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arxiv: 2508.10810 · v2 · submitted 2025-08-14 · 🧮 math.FA

Sampling theorems for inverse problems on Riemannian manifolds

Giovanni S. Alberti , Ernesto De Vito , Bianca Gariboldi , Giacomo Gigante This is my paper

Pith reviewed 2026-05-18 22:59 UTC · model grok-4.3

classification 🧮 math.FA
keywords sampling theoremsinverse problemsRiemannian manifoldsMarcinkiewicz-Zygmund familiesconvolution operatorstwo-dimensional sphereerror boundssignal reconstruction
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The pith

Sampling theorems give explicit reconstruction error bounds for inverse problems on Riemannian manifolds from noisy pointwise samples.

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

The paper establishes sampling theorems for recovering an unknown function f on a Riemannian manifold from noisy point samples of Ff, where F is a linear operator such as a convolution. The theorems supply concrete bounds on the error that depend on the number of samples n, the smoothness of f, and the nature of F. A detailed analysis covers convolutions on compact two-point homogeneous spaces and yields a specific result for the two-dimensional sphere, with four examples drawn from terrestrial and celestial measurement problems. These bounds matter because they turn discrete noisy data into guaranteed accuracy levels for signal recovery on curved spaces.

Core claim

We consider inverse problems of reconstructing f from y = Ff + noise on a Riemannian manifold M without boundary, using only pointwise samples y_j = (Ff)(x_j) + η_j where {x_j} is a Marcinkiewicz-Zygmund family. Sampling theorems are derived that supply explicit bounds on the reconstruction error in terms of n, the smoothness of f and properties of F. The case of F being a convolution on a compact two-point homogeneous space is studied in detail, yielding as a corollary a sampling theorem for convolutions on the two-dimensional sphere, illustrated with four examples of relevant measurements.

What carries the argument

Marcinkiewicz-Zygmund families of sampling points that satisfy the inequalities needed to control reconstruction error for the operator F.

If this is right

  • The error bound decreases explicitly with larger n or greater smoothness of f when the points form a Marcinkiewicz-Zygmund family.
  • Convolutions on compact two-point homogeneous spaces admit detailed error estimates from the sampling theorems.
  • A sampling theorem holds for convolutions on the two-dimensional sphere as a direct corollary.
  • Four concrete examples of terrestrial and celestial measurements obtain explicit error guarantees from the derived bounds.

Where Pith is reading between the lines

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

  • Similar explicit bounds could be obtained on other Riemannian manifolds once suitable Marcinkiewicz-Zygmund point families are identified.
  • The dependence on n and smoothness could be used to decide how many measurement locations are needed for a target accuracy in sphere-based data applications.
  • The proof techniques for the convolution case may extend to other classes of operators F on the same spaces.

Load-bearing premise

The chosen points must form a Marcinkiewicz-Zygmund family on the manifold so that the required sampling inequalities hold.

What would settle it

A calculation on the two-dimensional sphere showing that reconstruction error for a smooth test function under a known convolution operator does not improve with n at the rate given by the bounds would disprove the theorems.

Figures

Figures reproduced from arXiv: 2508.10810 by Bianca Gariboldi, Ernesto De Vito, Giacomo Gigante, Giovanni S. Alberti.

Figure 1
Figure 1. Figure 1: Plot of the sequence (1 + m(m + 1))3/4 |bm| for ϑ0 = 2π/41 and m = 1, . . . , 1400. Hence, by applying Theorem 3 and Proposition 1 with γ = ζ = 3/2, we obtain ∥f † − p β m∥2 ≤ c −1 0 ∥F f† − F pβ m∥ H 3 2 (Sd) ≤ c −1 0 r 1 + κ ω − 1 ∥F f† ∥ H ω+ 3 2 (S2) (1 + m(m + 1))− ω−1 2 + c −1 0 √ κβ(1 + m(m + 1)) 3 4 ≤ 3 3 4 2 (sin ϑ0) 1 2 c −1 0 r 1 + κ ω − 1 ∥f † ∥Hω(S2)(1 + m(m + 1))− ω−1 2 + c −1 0 √ κβ(1 + m(m … view at source ↗
Figure 2
Figure 2. Figure 2: The function g in (11), used to construct the convolution filter of the example from §3.3. and J1 is the Bessel function of the first kind and order 1, corresponding to the far-field beamshape of an ideal circular aperture with radius R m operating at a wavelength λ0 m. As usual, ρ(x, o) is the angular distance of x from the north pole o. In [62] two possibilities are studied, one with λ0 = 3 and R = 9, wh… view at source ↗
read the original abstract

We consider inverse problems consisting of the reconstruction of an unknown signal $f$ from noisy measurements $y=Ff+\text{noise}$, where $Ff$ is a function on a Riemannian manifold without boundary $\mathcal M$. We consider the case when only pointwise samples are available, namely $y_j = (Ff)(x_j)+\eta_j$, where $\{x_j\}_{j=1}^n\subseteq\mathcal M$ is a Marcinkiewicz-Zygmund family. We derive sampling theorems providing explicit bounds on the reconstruction error depending on $n$, the smoothness of $f$ and the properties of $F$. We study in detail the case when $F$ is a convolution on a compact two-point homogeneous space. As a corollary, we state a sampling theorem for convolutions on the two-dimensional sphere, and discuss four relevant examples related to terrestrial and celestial measurements.

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 paper develops sampling theorems for inverse problems on Riemannian manifolds without boundary. Given noisy pointwise samples y_j = (Ff)(x_j) + η_j where {x_j} forms a Marcinkiewicz-Zygmund family, it derives explicit bounds on the reconstruction error in terms of n, the smoothness of f, and properties of the operator F. Detailed analysis is provided for convolution operators F on compact two-point homogeneous spaces, yielding a corollary for convolutions on the 2-sphere together with four examples drawn from terrestrial and celestial measurements.

Significance. If the claimed explicitness of the error bounds holds without hidden manifold-dependent constants, the results would offer a useful theoretical tool for stable reconstruction from discrete samples in manifold signal processing. The specialization to two-point homogeneous spaces and the sphere corollary provide concrete, applicable statements that could inform measurement design in geophysics and astronomy. The manuscript's strength lies in linking abstract sampling inequalities to operator-specific bounds, but this value is conditional on the quantification of the underlying Marcinkiewicz-Zygmund constants.

major comments (2)
  1. [Abstract / MZ-family paragraph] Abstract and the paragraph introducing the Marcinkiewicz-Zygmund family: the reconstruction-error bounds are stated to depend explicitly on n, smoothness of f, and properties of F, yet they rest on sampling inequalities whose lower and upper constants are not shown to be expressible in terms of n and intrinsic geometric invariants of a general M (e.g., injectivity radius or curvature bounds). If these constants remain unspecified or absorbed, the explicitness claim does not hold for arbitrary Riemannian manifolds even though it may hold for the symmetric convolution cases.
  2. [General sampling theorem] Section deriving the general sampling theorem (presumably §3 or Theorem 3.1): the error estimate is obtained by combining the sampling inequality with an approximation or regularization term; the final expression must be checked to confirm that no manifold-specific constants from the MZ family enter the bound in a non-explicit way, otherwise the dependence claimed in the abstract is incomplete.
minor comments (2)
  1. [Introduction] Notation for the manifold M and the operator F should be introduced consistently in the first section rather than appearing first in the abstract.
  2. [Sphere corollary / examples] The four examples on the sphere would benefit from a short table summarizing the corresponding operators F, smoothness assumptions, and resulting error rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The points raised concern the precise meaning of 'explicit' bounds in the general Riemannian setting versus the specialized convolution cases. We address each major comment below and indicate the revisions we will make to clarify the dependence on Marcinkiewicz-Zygmund constants.

read point-by-point responses

  1. Referee: [Abstract / MZ-family paragraph] Abstract and the paragraph introducing the Marcinkiewicz-Zygmund family: the reconstruction-error bounds are stated to depend explicitly on n, smoothness of f, and properties of F, yet they rest on sampling inequalities whose lower and upper constants are not shown to be expressible in terms of n and intrinsic geometric invariants of a general M (e.g., injectivity radius or curvature bounds). If these constants remain unspecified or absorbed, the explicitness claim does not hold for arbitrary Riemannian manifolds even though it may hold for the symmetric convolution cases.

    Authors: We agree that the abstract's phrasing could be read as claiming fully explicit constants solely in terms of n and basic geometric invariants for arbitrary manifolds. In the general theorem the error bounds are expressed explicitly in terms of the lower and upper MZ constants of the given sampling family together with n, the smoothness parameter, and operator properties of F. These MZ constants are intrinsic to the chosen point set on M and are not further reduced to curvature bounds in the general case, as that would require additional assumptions on the distribution of points. In the subsequent analysis for compact two-point homogeneous spaces we do derive explicit upper and lower bounds on the MZ constants using the representation theory and volume growth of those spaces, yielding completely explicit expressions for the sphere corollary. We will revise the abstract and the introductory paragraph on MZ families to state that the general bounds depend explicitly on the MZ constants of the sampling set, while the convolution results on two-point homogeneous spaces provide fully explicit constants in terms of n and the geometry. revision: partial

  2. Referee: [General sampling theorem] Section deriving the general sampling theorem (presumably §3 or Theorem 3.1): the error estimate is obtained by combining the sampling inequality with an approximation or regularization term; the final expression must be checked to confirm that no manifold-specific constants from the MZ family enter the bound in a non-explicit way, otherwise the dependence claimed in the abstract is incomplete.

    Authors: We have re-examined the proof of the general sampling theorem. The reconstruction error is bounded by a term involving the MZ lower constant times the noise level plus a term involving the MZ upper constant times the approximation error of the regularized solution. No additional manifold-dependent constants are introduced beyond those already present in the MZ inequality itself. The final expression is therefore explicit once the MZ constants are fixed by the sampling family. To address the referee's concern we will add a short remark after the statement of the general theorem clarifying that the MZ constants are regarded as known quantities characterizing the sampling set and that their explicit evaluation is carried out in the later sections for the symmetric-space cases. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds derived from assumed MZ sampling inequalities

full rationale

The paper assumes {x_j} forms a Marcinkiewicz-Zygmund family on M and derives sampling theorems with error bounds depending on n, smoothness of f, and properties of F. No load-bearing step reduces by the paper's own equations or self-citation to its inputs by construction. The MZ property supplies the sampling inequalities as an external assumption; constants are absorbed into the stated bounds without redefinition or fitting. Special cases (convolutions on two-point homogeneous spaces and the sphere) are handled separately with the same structure. The derivation is self-contained against the given assumptions and does not exhibit self-definitional, fitted-prediction, or uniqueness-imported patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from Riemannian geometry, functional analysis, and sampling theory. No free parameters or new entities are introduced in the abstract description.

axioms (2)
  • domain assumption The collection {x_j} is a Marcinkiewicz-Zygmund family on the manifold M.
    Invoked to guarantee the sampling inequalities that control reconstruction error.
  • domain assumption F is a bounded linear operator (specifically a convolution when studied in detail) on the manifold.
    Central to the inverse-problem formulation and the convolution specialization.

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