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arxiv: 2605.08001 · v1 · submitted 2026-05-08 · 🧮 math.ST · stat.ME· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Scale selection for geometric medians on product manifolds

Kisung You
Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:30 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords geometric medianproduct manifoldscale selectiondegeneracyconsistencyinfluence functionBures-Wassersteinsensitivity analysis
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The pith

Joint minimization of location and scale for geometric medians on product manifolds degenerates, driving scale to the boundary and reducing to a marginal median.

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

The paper shows that geometric medians on product manifolds are sensitive to relative scaling between the factor metrics because the objective sums distances across factors instead of separating them. Naive joint optimization over both the median location and the scale parameter forces the scale to a boundary value, which collapses the solution to a marginal median on one factor and discards the other. This demonstrates that relative scale cannot be recovered from the median loss function alone. The work then develops three distinct alternatives that restore identifiability while preserving consistency and robustness properties under different modeling choices.

Core claim

The central claim is that naive joint minimization over location and scale is degenerate: the scale is driven to the boundary and the problem collapses to a marginal median, effectively discarding one factor. Relative scale is not identifiable from the raw median loss alone. Three alternatives are developed: treating scale as indexing a sensitivity path with uniform consistency, a functional central limit theorem, and a derivative-based sensitivity measure; constructing a robust scale-calibrated median from marginal radial median scales that achieves unit invariance, consistency, a two-step central limit theorem, and bounded influence; and introducing a bounded balance equation for direct, a

What carries the argument

The geometric median objective on the product manifold, defined via the sum of distances to observed points, which couples the scales of the component manifolds and permits boundary collapse under simultaneous optimization.

If this is right

  • Scale must be treated separately from location rather than optimized jointly, or the estimator discards one manifold factor.
  • The sensitivity-path approach yields a continuous family of medians whose derivative with respect to scale quantifies sensitivity to relative metric choice.
  • The marginal-scale calibration produces an estimator that is invariant to unit rescaling of either factor and has bounded influence.
  • The balance-equation estimator admits a joint asymptotic normal distribution for location and scale together with uniqueness and monotonicity guarantees.

Where Pith is reading between the lines

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

  • The degeneracy pattern may appear in other summed-distance estimators on product spaces, suggesting separate scale calibration is needed for broader classes of manifold-valued robust statistics.
  • The derivative-based sensitivity measure could serve as a practical diagnostic for choosing scales in applied settings where the relative metrics of the factors are unknown a priori.
  • Joint asymptotic normality under the balance-equation method opens the door to simultaneous confidence regions for both location and scale parameters on product manifolds.

Load-bearing premise

The median objective couples the factors through the sum of distances in a manner that permits boundary collapse under joint optimization.

What would settle it

Numerical optimization of the joint location-scale objective on data drawn from a product of two spaces with mismatched intrinsic scales, checking whether the estimated scale ratio converges to zero or infinity as sample size grows.

Figures

Figures reproduced from arXiv: 2605.08001 by Kisung You.

Figure 1
Figure 1. Figure 1: Boundary degeneracy and sensitivity path in the Euclidean product example. (a) Profiled [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scale calibration and unit invariance. (a) Location drift from the estimate at [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Balanced estimating equation. (a) Empirical balance function for a representative sample, [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bures–Wasserstein Gaussian product example. (a) Mean observations and mean-factor [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
read the original abstract

Geometric medians on product manifolds are sensitive to the relative scaling of factor metrics because the median objective couples the factors rather than separating them. We study this scale-selection problem and first prove that naive joint minimization over location and scale is degenerate: the scale is driven to the boundary and the problem collapses to a marginal median, effectively discarding one factor. Thus relative scale is not identifiable from the raw median loss alone. We develop three alternatives to mitigate this issue. The first treats scale as indexing a sensitivity path and establishes uniform consistency, a functional central limit theorem, and a derivative-based sensitivity measure. The second constructs a robust scale-calibrated median using marginal radial median scales, yielding unit invariance, consistency, a two-step central limit theorem, and bounded influence. The third introduces a bounded balance equation for direct scale estimation, with monotonicity, uniqueness, joint asymptotic normality, and bounded influence. Simulations illustrate boundary collapse, sensitivity, unit invariance, and balanced estimation in Euclidean and Bures-Wasserstein settings.

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 studies scale selection for geometric medians on product manifolds. It proves that naive joint minimization over location and relative scale is degenerate: the scale parameter is driven to the boundary (typically zero), collapsing the estimator to a marginal median on one factor and discarding the other. Three alternatives are developed: (i) treating scale as a sensitivity index with uniform consistency, a functional CLT, and a derivative-based sensitivity measure; (ii) a scale-calibrated median constructed from marginal radial median scales, yielding unit invariance, consistency, a two-step CLT, and bounded influence; (iii) direct scale estimation via a bounded balance equation, with monotonicity, uniqueness, joint asymptotic normality, and bounded influence. Theoretical results are illustrated by simulations on Euclidean and Bures-Wasserstein product spaces.

Significance. If the degeneracy result and the accompanying consistency/CLT/influence-function statements hold under the stated assumptions, the work identifies a fundamental identifiability obstruction in product-manifold geometric medians and supplies three theoretically grounded remedies. The degeneracy argument, the uniform consistency and FCLT for the sensitivity path, the two-step CLT and unit invariance for the calibrated median, and the monotonicity/uniqueness for the balance equation are all load-bearing contributions to robust statistics on manifolds. The explicit construction of bounded-influence estimators and the simulation evidence in both Euclidean and Bures-Wasserstein geometries strengthen the practical relevance.

major comments (2)
  1. [Introduction / degeneracy section] The degeneracy claim (naive joint minimization collapses to a marginal median) is central; the argument appears to rest on the partial derivative of the objective with respect to the scale parameter being strictly positive for scale > 0 whenever the second-factor distances are not identically zero. This should be stated with the precise product distance form used (e.g., sqrt(d1^2 + lambda^2 d2^2)) and the domain of lambda made explicit, because the sign of the derivative may change under other product metrics or when distances can be zero.
  2. [Section on scale-calibrated median] For the second method, the two-step CLT and bounded-influence claim rely on the marginal radial median scales being consistent and having their own CLT; the joint asymptotic covariance between the calibrated location and the estimated scales should be derived explicitly rather than asserted, because the calibration step introduces dependence that is not automatically negligible.
minor comments (2)
  1. [Throughout] Notation for the product manifold, the relative scale parameter, and the three estimators should be unified across sections to avoid re-definition of symbols.
  2. [Simulation section] The simulation section would benefit from a table reporting the empirical scale estimates and their variability under the three methods, rather than only qualitative illustrations of boundary collapse.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the contributions, and the recommendation for minor revision. The two major comments help clarify the presentation of the degeneracy argument and the asymptotic covariance in the scale-calibrated estimator. We address each point below.

read point-by-point responses

  1. Referee: [Introduction / degeneracy section] The degeneracy claim (naive joint minimization collapses to a marginal median) is central; the argument appears to rest on the partial derivative of the objective with respect to the scale parameter being strictly positive for scale > 0 whenever the second-factor distances are not identically zero. This should be stated with the precise product distance form used (e.g., sqrt(d1^2 + lambda^2 d2^2)) and the domain of lambda made explicit, because the sign of the derivative may change under other product metrics or when distances can be zero.

    Authors: We agree that explicitness improves clarity. The degeneracy proof in the manuscript is derived for the standard weighted product distance d((x,y),(x',y')) = sqrt(d_M(x,x')^2 + lambda^2 d_N(y,y')^2) with lambda in (0, infty). Under this metric the partial derivative of the objective with respect to lambda is strictly positive for lambda > 0 whenever the second-factor distances are not all zero (which is the generic case under continuous distributions). We will revise the introduction and the degeneracy section to state this distance form and domain explicitly, and to note that the result is tied to this natural relative-scaling metric rather than to arbitrary product metrics. revision: yes

  2. Referee: [Section on scale-calibrated median] For the second method, the two-step CLT and bounded-influence claim rely on the marginal radial median scales being consistent and having their own CLT; the joint asymptotic covariance between the calibrated location and the estimated scales should be derived explicitly rather than asserted, because the calibration step introduces dependence that is not automatically negligible.

    Authors: The two-step central limit theorem is obtained by first establishing marginal consistency and CLTs for the radial-median scales and then applying a continuous-mapping argument to the calibration map. The dependence between the location estimator and the estimated scales is accounted for in the proof, but the explicit joint asymptotic covariance matrix is only sketched. We will expand the relevant theorem statement and proof to display the full covariance expression, which incorporates the cross terms arising from the calibration step. This addition will make the bounded-influence property fully transparent without changing any of the stated results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation begins with a direct proof that joint minimization over location and scale on the product manifold is degenerate, driven by the strictly positive partial derivative of the median objective with respect to the scale parameter whenever the second-factor distances are non-zero; this follows immediately from the product distance form without any fitted inputs or self-referential definitions. The three alternatives are then constructed explicitly as new estimators (sensitivity path, marginal radial calibration, and bounded balance equation) whose consistency, CLTs, uniqueness, and influence bounds are derived from standard asymptotic arguments on the manifold rather than being equivalent to the input data or prior self-citations by construction. No equations reduce predictions to fitted parameters, no uniqueness theorems are imported from overlapping authors, and no ansatzes are smuggled via citation; the central claims remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the geometric median and on classical asymptotic theory for consistency and central limit theorems; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Geometric median on a product manifold is defined by joint minimization of summed distances under the product metric.
    Invoked to establish the coupling of factors and the boundary-collapse behavior.
  • standard math Standard regularity conditions for uniform consistency and functional central limit theorems hold on the manifold.
    Required for the stated asymptotic results on the sensitivity path and the two-step estimator.

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