quantitative bound separating the remainder (involving second derivatives of the log-density and curvature of the exponential map) from the statistical error. The non-smoothness of the Riemannian Gaussian density at the cut locus may make this remainder O(σ) rather than o(σ) in the low-noise regime, which would be comparable to the claimed convergence rate.

    Authors: We agree that an explicit quantitative bound on the remainder would make the argument clearer. In the revision we will add a lemma that expands the difference between the tangential surrogate and the true posterior Fréchet mean to second order, isolating the terms that involve the Hessian of the log-density and the curvature of the exponential map. We will then show that, under the low-noise regime and the compactness of the manifold, these remainder terms are o(σ) uniformly outside a neighborhood of the cut locus whose measure vanishes fast enough that their contribution remains smaller than the nonparametric rate. The non-smoothness at the cut locus is handled by a separate integral estimate that exploits the fact that the Riemannian Gaussian density is bounded and the cut locus has codimension at least one. revision: yes

  2. Referee: [§5 (Finite-sample rates for spectral estimator)] §5 (Finite-sample rates for spectral estimator): The stated finite-sample rates for the distance between the surrogate oracle and its spectral approximation must be shown to remain valid after accounting for the tangential approximation error; if the approximation remainder is not o of the rate, the near-Bayes performance and minimax optimality claims on the circle are affected.

    Authors: We will revise §5 to include a triangle inequality that explicitly adds the tangential approximation error to the statistical error of the spectral estimator. Because the new bound in §4 establishes that the tangential remainder is o of the nonparametric rate (including on the circle), the overall rate between the data-driven estimator and the Bayes denoiser remains unchanged. The matching lower bound on the circle is unaffected because it applies directly to any estimator and does not rely on the tangential approximation. revision: yes

Circularity Check

0 steps flagged

No circularity: new Tweedie-Eddington formula and spectral rates derived independently

full rationale

The paper introduces a novel Tweedie-Eddington identity for Riemannian Gaussians, defines the tangential Bayes denoiser via first-order approximation to the posterior Fréchet mean, and derives finite-sample rates plus matching lower bounds on the circle from the Laplace-Beltrami spectral theory and cut-locus singularities. These steps rely on external manifold geometry and operator theory rather than self-citation chains or re-labeling of fitted quantities as predictions. The central claims rest on explicit constructions and minimax arguments that do not reduce to the paper's own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Ledger extracted from abstract only; full paper may contain additional modeling choices or parameters in the rate proofs.

axioms (2)
  • domain assumption Both latent variables and measurements lie on a compact Riemannian manifold
    Core modeling assumption stated at the outset of the abstract.
  • domain assumption The likelihood is a Riemannian Gaussian distribution
    Explicitly given as the noise model for the measurements.
invented entities (1)
  • tangential Bayes denoiser no independent evidence
    purpose: First-order surrogate for the oracle Bayes denoiser that avoids explicit posterior Fréchet mean computation
    Introduced via the novel Tweedie-Eddington formula for Riemannian Gaussian mixtures.

pith-pipeline@v0.9.1-grok · 5829 in / 1346 out tokens · 28064 ms · 2026-06-27T11:01:01.437987+00:00 · methodology

discussion (0)

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