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arxiv: 2606.24771 · v2 · pith:3VIM6SWInew · submitted 2026-06-23 · 🧮 math.ST · math.PR· stat.TH

Autoregressive Processes on Riemannian Manifolds

Meshal Abuqrais , Davide Pigoli This is my paper

Pith reviewed 2026-06-25 21:42 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords autoregressive modelRiemannian manifoldFréchet meanergodic Markov chainstrong law of large numbersmanifold-valued dataconsistencytime series on manifolds
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The pith

R-AR(1) model on Riemannian manifolds has strongly consistent Fréchet mean and autoregressive estimators

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

This paper introduces the Riemannian autoregressive model of order one for data valued on manifolds. The model uses a Fréchet mean parameter μ and an autoregressive coefficient φ to capture central tendency and temporal dependence. A key contribution is proving a strong law of large numbers for the sample Fréchet mean of ergodic Markov chains in proper metric spaces, which enables strong consistency results for the estimators. This moves beyond independent and identically distributed assumptions to handle dependent manifold data. Readers should care if they work with time series on curved spaces where standard Euclidean methods fail.

Core claim

The central discovery is the R-AR(1) model defined by parameters μ representing the intrinsic central tendency as the Fréchet mean and φ controlling stationarity and ergodicity, together with the proof that the sample Fréchet mean set of ergodic Markov chains in proper metric spaces satisfies a strong law of large numbers, which establishes the strong consistency of the model estimators.

What carries the argument

The strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains in proper metric spaces, which provides the foundation for consistency in the R-AR(1) model.

If this is right

  • Estimators of the Fréchet mean and autoregressive parameter are strongly consistent.
  • The model can be applied to mean-reverting dynamics in nonlinear geometries such as the hyperbolic plane.
  • It enables analysis of manifold-valued data like aerosol size distributions on the Fisher-Rao manifold.
  • The approach generalizes classical discrete-time stochastic processes to manifold-valued observations.

Where Pith is reading between the lines

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

  • The SLLN result could support consistency proofs for other statistics on manifolds under Markov dependence.
  • Extensions to higher-order autoregressive models on manifolds may follow similar ergodicity arguments.
  • The framework suggests potential for forecasting or simulation of manifold time series with mean reversion.
  • Connections might exist to geometric statistics methods for dependent data.

Load-bearing premise

The observed process must be generated by an ergodic Markov chain defined on a proper metric space.

What would settle it

Finding an ergodic Markov chain on a proper metric space for which the sample Fréchet mean set does not converge almost surely would falsify the strong law of large numbers and the resulting consistency.

Figures

Figures reproduced from arXiv: 2606.24771 by Davide Pigoli, Meshal Abuqrais.

Figure 1
Figure 1. Figure 1: Trajectory evolution in the Poincar´e disk model of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geodesic rays from µ in the Poincar´e disk model of H2 for two R-AR models with the same µ and different values of ϕ. 3.2 Estimators Consider a stationary R-AR(µ, ϕ) process taking values in a Riemannian manifold M, and let {X1, . . . , Xn} denote observed data points from the process. In this subsection, we propose estima￾tors for the parameters µ and ϕ of the process. A natural estimator for the paramete… view at source ↗
Figure 3
Figure 3. Figure 3: Estimation error dH2 (µ, µˆn) of the sample Fr´echet mean as a function of the sample size n, for ϕ ∈ {0.8, 0.9, 0.95, 0.99}. Each curve corresponds to one independently simulated trajectory with random initialisation. Page 25 of 37 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Absolute error |ϕˆ n − ϕ| of the autoregressive parameter estimator as a function of the sample size n, for ϕ ∈ {0.8, 0.9, 0.95, 0.99}. Each curve corresponds to one independently simulated trajectory with random initialisation. Page 26 of 37 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boxplots of Fisher–Rao forecast errors across various estimation windows and [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Estimated trajectories of the autoregressive coefficient [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the E30 estimation for a five-day prediction scheme applied at [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
read the original abstract

This paper introduces a Riemannian autoregressive (R-AR) model of order one, generalising classical discrete-time stochastic processes to manifold-valued data. The model is based on two parameters, a parameter $\mu$ representing the intrinsic central tendency as the Fr\'echet mean and an autoregressive parameter $\phi$ controlling the stationarity and ergodic properties. Due to the inherent dependence structure of the R-AR process, the estimation procedure for these parameters necessitates new asymptotic results for dependent processes on manifolds. Thus, we establish a strong law of large numbers for the sample Fr\'echet mean set of ergodic Markov chains in proper metric spaces. By proving this general consistency result, we move beyond the limitations of classical i.i.d. theory to provide the mathematical foundation required for the strong consistency of our proposed estimators. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.

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 / 1 minor

Summary. The paper introduces a Riemannian autoregressive (R-AR) model of order one on a manifold, parameterized by the Fréchet mean μ and autoregressive coefficient φ that controls stationarity and ergodicity. It establishes a strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains on proper metric spaces, which is invoked to obtain strong consistency of the estimators for μ and φ. The claims are illustrated by simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold.

Significance. If the SLLN holds under the stated ergodicity condition, the work supplies a useful extension of classical AR(1) models to manifold-valued time series and a general consistency tool for dependent data on metric spaces that could apply beyond this model.

major comments (2)
  1. [Abstract] Abstract: the SLLN result is asserted as the foundation for estimator consistency, but the text supplies no explicit statement of the theorem, no proof details, and no list of assumptions beyond ergodicity of the Markov chain; the central consistency claim therefore cannot be verified.
  2. [Model definition] Model definition: ergodicity is stated to be controlled by φ, yet the precise contraction or contraction-mapping condition on φ (required for the SLLN to apply) is not given; this assumption is load-bearing for both the Markov-chain property and the consistency conclusion.
minor comments (1)
  1. [Abstract] The abstract mentions numerical simulations and one application but provides no quantitative results, figures, or tables; these should be summarized with key metrics if they support the claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the SLLN result is asserted as the foundation for estimator consistency, but the text supplies no explicit statement of the theorem, no proof details, and no list of assumptions beyond ergodicity of the Markov chain; the central consistency claim therefore cannot be verified.

    Authors: The abstract is a concise summary of the contributions. The explicit statement of the SLLN theorem (including its proof and the full set of assumptions, which center on ergodicity of the Markov chain on a proper metric space) is given in the main body of the paper. The estimator consistency follows from applying this theorem to the R-AR process. To address the concern about verifiability from the abstract alone, we will revise the abstract to briefly reference the ergodicity assumption and the proper metric space setting. revision: yes

  2. Referee: [Model definition] Model definition: ergodicity is stated to be controlled by φ, yet the precise contraction or contraction-mapping condition on φ (required for the SLLN to apply) is not given; this assumption is load-bearing for both the Markov-chain property and the consistency conclusion.

    Authors: We agree that the precise condition on φ should be stated explicitly. The model requires that the autoregressive map induced by φ is a contraction with Lipschitz constant strictly less than 1 with respect to the Riemannian distance; this guarantees the Markov chain is ergodic and thereby enables the SLLN. We will add this explicit contraction-mapping condition (and the resulting admissible range for φ) to the model definition section in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation defines the R-AR(1) model via Fréchet mean μ and coefficient φ, establishes that the process is an ergodic Markov chain (with ergodicity controlled by φ), and invokes a separately proved general SLLN for the sample Fréchet mean set of ergodic Markov chains on proper metric spaces to obtain strong consistency. The SLLN is presented as an independent general theorem whose hypotheses are the standard ergodicity condition; it is not obtained by fitting parameters from the target estimators or by self-citation chains. No equation or step reduces the claimed consistency result to a tautology or to a fitted input renamed as a prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model introduces two parameters whose estimation rests on the ergodicity assumption. Information is limited to the abstract.

free parameters (2)
  • μ
    Fréchet mean parameter representing intrinsic central tendency; estimated from data.
  • φ
    Autoregressive coefficient controlling stationarity and dependence; estimated from data.
axioms (1)
  • domain assumption The observed process is an ergodic Markov chain on a proper metric space.
    Required to invoke the SLLN for consistency of the sample Fréchet mean.

pith-pipeline@v0.9.1-grok · 5700 in / 1351 out tokens · 23963 ms · 2026-06-25T21:42:01.820331+00:00 · methodology

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Reference graph

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