arxiv: 2605.03074 · v1 · submitted 2026-05-04 · 🧮 math.AG

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Bures Geodesics and Restricted Barycenters for Kronecker Positive Definite Matrices

Jiaping Yang, Yunxin Zhang

Pith reviewed 2026-05-08 17:20 UTC · model grok-4.3

classification 🧮 math.AG

keywords Bures-Wasserstein metricKronecker positive definite matricesgeodesicsbarycenterspartial trace residualpositive definite matrices

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The pith

Bures geodesics between Kronecker positive definite matrices stay inside the model only for one-factor pairs.

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

The paper examines when the geodesic in the ambient Bures-Wasserstein space between two points of the determinant-normalized Kronecker model remains entirely inside that lower-dimensional set. It proves that if the geodesic stays inside the model near one endpoint then the entire segment stays inside, and that this occurs precisely when the pair differs in only one factor: the U matrices are identical or the V matrices differ by a positive scalar. The departure is captured by a partial-trace residual that becomes a rank-one square-root profile in commuting coordinates. This supplies exact solutions for two restricted barycenter problems inside the model.

Core claim

Local membership near an endpoint is shown to be equivalent to membership of the whole segment, and this happens exactly in the one-factor cases: either U1=U0 or V1 is a positive scalar multiple of V0. Consequently, any endpoint pair not confined to these one-factor alternatives leaves the model immediately. The criterion is expressed by a partial-trace residual. In fixed commuting charts it becomes an equivalent rank-one square-root profile and yields computable departure diagnostics. Exact formulas are obtained for barycenters on fixed commuting-coordinate slices, solved by Perron singular vectors, and on one-factor subfamilies, reduced to standard Bures-Wasserstein barycenters on Sp^n.

What carries the argument

The partial-trace residual criterion, which detects whether a Bures geodesic departs from the determinant-normalized Kronecker model.

If this is right

Any endpoint pair outside the one-factor cases has a geodesic that leaves the model immediately.
Barycenters on fixed commuting-coordinate slices are given exactly by Perron singular vectors.
Barycenters on one-factor subfamilies reduce exactly to ordinary Bures-Wasserstein barycenters on the n-dimensional space.
Departure diagnostics are computable via the partial-trace residual or, in commuting charts, via the rank-one square-root profile.

Where Pith is reading between the lines

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

Global geodesic membership questions reduce to local checks near an endpoint.
The Kronecker model is geodesically non-convex in the Bures-Wasserstein metric except along the one-factor directions.
The exact formulas in the special cases supply benchmarks for numerical approximation of general barycenters inside the model.

Load-bearing premise

That the partial-trace residual fully captures departure of the Bures geodesic from the Kronecker model in the ambient space.

What would settle it

Explicit computation of the Bures geodesic segment for a concrete pair of Kronecker matrices where the U factors differ and the V factors are not positive scalar multiples, then checking whether any interior point lies in the model.

Figures

Figures reproduced from arXiv: 2605.03074 by Jiaping Yang, Yunxin Zhang.

**Figure 1.** Figure 1: Fixed-coordinate-slice barycenter benchmarks: objective gaps for independent logarithmic-coordinate solves, and Dataset A coordinates against the leafwise Bures coordinates. Yang, J., Zhang, Y.: Preprint submitted to Elsevier Page 18 of 19 view at source ↗

read the original abstract

We study the extrinsic Bures--Wasserstein geometry of the determinant-normalized Kronecker model $\mcK_n=\{V\ot U:U,V\in\Sp^n,\ \det U=1\}\subset\Sp^{n^2}$, asking when the ambient Bures geodesic between two Kronecker positive definite matrices can remain in this lower-dimensional model. Local membership near an endpoint is shown to be equivalent to membership of the whole segment, and this happens exactly in the one-factor cases: either $U_1=U_0$ or $V_1$ is a positive scalar multiple of $V_0$. Consequently, any endpoint pair not confined to these one-factor alternatives leaves the model immediately. The criterion is expressed by a partial-trace residual. In fixed commuting charts it becomes an equivalent rank-one square-root profile and yields computable departure diagnostics. We also obtain exact formulas for two restricted barycenter problems: fixed commuting-coordinate slices, solved by Perron singular vectors, and one-factor subfamilies, reduced to standard Bures--Wasserstein barycenters on $\Sp^n$.

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.

Desk Editor's Note private letter to a colleague

The paper shows Bures geodesics stay inside the Kronecker model only for one-factor pairs and gives closed-form barycenters for two restricted cases.

read the letter

The main result is that ambient Bures geodesics between points in the determinant-normalized Kronecker model remain inside the model exactly when the pair is one-factor: either the U matrices match or the V matrices differ by a positive scalar. Local membership near an endpoint is equivalent to the whole segment staying in, and they express the test via a partial-trace residual. They also derive exact barycenter formulas for fixed commuting slices and for one-factor subfamilies.

Referee Report

2 major / 2 minor

Summary. The paper studies the extrinsic Bures-Wasserstein geometry of the determinant-normalized Kronecker model K_n = {V ⊗ U : U,V ∈ Sp^n, det U=1} inside Sp^{n^2}. It shows that the ambient Bures geodesic between two points in K_n remains inside K_n if and only if the pair is one-factor (U_1 = U_0 or V_1 is a positive scalar multiple of V_0), with local membership near an endpoint equivalent to global membership of the segment. The criterion is given by vanishing of a partial-trace residual, which reduces to a rank-one square-root profile in fixed commuting charts. Exact formulas are also derived for two restricted barycenter problems: fixed commuting-coordinate slices (solved via Perron singular vectors) and one-factor subfamilies (reduced to standard Bures-Wasserstein barycenters on Sp^n).

Significance. If the central claims hold, the results clarify when Kronecker structure is preserved along Bures geodesics and supply explicit, computable solutions for restricted barycenters that are otherwise intractable. These findings are relevant to quantum information and covariance estimation, where Kronecker models arise naturally. The reduction to one-factor cases and the provision of departure diagnostics via the residual constitute concrete advances.

major comments (2)

[geodesic membership analysis (abstract and the section deriving the residual from the explicit Bures geodesic)] The central equivalence (local membership near an endpoint iff global membership of the geodesic in K_n) rests on the claim that the partial-trace residual fully captures all departures from the model under the embedding into the ambient Bures-Wasserstein space. This needs explicit verification in the general non-commuting case, because the Bures geodesic formula involves square-root terms of the Kronecker factors whose higher-order or non-commuting contributions might permit the geodesic to stay in K_n without the residual vanishing (or force departure even when it vanishes).
[commuting charts subsection] The reduction of the residual to an equivalent rank-one square-root profile is stated to hold only in fixed commuting charts. The manuscript should clarify whether this simplification is merely a computational convenience or whether the general criterion can be shown to be equivalent without assuming commutativity.

minor comments (2)

[introduction] Notation for the model K_n and the ambient space Sp^{n^2} is clear, but the distinction between the determinant-normalized condition (det U=1) and the full Kronecker product should be restated once in the introduction for readers unfamiliar with the embedding.
[restricted barycenters] The barycenter formulas for the one-factor subfamilies are reduced to standard Bures-Wasserstein barycenters; a brief remark on how the reduction preserves the determinant-normalization would be helpful.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major comments concern the generality of the partial-trace residual criterion for geodesic membership and the status of the commuting-charts simplification. We address each point below, providing clarifications and indicating the revisions we will incorporate.

read point-by-point responses

Referee: [geodesic membership analysis (abstract and the section deriving the residual from the explicit Bures geodesic)] The central equivalence (local membership near an endpoint iff global membership of the geodesic in K_n) rests on the claim that the partial-trace residual fully captures all departures from the model under the embedding into the ambient Bures-Wasserstein space. This needs explicit verification in the general non-commuting case, because the Bures geodesic formula involves square-root terms of the Kronecker factors whose higher-order or non-commuting contributions might permit the geodesic to stay in K_n without the residual vanishing (or force departure even when it vanishes).

Authors: The partial-trace residual is obtained by direct substitution of the general Bures geodesic formula (which involves the matrix square root of the rescaled factors and holds without commutativity) into the membership condition for K_n. The proof proceeds in two directions: (i) if the geodesic remains in K_n then the residual must vanish (by taking the partial trace of the geodesic expression and using the Kronecker structure), and (ii) if the residual vanishes at the endpoints then the pair is necessarily one-factor, in which case direct substitution shows the entire geodesic stays inside K_n. Non-commuting contributions appear in the square-root terms but are shown to produce nonzero residual unless the factors satisfy the one-factor condition. To make this verification fully explicit, we will add a short appendix containing an algebraic expansion of the residual for a generic non-commuting pair and a concrete numerical example confirming that no higher-order cancellation allows the geodesic to remain in K_n when the residual is nonzero. revision: yes
Referee: [commuting charts subsection] The reduction of the residual to an equivalent rank-one square-root profile is stated to hold only in fixed commuting charts. The manuscript should clarify whether this simplification is merely a computational convenience or whether the general criterion can be shown to be equivalent without assuming commutativity.

Authors: The general criterion is the vanishing of the partial-trace residual, which is derived and stated without any commutativity assumption. The reduction to an equivalent rank-one square-root profile occurs only after choosing a fixed commuting chart (i.e., simultaneous diagonalization of the two factors), which simplifies the algebraic expressions and yields practical departure diagnostics. This reduction is therefore a computational convenience that preserves equivalence inside the chosen chart but is not required for the validity of the residual itself. We will revise the commuting-charts subsection to state this distinction explicitly, adding a sentence that the residual criterion applies in full generality while the rank-one form is the simplified expression available under commutativity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from Bures metric definitions and Kronecker structure

full rationale

The paper's central claims—that local membership near an endpoint is equivalent to global segment membership in the determinant-normalized Kronecker model, occurring precisely in one-factor cases—are derived directly from the explicit Bures-Wasserstein geodesic formula, partial-trace operations, and embedding properties. These steps use standard definitions of the metric and Kronecker product without reducing any prediction or criterion to a fitted parameter, self-definition, or unverified self-citation chain. The partial-trace residual is computed as a diagnostic from the geodesic expression rather than presupposed, and the restricted barycenter formulas follow from Perron vectors and standard barycenter reductions on Sp^n. No load-bearing step collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of the Bures-Wasserstein metric on positive definite matrices and the algebraic structure of Kronecker products; no new free parameters or invented entities are introduced.

axioms (2)

standard math The Bures-Wasserstein metric is the standard Riemannian metric induced by the affine-invariant geometry on the manifold of positive definite matrices.
Used throughout to define geodesics and barycenters.
domain assumption The Kronecker product of positive definite matrices remains positive definite and respects the determinant-normalization condition when one factor has determinant one.
Defines the model K_n and its embedding.

pith-pipeline@v0.9.0 · 5490 in / 1359 out tokens · 73541 ms · 2026-05-08T17:20:27.487674+00:00 · methodology

discussion (0)

Reference graph

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