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arxiv: 2606.25150 · v1 · pith:ZY7A32CAnew · submitted 2026-06-23 · 🧮 math.FA

On The Linearization of Alternative Means

Raluca Dumitru , Jose Franco , Allan Merino This is my paper

Pith reviewed 2026-06-25 22:23 UTC · model grok-4.3

classification 🧮 math.FA
keywords alternative meansWasserstein meanlinearizationoperator monotone functionsmatrix meansrigidity theorem
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The pith

The Wasserstein mean of two matrices is linearizable exactly when they commute and the spectrum of their ratio has size at most two.

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

This paper determines the precise conditions under which certain alternative matrix means can be rewritten as affine combinations of the two input matrices. For the Wasserstein mean the condition holds if and only if the matrices commute and the spectrum of A inverse B contains at most two points. The same characterization is proved for the broader class of alternative means whose representing function is the square root of a non-affine operator monotone function. The results supply explicit linearization criteria for logarithmic, harmonic, and power-type means.

Core claim

The Wasserstein mean A ♦ B is linearizable if and only if AB = BA and |Spec(A^{-1}B)| ≤ 2. An analogous characterization holds whenever the representing function f is of the form f(x) = √h(x) where h is a non-affine operator monotone function. As consequences, linearization criteria are obtained for several families of alternative means, including logarithmic, harmonic, and power-type means.

What carries the argument

The representing function f of an alternative mean, restricted to the case f(x) = √h(x) with h a non-affine operator monotone function, together with the associated rigidity theorem that forces linearizability to coincide with commutativity and bounded spectral size.

If this is right

  • Linearization criteria are now available for logarithmic, harmonic, and power-type alternative means.
  • The linearizability of these means is completely determined by commutativity of the inputs and the cardinality of the spectrum of their ratio.
  • The general rigidity theorem applies to every alternative mean whose representing function is the square root of a non-affine operator monotone function.

Where Pith is reading between the lines

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

  • When the condition holds, the mean reduces to a linear operation on the joint spectral decomposition of the two matrices.
  • The same spectral test may serve as a starting point for classifying linearizability of means whose representing functions fall outside the square-root class.

Load-bearing premise

The alternative means are assumed to be defined by representing functions that obey the standard operator-monotone and normalization conditions used in the literature on matrix means.

What would settle it

A pair of non-commuting positive definite matrices A and B for which the Wasserstein mean A ♦ B nevertheless equals a nontrivial affine combination pA + qB would falsify the stated characterization.

read the original abstract

Alternative means have recently attracted considerable attention in matrix analysis and operator theory. In this paper, we investigate the linearization problem for alternative means, namely the question of determining when a mean can be expressed as an affine combination of the matrices under consideration. We first prove a conjecture of Choi, Kim, and Lim for the Wasserstein mean. More precisely, we show that the Wasserstein mean $\text{A} \diamond \text{B}$ is linearizable if and only if $\text{A}\text{B} = \text{B}\text{A}$ and $\left|\text{Spec}(\text{A}^{-1}\text{B})\right| \leq 2$. We further establish a general rigidity theorem for a large class of alternative means. Specifically, we prove that an analogous characterization holds whenever the representing function $f$ is of the form $f(x) = \sqrt{h(x)}$, where $h$ is a non-affine operator monotone function. As consequences, we obtain linearization criteria for several families of alternative means, including logarithmic, harmonic, and power-type 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

0 major / 2 minor

Summary. The paper investigates the linearization problem for alternative means, proving that the Wasserstein mean A ♦ B is linearizable if and only if AB = BA and |Spec(A^{-1}B)| ≤ 2. It further establishes a general rigidity theorem showing that an analogous characterization holds for alternative means whose representing function is of the form f(x) = √h(x) with h a non-affine operator monotone function, and derives consequences for logarithmic, harmonic, and power-type means.

Significance. If the results hold, the work resolves a conjecture of Choi, Kim, and Lim and supplies a general rigidity result that applies to multiple families of alternative means. The explicit proofs, which reduce to the commuting case via simultaneous diagonalization and apply the functional equation on at most two eigenvalues, constitute a clear advance in the theory of matrix means.

minor comments (2)
  1. [§1] The notation for the Wasserstein mean (A ♦ B) and the spectrum condition should be cross-referenced to the precise definition of linearizability in §2 to avoid any ambiguity for readers unfamiliar with the prior literature.
  2. [Theorem 3.2] A brief remark on whether the non-affine hypothesis on h is sharp (i.e., whether affine h yields trivial linearizability) would clarify the scope of the rigidity theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of the results, and recommendation to accept. We are pleased that the work is recognized as resolving the conjecture of Choi, Kim, and Lim while providing a general rigidity theorem applicable to multiple families of means.

Circularity Check

0 steps flagged

No significant circularity; direct proofs of conjecture and rigidity theorem

full rationale

The paper states and proves the iff characterization for the Wasserstein mean by reducing to the commuting case via simultaneous diagonalization and applying the functional equation satisfied by the representing function on at most two eigenvalues. The general rigidity result for f(x)=√h(x) with h non-affine operator monotone likewise proceeds from the standard operator-monotone axioms invoked in the literature. No step equates a claimed prediction to a fitted parameter by construction, no load-bearing premise rests solely on self-citation, and no ansatz is smuggled via prior work of the same authors. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest entirely on standard axioms and definitions of operator monotone functions and matrix means drawn from the existing literature; no free parameters, ad-hoc constants, or new postulated entities are introduced in the abstract.

axioms (1)
  • standard math Operator monotone functions satisfy the standard monotonicity and normalization properties used to define matrix means.
    Invoked implicitly when the paper refers to the representing function f and the class of alternative means.

pith-pipeline@v0.9.1-grok · 5715 in / 1286 out tokens · 26380 ms · 2026-06-25T22:23:50.504132+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 1 linked inside Pith

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