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arxiv: 2605.16876 · v1 · pith:QVTJYM3Qnew · submitted 2026-05-16 · 🧮 math.FA

A multivariable mean equation arising from the spectral geometric mean

Sejong Kim , Vatsalkumar N. Mer This is my paper

Pith reviewed 2026-05-19 19:17 UTC · model grok-4.3

classification 🧮 math.FA
keywords spectral geometric meanpositive definite operatorsmultivariable meansnonlinear equationsoperator meansKarcher mean
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The pith

The unique positive definite solution to a proposed nonlinear equation is the spectral geometric mean in the two-variable case.

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

This paper proposes a nonlinear equation as an initial way to define a multivariable spectral geometric mean for positive definite operators. It establishes that for two operators, this equation has a unique positive definite solution which is exactly the spectral geometric mean. For more than two operators, the equation can have multiple solutions, whose properties the authors investigate and compare against other means such as the Karcher mean. They further derive similar equations based on alternative mean theories recently developed for positive definite operators.

Core claim

We formulate a multivariable extension of the spectral geometric mean via the nonlinear equation whose solutions are studied as candidates for the mean; in the two-variable case the unique positive definite solution coincides with the weighted spectral geometric mean defined using the geometric mean operation.

What carries the argument

The nonlinear multivariable mean equation derived from the definition of the spectral geometric mean A ♮_t B.

If this is right

  • The solutions provide a candidate definition for the multivariable spectral geometric mean when uniqueness holds.
  • Properties of the solutions include those shared with other operator means such as monotonicity and invariance under congruence.
  • Comparisons with least squares means highlight similarities and differences in the multi-variable setting.
  • The approach extends to equations arising from other alternative means like the Wasserstein mean.

Where Pith is reading between the lines

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

  • If additional conditions ensure uniqueness, this equation could serve as a practical definition for computing multivariable spectral geometric means.
  • Connections to the Karcher mean suggest possible inequalities or relations between these different means for the same set of operators.
  • Testing the equation on specific low-dimensional matrices could reveal patterns in the number of solutions.

Load-bearing premise

The chosen nonlinear equation is the natural way to extend the two-variable spectral geometric mean to more variables.

What would settle it

For three given positive definite matrices, solve the equation numerically and check whether the solutions match expected properties of a geometric mean or coincide with known means; multiple distinct positive definite solutions would confirm the non-uniqueness claim.

read the original abstract

In the 1980s, Kubo and Ando introduced operator means on $\mathbb{P}$, the open convex cone of positive definite operators. One significant example is the weighted geometric mean $$ A \sharp_{t} B = A^{1/2} (A^{-1/2} B A^{-1/2})^{t} A^{1/2}, \qquad A,B \in \mathbb{P}. $$ The Karcher mean serves as a natural multivariable extension of this mean by minimizing the sum of squared Riemannian trace distances of positive definite matrices. It coincides a unique positive definite solution to the Karcher equation, which allows us to define the Karcher mean on $\mathbb{P}$. The weighted spectral geometric mean is defined as another geometric mean of two positive definite operators as follows: $$ A \natural_t B = (A^{-1} \sharp B)^{t} A (A^{-1} \sharp B)^{t}, $$ where $A \sharp B = A \sharp_{1/2} B$. In this paper, we make an initial attempt to formulate a multivariable spectral geometric mean through a nonlinear equation. In the two-variable case, the unique positive definite solution of this equation is precisely the spectral geometric mean. However, in the multi-variable case, the equation need not have a unique solution. We study properties of its solutions and compare them with other least squares means of positive definite matrices. Recently, a new theory of alternative means for positive definite operators has been developed, which includes the spectral geometric mean and the Wasserstein mean. We also consider multivariable equation arising from the alternative means.

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 proposes a nonlinear equation as an initial attempt to define a multivariable spectral geometric mean for positive definite operators on the cone P. It shows that the two-variable restriction of this equation has the unique positive definite solution equal to the spectral geometric mean A ♮_t B. For n>2 variables the equation may admit multiple solutions; the authors study their properties, compare them to least-squares means, and formulate analogous equations arising from other alternative means.

Significance. If the central construction can be placed on firmer footing, the work supplies a concrete mean equation that correctly recovers the spectral geometric mean in the two-variable case and thereby extends the recent theory of alternative means (including the Wasserstein mean) to the multivariable setting. Such an equation-based approach parallels the Karcher mean and could facilitate further analytic and numerical study of operator means.

major comments (2)
  1. [Abstract and the definition of the proposed equation (presumably §2 or §3)] The choice of the particular nonlinear equation is justified almost exclusively by the two-variable reduction property stated in the abstract. No independent characterizing property (invariance under unitary conjugation, monotonicity with respect to the Loewner order, or derivation from a variational principle or limit process) is supplied that would single out this equation among other candidates that also reduce to A ♮_t B when n=2. This weakens the claim that the equation constitutes the natural multivariable extension.
  2. [Multi-variable analysis section] Existence and uniqueness are not addressed in detail for the multivariable case. The abstract and subsequent discussion note non-uniqueness but provide neither sufficient conditions guaranteeing at least one positive definite solution nor any quantitative control on the set of solutions (e.g., diameter bounds or convergence of iterative schemes).
minor comments (2)
  1. [Definition of the mean equation] Clarify the precise statement of the proposed nonlinear equation (including the precise weighting and the role of the two-variable spectral geometric mean inside the equation) so that readers can verify the two-variable reduction without ambiguity.
  2. [Numerical comparisons] Add a short comparison table or numerical example illustrating how the solutions of the new equation differ from the Karcher mean and from the least-squares mean for a concrete triple of 2×2 positive definite matrices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our initial attempt to define a multivariable spectral geometric mean via a nonlinear equation. We address the major comments point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract and the definition of the proposed equation (presumably §2 or §3)] The choice of the particular nonlinear equation is justified almost exclusively by the two-variable reduction property stated in the abstract. No independent characterizing property (invariance under unitary conjugation, monotonicity with respect to the Loewner order, or derivation from a variational principle or limit process) is supplied that would single out this equation among other candidates that also reduce to A ♮_t B when n=2. This weakens the claim that the equation constitutes the natural multivariable extension.

    Authors: The referee is correct that the primary justification is the two-variable reduction to the spectral geometric mean. The manuscript presents this explicitly as an 'initial attempt' rather than a fully characterized natural extension. To strengthen the presentation, we will revise the abstract and the section introducing the equation to highlight additional properties that any solution satisfies, including invariance under simultaneous unitary conjugation of all arguments and monotonicity in the Loewner order with respect to each variable (when the other variables are fixed). These properties are verified directly from the equation and help distinguish the construction. We do not yet have a variational or limit-process derivation, which we will note as a direction for future work rather than claiming completeness. revision: partial

  2. Referee: [Multi-variable analysis section] Existence and uniqueness are not addressed in detail for the multivariable case. The abstract and subsequent discussion note non-uniqueness but provide neither sufficient conditions guaranteeing at least one positive definite solution nor any quantitative control on the set of solutions (e.g., diameter bounds or convergence of iterative schemes).

    Authors: We agree that the current treatment is limited: the paper shows non-uniqueness via counterexamples for n>2 and analyzes properties of existing solutions, but does not supply general existence results or quantitative bounds. In the revision we will add a subsection providing sufficient conditions for existence (for instance, when the operators commute or lie in a sufficiently small ball in the Riemannian metric) via a fixed-point argument on a compact convex set in the positive definite cone. We will also include operator-norm diameter bounds for the solution set in terms of the input operators and establish local convergence of a natural fixed-point iteration under those conditions. This directly addresses the request for more detailed analysis. revision: yes

Circularity Check

1 steps flagged

Equation posited as multivariable extension with built-in reduction to two-variable spectral geometric mean

specific steps
  1. self definitional [Abstract]
    "In this paper, we make an initial attempt to formulate a multivariable spectral geometric mean through a nonlinear equation. In the two-variable case, the unique positive definite solution of this equation is precisely the spectral geometric mean."

    The nonlinear equation is introduced specifically to serve as the definition of the multivariable mean, yet it is constructed such that its solution reduces exactly to the already-known two-variable spectral geometric mean; the 'derivation' of the mean as solution to the equation therefore encodes the target object by design rather than deriving it independently.

full rationale

The paper explicitly frames its contribution as an 'initial attempt' to define a multivariable mean via a nonlinear equation chosen so that the two-variable restriction yields the known spectral geometric mean as unique solution. This is a consistency check rather than a load-bearing derivation from independent first principles, and the manuscript notes non-uniqueness in the multivariable setting while comparing solutions to other means. No self-citation chain, fitted parameter renamed as prediction, or ansatz imported via prior work is required for the central claim; the construction is transparent and self-contained as a proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard properties of positive definite operators and operator means; the main addition is the new equation itself rather than new free parameters or external entities.

axioms (1)
  • domain assumption Positive definite operators form an open convex cone equipped with the Riemannian trace metric used for the Karcher mean.
    Invoked to motivate the Karcher mean and the new spectral extension.
invented entities (1)
  • Multivariable spectral geometric mean defined by the nonlinear equation no independent evidence
    purpose: To extend the two-variable spectral geometric mean to an arbitrary number of positive definite operators.
    Introduced in the paper as the solution set of the proposed equation; no independent external validation is mentioned.

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Works this paper leans on

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